On the expansion formulas of cluster varieties from surfaces and their combinatorial properties

On the expansion form ulas of cluster v arieties from surfaces and their com binatorial prop erties V u T ung Lam Dinh*, Iv an Chi-Ho Ip** F ebruary 26, 2026 Abstract This pap er explores the cluster algebra structure of the mo duli space A SL n +1 , S of twisted SL n +1 -lo cal systems on a surface. W e derive general recurrence relations for cluster v ariables arising from flips of a triangulation, corresp onding to sp ecific sequences of mutations. Our approac h is grounded in a detailed com binatorial analysis ov er the standard n -triangulated m -gon (with explicit calculations for n = 1 , 2). As a generalization, the non-simply-laced G 2 t yp e is also considered. W e pro ve the wel l-triangulate d prop ert y for cluster mutations under flips, which provides a combinatorial framework for understanding the stability and transformation rules of these cluster algebra structures, and compute the monomial coun ts for the cluster expansion formula. Con ten ts 1 In tro duction 2 2 Preliminaries 4 2.1 Cluster algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Mo duli spaces of G -local systems from surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Cluster realization of A G, S in type A n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Cluster expansion formulas for general flips 14 3.1 Unpunctured surface case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Com binatorial interpretation via stair paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Punctured surface case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Solving recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Cluster expansion formulas of n -triangulated m -gon 30 4.1 Notation and lab eling con ven tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 W ell-triangulated Preserv ation and Number of Monomials Theorem . . . . . . . . . . . . . . 33 4.3 General recursive formulas for particular v alues of n . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 General num ber of terms prop ert y and examples . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 General expansion formulas of cluster realization of type G 2 in 4 -gon 59 5.1 Cluster realization of type G 2 lo cal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2 Detailed calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Recursion formula via go o d lattice 69 B Maple co de for computation of recurrence relation 72 Departmen t of Mathematics, Hong Kong Universit y of Science and T echnology *Email: vtldinh@connect.ust.hk **Email: iv an.ip@ust.hk 1 1 In tro duction Cluster algebras, introduced by F omin and Zelevinsky [ 7 ], ha ve b ecome a fundamen tal structure in mo dern mathematics. These comm utativ e rings, equipped with a combinatorial seed structure go verned by m utation, ha ve found significan t applications in div erse areas such as represen tation theory [ 11 , 18 , 19 , 32 ], Poisson geometry [ 8 ], and integrable systems [ 9 , 17 ]. A cornerstone result, the Lauren t phenomenon [ 7 ], guaran tees that every cluster v ariable can b e expressed as a Laurent p olynomial in the initial cluster v ariables. The p ositivit y conjecture, asserting the non-negativity of co efficients in these polynomials, was resolved in full generalit y [ 12 , 24 ]. Simpler, com binatorial pro ofs exist in sp ecial cases, such as for cluster algebras from surfaces using p erfect matc hings and snake graphs [ 25 , 26 ]. In this pap er, w e in vestigate the expansion form ulas asso ciated with the cluster algebra structure on the mo duli space A G, S of twisted G -lo cal systems on a decorated Riemann surface S as introduced by F o ck, Gonc harov, and Shen [ 5 , 11 ] for a split semi-simple simply-connected algebraic group G . The case G = SL 2 reco vers the well-studied cluster algebra from surfaces, whic h is also connected to T eichm¨ uller theory [ 6 , 27 ]. In this paper, our primary fo cus is on the case G = SL n +1 . W e aim to derive general form ulas b y analyzing configurations corresponding to the standard n -triangulated m -gon. Subsequen tly , we also examine mutation sequences for quadrilateral flips in the case of quivers asso ciated with type G 2 , leading to some elementary p olynomial iden tities. The comprehensive framework dev elop ed by Goncharo v and Shen [ 11 ] provides the geometric foundation for our inv estigation. The work of Shapiro–Sc hrader [ 32 ] provides the crucial quan tum group connection that guides our approach. Their explicit embedding of the quantum group U q ( sl n +1 ) in to quantum cluster algebras asso ciated to mo duli spaces of framed SL n +1 -lo cal systems reveals the geometry underlying the represen tation-theoretic asp ects of these cluster structures. Notably , the R -matrix action is realized as a sequence of quiver m utations that pro duces the geometric action of a half-Dehn twist in a twice-punctured disk. The cluster realization of quantum group em b eddings and R -matrices was later generalized to all Lie t yp es in [ 18 ]; in particular, one of the explicit realizations motiv ates the computation for the case of type G 2 in Section 5 . The G = SL 2 case associates the cluster algebra from surfaces with a fundamen tal combinatorial mo del via p erfect matchings on snak e graphs [ 25 , 26 ], which yield explicit expansion formulas for the cluster v ariables. F or unpunctured surfaces with an initial triangulation T , the cluster v ariable x γ asso ciated to an arc γ ∈ T admits the expansion formula [ 25 , 28 ]: x γ = 1 cross( γ ) X P ∈ Match( e G γ ) x ( P ) y ( P ) , (1) where e G γ is the snake graph associated to the arc γ , cross( γ ) is a product of cluster v ariables corresp onding to arcs crossed by γ , the sum is ov er all p erfect matc hings P of e G γ , and x ( P ) and y ( P ) are monomials in the initial cluster v ariables and co efficien ts resp ectiv ely . This formula establishes the Laurent p ositivit y prop ert y for cluster v ariables in unpunctured surfaces. Alternativ ely , a combinatorial mo del using T -paths also yields an equiv alen t expansion [ 4 ]. F or a diagonal M a,b in a p olygon (considered as a disk with m marked points): x M = X α ∈ W T ( a,b ) x ( α ) , (2) where each T -path α contributes a monomial x ( α ) with co efficien ts either 0 or 1. This illustrates the fundamen tal connection be t w een cluster v ariables and combinatorial paths in triangulated p olygons. In this w ork, we attempt to generalize the com binatorial mo del to higher rank, and deriv e explicit expansion formulas for the cluster algebra structure of A G, S asso ciated to a sequence of cluster mutations corresp onding to flips of diagonal in higher triangulations , with particular emphasis on G = SL n +1 . Belo w, w e summarize the k ey innov ations and main theorems as follows. (i) Wel l-triangulate d pr op erty : W e in tro duce and study the wel l-triangulate d property for triangles in n -triangulated p olygons (Definition 4.6 ). This combinatorial condition is characterized by a sp ecific monomial pattern x bc y ca z ab for the v ertex v alues, and the main result of the pap er shows that the prop ert y is preserved under arbitrary sequences of flips. 2 (ii) F r amework for n -triangulate d m -gon : W e develop a comprehensive framew ork for n -triangulated m - gons (Definitions 2.16 ) with explicit lab eling and m utation sequences, generalizing the type A 1 case to higher rank. (iii) Stair p ath c ombinatorics : W e in tro duce n -stair p aths and n -r everse d stair p aths (Definitions 3.6 , 3.8 ) as fundamen tal combinatorial ob jects encoding the structure of cluster v ariable expansions, visualized as alternating sequences of left/up or righ t/do wn segments. The recurrence relation established resembles the discrete 4D Hirota–Miwa equation (Proposition 3.14 ). (iv) Mutation se quenc es for gener al flips : W e establish explicit m utation sequences for flips within n - triangulated quadrilaterals (Definition 2.18 ), revealing the intricate lay ered structure of these transfor- mations. (v) R efle cte d p olygon c onstruction : Definition 4.11 in tro duces r efle cte d p olygons P ( p 1 , p 2 , . . . , p k ), provid- ing a symmetry reduction to ol for calculating the expansion formulas. The Main Theorems in this pap er are as follows. Theorem 1.1 (W ell-triangulated preserv ation (Theorem 4.7 )) . F or a triangulation T of a wel l-triangulate d p olygon P with the cluster r e alization of typ e A n , after any se quenc e of flips, the r esulting triangulation is also wel l-triangulate d. Next, w e obtain a closed-form expression for the exponents of the expansion formula of cluster v ariables along the diagonal of a triangulated square, whic h serves as a building blo c k. Theorem 1.2 (Exp onen t F orm ula (Theorem 3.4 )) . The exp onents in the exp ansion formula at unity satisfy a i,j = j ( i − j + 1) for al l i ≥ j ≥ 1 , le ading to the explicit evaluation: K n = (2 n , 2 2 n − 2 , . . . , 2 t ( n − t +1) , . . . , 2 n ) . (3) Finally , we further generalize our calculations to arbitrary m -gons: Theorem 1.3 (Num b er of monomials (Theorem 4.8 )) . Consider the cluster r e alization of A SL n +1 , S for a tri- angulation T of a surfac e S with b oundary and without punctur es, and fix any non T -diagonal γ . If the num- b er of monomials (c ounting multiplicities) in the L aur ent p olynomial of the exp ansion formula of γ for the c ase n = 1 is K , then the numb er of monomials (c ounting multiplicities) in the L aur ent p olynomial in e ach c o or di- nate of the exp ansion formula of γ for gener al n is of the form ( K n , K 2 n − 2 , . . . , K t ( n +1 − t ) , . . . , K 2 n − 2 , K n ) . After v erifying the main theorems, we provide w ays to lab el all vertices for conv enience in setting up recurrences, hence w e need to separate in to 12 different cases (dep ending on parity mo dulo 3 and mo dulo 2) where the order of diagonals in a p olygon changes, allo wing us to deduce the following formulas more concisely . Theorem 1.4 (Recurrence F ormulas for General Polygons (Theorems 4.16 , 4.17 )) . We establish c omplete r e curr enc e r elations for exp ansion formulas in the gener al 1 -triangulate d (The or em 4.16 ) and 2 -triangulate d (The or em 4.17 ) m -gons, pr oviding c omputational algorithms for cluster variables in arbitr ary n -triangulate d p olygons. F rom the num b er theory p erspective, we further find the connections b et ween the num b er of terms in the expansion formulas and c ontinue d fr actions . Corollary 1.5 (T erm Coun t F ormulas (Corollaries 4.18 , 4.19 )) . F or any n -triangulate d p olygon P ( p 1 , p 2 , . . . , p N ) : (i) The diagonal formula D [ p 1 ,p 2 ,...,p N ] l has pr e cisely a ( p 1 , p 2 , . . . , p N ) l ( n +1 − l ) terms, (ii) The inner vertex formula I [ p 1 ,p 2 ,...,p N ] ( i,j ) has pr e cisely a ( p 1 , p 2 , . . . , p N ) i ( n +1 − i − j ) a ( p 1 , p 2 , . . . , p N − 1 ) j ( n +1 − i − j ) terms, wher e a ( p 1 , p 2 , . . . , p N ) is the numer ator of the c ontinue d fr action [1; p 1 , p 2 , . . . , p N ] . 3 W e provide a comprehensive combinatorial in terpretation of cluster expansions through explicit form ulas for ψ t,k ( x i,j ) in terms of stair path sums (Prop ositions 3.7 , 3.9 ), and the recurrence relations connecting dif- feren t lev els of the triangulation hierarc hy , mediated via enumerating functions S ( x i,j , x k,l ) and R ( x i,j , x k,l ) coun ting stair paths and reversed-stair paths (Section 3.2 ). These results offer a complete combinatorial description of cluster v ariable expansions in t yp e A n lo cal system from surfaces, with explicit algorithms for computing the Laurent expansions in arbitrary n -triangulated p olygons. This also la ys the groundwork for further exploration in G 2 and other finite type cluster algebras. By thoroughly examining the standard n -triangulated m -gon, this work makes progress to w ard a univ ersal understanding of expansion formulas within the A G, S framew ork. The specific cases examined hav e rev ealed the deep links b etw een the algebraic and the graphical asp ects, providing a foundation for future research and to ols for theoretical and computational applications in cluster algebra theory . In particular, the well- triangulated prop ert y plays a key role in understanding the stability and transformation rules within these cluster algebraic structures. F uture research will aim to derive complete expansion formulas for single flips across all finite irreducible Dynkin types, seeking to elucidate b oth universal and type-dep endent b eha viors. This will in volv e devel- oping refined combinatorial strategies to solve recurrence relations, esp ecially for other Lie types, thereby adv ancing the understanding of the cluster structure on A G, S in general. Organization of the pap er. The pap er is organized as follo ws. Section 2 reviews the basic definitions and prop erties of cluster algebras and the cluster structure on mo duli spaces of t wisted G -lo cal systems detailing the type A n case. Section 3 pro vides the combinatorial interpretation of the expansion formula asso ciated to general flip of diagonal ov er a quadrilateral via the n -stair path mo del and its asso ciated re- currence relations. Section 4 presents a comprehensive analysis of the cluster structure of A G, S o ver the standard n -triangulated m -gon, including the w ell-triangulated prop ert y , monomial c oun ts, recurrence rela- tions of the expansion formulas, and its connections to contin ued fractions. Section 5 details the study of cluster structure o v er mutation sequences for quadrilateral flips in type G 2 quiv ers, revealing new polynomial iden tities. Finally , the App endices provide further observ ations and verification related to the recurrences discussed in Prop osition 3.15 . Ac knowledgmen t. This study w as conducted under the Undergraduate Research Opp ortunities Pro- gram (UROP) at The Hong Kong Universit y of Science and T ec hnology . The second author is supp orted by the Hong Kong RGC General Researc h F unds [GRF #16305122]. 2 Preliminaries 2.1 Cluster algebra In this section we summarize the definitions and notation on cluster algebra [ 7 ] needed in this pap er. Definition 2.1. A quiver is a tuple Q = ( Q 0 , Q 1 , s, t ), where Q 0 and Q 1 are finite sets, and s, t : Q 1 → Q 0 are set maps. Q 0 is the set of vertic es , Q 1 is the set of arr ows , s is the sour c e map, and t is the tar get map. W e assume Q has no 1-cycles (or simply sa y Q has no lo ops), i.e. s ( α )  = t ( α ) for any α ∈ Q 1 , and Q has no 2-cycles, i.e. for any α 1 , α 2 ∈ Q 1 suc h that t ( α 1 ) = s ( α 2 ), w e require t ( α 2 )  = s ( α 1 ). W e will presen t an arro w α ∈ Q 1 with s ( α ) = i and t ( α ) = j b y i → j . A quiv er Q is called an ic e quiver if we further partition the v ertex set Q 0 = M ⊔ P in to tw o sets (possibly empt y), called mutable and fr ozen vertices respectively , such that no arro ws connect tw o frozen vertices. In this section, let us denote n := | Q 0 | . Definition 2.2. The signe d adjac ency matrix of Q is the n × n integer matrix B = B ( Q ) = ( b ij ) indexed b y Q 0 , such that b ij := # { arro ws from i → j } − # { arrows from j → i } . (4) By definition B is a sk ew-symmetric matrix, i.e. B = − B T . In particular, all diagonal entries satisfy b ii = 0 for all i ∈ Q 0 . 4 Definition 2.3. Giv en an ice quiver Q and a m utable vertex k ∈ Q 0 , a quiver mutation µ k is an op eration transforming Q into a new quiv er µ k ( Q ) by the follo wing three steps: (1) F or t wo arrows i → k → j where not b oth i, j are frozen, add a new arrow i → j , counting with m ultiplicities. (2) Rev erse the direction of all arrows inciden t to k . (3) Remo v e all oriented 2-cycles. If Q can b e transformed in to Q ′ b y a finite sequence of mutations up to p erm utation of the v ertices, w e call those 2 quivers mutation e quivalent and write Q ∼ Q ′ . The set of all quivers that are m utation equiv alent to Q is the mutation e quivalenc e class [ Q ]. Lemma 2.4. The signe d adjac ency matrix B ′ = ( b ′ ij ) of the mutate d quiver Q ′ := µ k ( Q ) satisfies: b ′ ij =          − b ij if k ∈ { i, j } , b ij + b ik .b kj if b ik > 0 , b kj > 0 , b ij − b ik .b kj if b ik < 0 , b kj < 0 , b ij otherwise. (5) Equivalently, we c an r ewrite this as b ′ ij = ( − b ij if k ∈ { i, j } , b ij + | b ik | b kj + b ik | b kj | 2 otherwise. (6) Let F := Q ( u 1 , u 2 , ..., u n ) b e the am bient field of rational functions in n indep enden t v ariables. Definition 2.5. A cluster is an n -tuple x = ( x 1 , x 2 , ..., x n ) ∈ F n of algebraically indep enden t v ariables. The elemen ts in a cluster x ∈ F n are called cluster variables . A se e d is a pair ( x , Q ) where x ∈ F n is a cluster and Q is an ice quiver with v ertices Q 0 = { 1 , 2 , ..., n } (without lo ops and 2-cycles). A cluster v ariable x i is mutable (or fr ozen ) if the vertex i ∈ Q 0 is mutable (or frozen) resp ectiv ely . Definition 2.6. Let k ∈ Q 0 b e mutable. The mutation of se e d ( x , Q ) at k is a seed µ k ( x , Q ) := ( µ k ( x ) , µ k ( Q )) where µ k ( Q ) is the m utation of Q defined in Definition 2.3 , µ k ( x ) := ( x ′ 1 , x ′ 2 , ..., x ′ n ) ∈ F n is the cluster defined b y: x ′ l :=        x l if l  = k , 1 x k   Y k → i x i + Y j → k x j   if l = k (7) where the pro duct is tak en ov er all arrows in Q 1 that start or end in vertex k , resp ectiv ely (counted with m ultiplicity); the pro duct is understoo d to b e 1 if there are no such arro ws. Equiv alen tly , if B := B ( Q ) is the signed-adjacency matrix of Q , the equation can b e rewritten as: x k x ′ k = Y i ∈{ 1 , 2 ,...,n } b ik > 0 x b ik i + Y i ∈{ 1 , 2 ,...,n } b ik < 0 x − b ik i . (8) W e shall call this equation the exchange r elation of the cluster v ariables. Definition 2.7. Let ( x , Q ) b e a seed. The Z -subalgebra A ( Q ) ⊂ F generated by all the cluster v ariables ob- tained by all possible finite sequences of cluster m utations from ( x , Q ) is called the cluster algebr a asso ciated to the initial seed ( x , Q ). The algebra is indep enden t of the m utation equiv alence class of Q . 5 Let ( x , Q ) b e an initial seed and A ( Q ) the corresp onding cluster algebra. Let { 1 , ..., n } b e mutable and { n + 1 , ..., m } b e frozen. The Laurent phenomenon [ 7 ] states that A ( Q ) ⊂ Z [ x ± 1 1 , x ± 1 2 , ..., x ± 1 n , x n +1 , x n +2 , ..., x m ] . (9) One of the main recen t breakthroughs is the Positivity The or em , whic h w as pro ven in full generality in [ 12 ]. Theorem 2.8 (P ositivity Theorem [ 12 ]) . F or any cluster variables written as L aur ent p olynomial of the initial se e d as ab ove, al l the c o efficients lie in Z > 0 . The sp ecial case of the theorem which motiv ates the com binatorial understanding of the curren t work comes from cluster algebras arising from surfaces, which was prov ed in [ 25 , 26 ] by describing the co efficients as counting the num b er of p erfect matchings asso ciated to certain graphs called snake diagr ams . It turns out that they can also b e describ ed in terms of contin ued fractions [ 4 ]. W e recall some notations that are needed in this pap er. Definition 2.9. The standar d c ontinue d fr action representation of x ∈ R is defined as: x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + · · · (10) understo od as the limit of the sequence of c onver gents x n := b 0 + a 1 b 1 + a 2 b 2 + a 3 . . . + a n b n = A n B n . (11) A sp ecific example with n umerical v alues for π is: π = 3 + 1 2 6 + 3 2 6 + 5 2 6 + 7 2 6 + · · · . (12) When all a i = 1, we will write the contin ued fraction as x = [ b 0 ; b 1 , b 2 , . . . ] . (13) Prop osition 2.10 (F undamental theorem of con tinued fractions) . The c onver gents A n B n satisfy the initial c onditions: A − 1 = 1 , A 0 = b 0 , B − 1 = 0 , B 0 = 1 and for n ≥ 1 , the r e curr enc e r elations: A n = b n A n − 1 + a n A n − 2 , B n = b n B n − 1 + a n B n − 2 . (14) 6 2.2 Mo duli spaces of G -lo cal systems from surfaces W e recall some definitions and terminology related to the mo duli spaces of G -local systems ov er a b ordered surface with marked points. Definition 2.11. A pair ( S , M ) is called a b or der e d surfac e with marke d p oints if it satisfies: (i) S is a connected, orien table 2-dimensional surface but p ossibly with boundary ∂ S ; (ii) M ⊆ ∂ S is a non-empty finite set of marked points; (iii) Eac h connected b oundary component con tains at least one mark ed p oin t. A disk with n marked points is called an n -gon . In this article, we will assume that S has no punctures. See Section 3.3 for further discussions on the punctured surface cases. Definition 2.12. An ar c in ( S , M ) is an isotopy class of curv es γ satisfying: (i) The endpoints of γ lie in M ; (ii) γ do es not self-in tersect except at endp oin ts; (iii) γ is disjoint from M and ∂ S except at endp oin ts. Tw o arcs are called c omp atible if they do not intersect in the in terior of S . The set of all arcs is denoted b y A 0 ( S , M ). A maximal collection of distinct pairwise compatible arcs is called an ide al triangulation . The arcs of an ideal triangulation partition S in to ide al triangles T ∈ T . Since we only consider surfaces without punctures, there will b e no self-folded triangles. Prop osition 2.13. A ny ide al triangulation T of S c ontains exactly n = 6 g + 3 b + c − 6 (15) ar cs, wher e g denotes the genus of S , b the numb er of b oundary components , and c the numb er of marked p oin ts on ∂ S . Definition 2.14. Let T b e a triangulation containing a diagonal γ . Then there exists a unique quadrilateral in T having γ as a diagonal. The operation of replacing γ with the other diagonal γ ′ of this quadrilateral to obtain a new triangulation T ′ is called a flip (see Figure 1 ). γ A B C D A B C D γ ′ flip Figure 1: The flip of a quadrilateral Let G be a split semisimple simply-connected group ov er Q and S a bordered surface with marked p oin ts as ab o ve. The mo duli sp ac e of G -lo c al systems A G, S and its v arian ts are first considered b y F ock–Gonc harov [ 5 ] and further refined in Goncharo v–Shen [ 11 ]. T ypically a G -lo cal system consists of a principal G -bundle L on S with flat connection and certain extra data pro vided by decorated flags in A ∈ G/U , where U is the unip oten t radical of a fixed Borel subgroup B ⊂ G . T ogether with another decorated mo duli space P G ad , S where G ad is the adjoint group, this carries a cluster Poisson structure, i.e. corresp onds to a cluster X -v ariety , and the pair ( P G ad , S , A G, S ) forms a so- called cluster ensemble. There is an atlas for each c hoice of ideal triangulation for S , along with a choice of 7 sp ecial v ertex and also a choice of reduced w ord of the longest element w 0 of the W eyl group of G for each triangle of the ideal triangulation which enco des the seed of the cluster structure. Eac h of these atlases is related by a sequence of mutations. Consequen tly , in the simply-laced case, the cluster structure of A G, S ab o v e can b e enco ded by a generalized quiv er Q , which is the result of amalgamation of the b asic quivers , first constructed in [ 5 ] for type A n and [ 18 , 23 , 11 ] for other Lie types, asso ciated to each triangle T ∈ T of the triangulation of S . They satisfy certain features: (1) F or eac h triangle T ∈ T the b asic quiver Q T dep ends on the choice of reduced word of w 0 and the sp ecial v ertex, and different c hoices are related by a sequence of quiver m utations. (2) The basic quiver consists of n frozen vertices on each b oundary edge of T , where n is the rank of G . F urthermore, we allow “half-weigh ted arro ws” b et ween frozen vertices, so that the signed-adjacency matrix B has 1 2 Z co efficien ts. (3) The quiv er Q encoding the cluster structure of A G, S is obtained b y amalgamation , i.e. gluing the basic quiv ers, according to the triangulations, so that the frozen vertices on the glued b oundary b ecome m utable, and the w eights of the arrows add accordingly . The half arro ws b et ween frozen vertices facilitate the amalgamation b et ween basic quivers, but otherwise they do not play a role in the cluster structure of the frozen v ariables of A G, S . Remark 2.15. The non-simply-laced case is also a v ailable, which is described using quivers with m ultipliers. In particular, when G = SL 2 , the cluster structure is the same as the standard cluster algebr a fr om surfac es [ 6 ], where the basic quiver assigned to eac h triangle T is just a counterclockwise quiver with vertex on the midp oin t of each b oundary edge of T . The flipping of a diagonal corresp onds to a single cluster m utation, and the combinatorics of cluster v ariables and their expansion formula under these sequences of m utations are completely understoo d via the com binatorics of snake diagrams and p erfect matching [ 25 , 26 ]. In this pap er, w e will b e in terested in understanding the combinatorics of the cluster v ariables and their expansion formula for higher rank algebraic group G under sequences of cluster m utations arising from flipping the diagonal of triangulations on the surface of m -gons, which is considerably muc h more complicated. 2.3 Cluster realization of A G, S in t yp e A n Recall the mo duli space A G, S of twisted G -lo cal systems, whose cluster A -v ariety structure is enco ded by a certain quiv er Q asso ciated to the triangulation of the surface. In this subsection, we explain how the quiv ers are constructed for G = SL n +1 of type A n , and also recall the mutation sequences asso ciated with flips. W e first consider a single triangle, i.e. a disk with 3 marked points on the b oundary . Definition 2.16. An n -triangulate d quiver of a triangle T is constructed through the follo wing pro cedure: (1) Subdivide each edge of T by placing n equally spaced v ertices, creating n + 1 equal segmen ts per edge. These vertices are frozen. (2) Construct a triangulation b y dra wing lines parallel to the edges betw een corresponding v ertices. Their in tersections form the m utable vertices. (3) Classify triangles in to tw o types: (a) The three corner triangles (con taining v ertices of the original triangle) are designated as unshade d . (b) The remaining triangles alternate b et ween shade d and unshade d , with adjacen t triangles having opp osite t yp es. (4) Construct the corresponding quiver where: (a) Shaded triangles con tribute 3 arrows forming coun terclo c kwise cycles. 8 (b) Unshaded triangles con tribute 3 arrows forming clockwise cycles. (c) Red dotted arro ws represent half-arro ws in the quiver along the b oundaries. W e can now construct the quiver on a general mark ed surface by amalgamating (gluing) the corresponding quiv ers asso ciated to the triangles of a giv en triangulation. Definition 2.17. An n -triangulate d quiver of a quadrilater al is formed by gluing t w o n -triangulated triangles along a common edge. An n -triangulate d quiver of a surfac e is obtained by assem bling n -triangulated triangles according to a surface triangulation. Definition 2.18. A flip of an n -triangulated quiver of a quadrilateral is a sequence of quiv er mutations p erformed in n steps. F or eac h step k with 1 ≤ k ≤ n : (1) Inscribe a k × ( n + 1 − k ) rectangle such that its vertices coincide with b oundary v ertices of the n -triangulated quadrilateral, while the side of length n + 1 − k lies along the diagonal to b e flipped. (2) Divide this rectangle in to k ( n + 1 − k ) unit squares. (3) P erform quiv er mutations at the center of eac h square. The order of mutations within eac h step is irrelev ant, since no arrows exist b etw een the m utated v ertices after the previous steps, hence the mutations comm ute with eac h other. Figure 2 shows the v ertices (in blue) to mutate at eac h step in case n = 7. (a) Step 1 (b) Step 2 (c) Step 3 (d) Step 4 (e) Step 5 (f ) Step 6 (g) Step 7 (h) Final Figure 2: Step-by-step construction of a flip in a 7-triangulated quadrilateral Example 2.19. F or n = 2, each edge of a triangle is divided into 3 equal parts using 2 v ertices. W e get the 2-triangulated triangle with its corresp onding quiv er (see Figure 3 ) and the resulting quadrilateral after flip (see Figure 4 ). 9 Figure 3: A 2-triangulated triangle and the corresponding quiv er Figure 4: Original quadrilateral (left) and after flip (righ t) Note that no arrows exist b et ween v ertices lying on the same main diagonal. The flip is achiev ed through sequences of mutations in t wo steps (refer to the lab eling in Figure 5 ): Step 1: Mutate at x 1 and x 2 ; Step 2: Mutate at x 3 and x 4 . Since no edges connect v ertices within the same step, the m utation order do es not affect the result. Without loss of generality , w e can let the sequence b e µ = { x 2 → x 1 → x 4 → x 3 } . µ y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 x 2 x 4 x 3 x 1 y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 x ′ 2 x ′ 4 x ′ 3 x ′ 1 Figure 5: Quivers corresponding to the quadrilaterals in Figure 4 y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 y 7 y 8 y 6 y 5 y 4 y 3 y 1 y 2 x ′ 2 x ′ 1 x ′ 4 x ′ 3 x 2 x ′ 2 x ′ 2 x ′ 2 x 4 x 4 x 4 x ′ 4 x 3 x 3 x 3 x 3 x 1 x 1 x ′ 1 x ′ 1 Figure 6: Mutation sequence for the flip. Starting with the initial cluster v ariables x = ( y 1 , ..., y 8 , x 1 , ..., x 4 ), the resulting cluster v ariables after 10 the flip can b e expressed in terms of the initial cluster v ariables in Laurent p olynomials as: x ′ 2 = y 7 x 3 + y 6 x 4 x 2 ; x ′ 1 = y 3 x 4 + y 2 x 3 x 1 ; x ′ 4 = y 8 x ′ 1 + y 1 x ′ 2 x 4 = y 3 y 8 x 2 x 4 + y 2 y 8 x 2 x 3 + y 1 y 7 x 1 x 3 + y 1 y 6 x 1 x 4 x 1 x 2 x 4 ; x ′ 3 = y 4 x ′ 2 + y 5 x ′ 1 x 3 = y 4 y 7 x 1 x 3 + y 4 y 6 x 1 x 4 + y 3 y 5 x 2 x 4 + y 2 y 5 x 2 x 3 x 1 x 2 x 3 . Example 2.20. Consider a p en tagon with a giv en triangulation. Each triangle is assigned the 2-triangulated quiv er, yielding the initial quiver: A B C D E y 5 y 6 y 7 y 8 y 9 y 10 x 1 x 2 x 3 x 5 x 6 x 7 y 1 y 2 y 3 y 4 x 4 Figure 7: Pen tagon with triangulation (left) and the initial quiver for the p en tagon (right) T o obtain the diagonal B E , we consider t wo flip sequences: µ 1 : µ 2 : A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E Figure 8: Both p ossible flip sequences Note that in the 2-triangulated construction, w e do not assume that the main diagonal is equally divided in to 3 parts. The mutation sequences for both flip paths are as follow (the order of eac h flip sequence is not 11 necessary unique): µ 1 = { x 2 → x 7 → x 4 → x 5 → x 1 → x 6 → x 3 → x 2 } ; µ 2 = { x 1 → x 6 → x 3 → x 5 → x 2 → x 7 → x 4 → x 1 } . Belo w we will verify that the cluster expansion of the resulting cluster v ariables corresponding to b oth m utation sequences coincides up to reindexing of the v ariables. F or the first flip sequence µ 1 (top of Figure 8 ), we calculate: x ′ 2 = x 1 x 4 + y 1 x 5 x 2 ; x ′ 7 = y 4 x 5 + y 5 x 4 x 7 ; x ′ 4 = y 3 x ′ 2 + y 2 x ′ 7 x 4 = y 3 x 1 x 4 x 7 + y 3 y 1 x 5 x 7 + y 2 y 4 x 2 x 5 + y 2 y 5 x 2 x 4 x 2 x 4 x 7 ; x ′ 5 = x 6 x ′ 7 + y 6 x ′ 2 x 5 = y 4 x 2 x 5 x 6 + y 5 x 2 x 4 x 6 + y 6 x 1 x 4 x 7 + y 1 y 6 x 5 x 7 x 2 x 5 x 7 ; x ′ 1 = y 1 x 3 + y 10 x ′ 2 x 1 = y 1 x 2 x 3 + y 10 x 1 x 4 + y 1 y 10 x 5 x 1 x 2 ; x ′ 6 = x 3 x ′ 5 + y 7 x ′ 2 x 6 = y 4 x 2 x 3 x 5 x 6 + y 5 x 2 x 3 x 4 x 6 + y 6 x 1 x 3 x 4 x 7 + y 1 y 6 x 3 x 5 x 7 + y 7 x 1 x 4 x 5 x 7 + y 1 y 7 x 2 5 x 7 x 2 x 5 x 6 x 7 ; x ′ 3 = y 8 x ′ 1 + y 9 x ′ 6 x 3 = y 1 y 8 x 2 x 3 x 5 x 6 x 7 + y 8 y 10 x 1 x 4 x 5 x 6 x 7 + y 1 y 8 y 10 x 2 5 x 6 x 7 + y 4 y 9 x 1 x 2 x 3 x 5 x 6 x 1 x 2 x 3 x 5 x 6 x 7 + y 5 y 9 x 1 x 2 x 3 x 4 x 6 + y 6 y 9 x 2 1 x 3 x 4 x 7 + y 1 y 6 y 9 x 1 x 3 x 5 x 7 + y 7 y 9 x 2 1 x 4 x 5 x 7 + y 1 y 7 y 9 x 1 x 2 5 x 7 x 1 x 2 x 3 x 5 x 6 x 7 ; x ′′ 2 = y 2 x ′ 6 + x ′ 4 x ′ 1 x ′ 2 = y 2 y 4 x 2 2 x 3 x 5 x 6 + y 2 y 5 x 2 2 x 3 x 4 x 6 + y 2 y 6 x 1 x 2 x 3 x 4 x 7 + y 2 y 7 x 1 x 2 x 4 x 5 x 7 + y 3 y 10 x 1 x 4 x 5 x 6 x 7 x 1 x 2 x 4 x 5 x 6 x 7 + y 1 y 3 y 10 x 2 5 x 6 x 7 + y 2 y 4 y 10 x 2 x 2 5 x 6 + y 2 y 5 y 10 x 2 x 4 x 5 x 6 + y 1 y 3 x 2 x 3 x 5 x 6 x 7 x 1 x 2 x 4 x 5 x 6 x 7 . F or the second flip sequence µ 2 (b ottom of Figure 8 ), w e calculate: x ′ 1 = x 2 x 3 + y 10 x 5 x 1 ; x ′ 6 = y 7 x 5 + y 6 x 3 x 6 ; x ′ 3 = y 8 x ′ 1 + y 9 x ′ 6 x 3 = y 8 x 2 x 3 x 6 + y 8 y 10 x 5 x 6 + y 7 y 9 x 1 x 5 + y 6 y 9 x 1 x 3 x 1 x 3 x 6 ; x ′ 5 = x 7 x ′ 6 + y 5 x ′ 1 x 5 = y 7 x 1 x 5 x 7 + y 6 x 1 x 3 x 7 + y 5 x 2 x 3 x 6 + y 5 y 10 x 5 x 6 x 1 x 5 x 6 ; x ′ 2 = y 10 x 4 + y 1 x ′ 1 x 2 = y 10 x 1 x 4 + y 1 x 2 x 3 + y 1 y 10 x 5 x 1 x 2 ; x ′ 7 = x 4 x ′ 5 + y 4 x ′ 1 x 7 = y 7 x 1 x 4 x 5 x 7 + y 6 x 1 x 3 x 4 x 7 + y 5 x 2 x 3 x 4 x 6 + y 5 y 10 x 4 x 5 x 6 + y 4 x 2 x 3 x 5 x 6 + y 4 y 10 x 2 5 x 6 x 1 x 5 x 6 x 7 ; x ′ 4 = y 3 x ′ 2 + y 2 x ′ 7 x 4 = y 3 y 10 x 1 x 4 x 5 x 6 x 7 + y 1 y 3 x 2 x 3 x 5 x 6 x 7 + y 1 y 3 y 10 x 2 5 x 6 x 7 + y 2 y 7 x 1 x 2 x 4 x 5 x 7 x 1 x 2 x 4 x 5 x 6 x 7 + y 2 y 6 x 1 x 2 x 3 x 4 x 7 + y 2 y 5 x 2 2 x 3 x 4 x 6 + y 2 y 5 y 10 x 2 x 4 x 5 x 6 + y 2 y 4 x 2 2 x 3 x 5 x 6 + y 2 y 4 y 10 x 2 x 2 5 x 6 x 1 x 2 x 4 x 5 x 6 x 7 ; x ′′ 1 = y 9 x ′ 7 + x ′ 3 x ′ 2 x ′ 1 = y 5 y 9 x 1 x 2 x 3 x 4 x 6 + y 7 y 9 x 2 1 x 4 x 5 x 7 + y 6 y 9 x 2 1 x 3 x 4 x 7 + y 4 y 9 x 1 x 2 x 3 x 5 x 6 + y 8 y 10 x 1 x 4 x 5 x 6 x 7 x 1 x 2 x 3 x 5 x 6 x 7 + y 1 y 8 x 2 x 3 x 5 x 6 x 7 + y 1 y 8 y 10 x 2 5 x 6 x 7 + y 1 y 7 y 9 x 1 x 2 5 x 7 + y 1 y 6 y 9 x 1 x 3 x 5 x 7 x 1 x 2 x 3 x 5 x 6 x 7 . F or the first sequence, the dotted diagonal corresp onds to expansion v ariables x ′′ 2 and x ′ 3 ; for the second sequence, it corresp onds to x ′′ 1 and x ′ 4 . Our calculations show that x ′ 3 = x ′′ 1 and x ′′ 2 = x ′ 4 , confirming that b oth sequences yield the same result up to vertex relabeling. Example 2.21. F or n = 3, each edge of a triangle is divided into 4 equal parts using 3 v ertices. The construction follows similarly: 12 Figure 9: A triangle and its corresp onding 3-triangulated triangle Consider the 3-triangulated quiver of a quadrilateral, lab eled as in the left of Figure 10 . µ y 10 y 11 y 12 y 9 y 8 y 7 y 6 y 5 y 4 y 1 y 2 y 3 x 3 x 8 x 4 x 9 x 2 x 7 x 5 x 6 x 1 y 10 y 11 y 12 y 9 y 8 y 7 y 6 y 5 y 4 y 1 y 2 y 3 x 3 x 8 x 4 x 9 x 2 x 7 x 5 x 6 x 1 Figure 10: Quivers before and after flipping a quadrilateral The flip of the quiver corresp onding to the flip of diagonal of the quadrilateral can b e p erformed in 3 m utation steps: Step 1: Mutate at x 1 , x 2 , x 3 ; Step 2: Mutate at x 6 , x 7 , x 8 , x 9 ; Step 3: Mutate at x 2 , x 4 , x 5 . Without loss of generality , w e can consider the follo wing flip sequence: µ = { x 3 → x 2 → x 1 → x 9 → x 8 → x 6 → x 7 → x 5 → x 4 → x 2 } . The resulting cluster v ariables are then given b y: x ′ 3 = y 10 x 8 + y 9 x 9 x 3 ; x ′ 2 = x 9 x 7 + x 6 x 8 x 2 ; x ′ 1 = x 6 y 4 + x 7 y 3 x 1 ; x ′ 9 = y 11 x ′ 2 + x 5 x ′ 3 x 9 = x 3 x 9 x 7 y 11 + x 3 x 8 x 6 y 11 + x 2 x 8 x 5 y 10 + x 2 x 9 x 5 y 9 x 3 x 2 x 9 ; x ′ 8 = y 8 x ′ 2 + x 4 x ′ 3 x 8 = x 3 x 9 x 7 y 8 + x 3 x 8 x 6 y 8 + x 2 x 8 x 4 y 10 + x 2 x 9 x 4 y 9 x 3 x 2 x 8 ; x ′ 6 = x 5 x ′ 1 + y 2 x ′ 2 x 6 = x 2 x 6 x 5 y 4 + x 2 x 7 x 5 y 3 + x 1 x 9 x 7 y 2 + x 1 x 8 x 6 y 2 x 2 x 1 x 6 ; x ′ 7 = x 4 x ′ 1 + y 5 x ′ 2 x 7 = x 2 x 6 x 4 y 4 + x 2 x 7 x 4 y 3 + x 1 x 9 x 7 y 5 + x 1 x 8 x 6 y 5 x 2 x 1 x 7 ; 13 x ′ 5 = y 12 x ′ 6 + y 1 x ′ 9 x 5 = x 3 x 1 x 8 x 2 6 y 1 y 11 + x 3 x 1 x 2 9 x 7 y 2 y 12 + x 3 x 1 x 9 x 6 x 7 y 1 y 11 + x 3 x 1 x 9 x 8 x 6 y 2 y 12 x 3 x 2 x 1 x 9 x 6 x 5 + x 2 x 1 x 8 x 6 x 5 y 1 y 10 + x 2 x 1 x 9 x 6 x 5 y 1 y 9 + x 3 x 2 x 9 x 6 x 5 y 4 y 12 + x 3 x 2 x 9 x 7 x 5 y 3 y 12 x 3 x 2 x 1 x 9 x 6 x 5 ; x ′ 4 = y 6 x ′ 8 + y 7 x ′ 7 x 4 = x 3 x 1 x 9 x 2 7 y 8 y 6 + x 3 x 1 x 2 8 x 6 y 7 y 5 + x 3 x 1 x 8 x 9 x 7 y 7 y 5 + x 3 x 1 x 8 x 7 x 6 y 8 y 6 x 3 x 2 x 1 x 8 x 7 x 4 + x 2 x 1 x 8 x 7 x 4 y 10 y 6 + x 2 x 1 x 9 x 7 x 4 y 9 y 6 + x 3 x 2 x 8 x 6 x 4 y 7 y 4 + x 3 x 2 x 8 x 7 x 4 y 7 y 3 x 3 x 2 x 1 x 8 x 7 x 4 ; x ′′ 2 = x ′ 9 x ′ 7 + x ′ 6 x ′ 8 x ′ 2 = x 3 x 1 x 8 x 6 ( x 9 x 7 + x 6 x 8 ) y 11 y 5 + x 3 x 2 x 8 x 2 6 x 4 y 4 y 11 + x 3 x 2 x 8 x 6 x 7 x 4 y 11 y 3 x 3 x 2 x 1 x 9 x 8 x 6 x 7 + x 2 x 1 x 2 8 x 6 x 5 y 10 y 5 + x 2 2 x 8 x 6 x 5 x 4 y 10 y 4 + x 2 2 x 8 x 7 x 5 x 4 y 10 y 3 + x 2 x 1 x 9 x 8 x 6 x 5 y 5 y 9 x 3 x 2 x 1 x 9 x 8 x 6 x 7 + x 2 2 x 9 x 6 x 5 x 4 y 4 y 9 + x 2 2 x 9 x 7 x 5 x 4 y 3 y 9 + x 3 x 2 x 9 x 6 x 7 x 5 y 4 y 8 + x 3 x 2 x 9 x 2 7 x 5 y 8 y 3 x 3 x 2 x 1 x 9 x 8 x 6 x 7 + x 3 x 1 x 9 x 7 ( x 9 x 7 + x 6 x 8 ) y 2 y 8 + x 2 x 1 x 9 x 8 x 7 x 4 y 10 y 2 + x 2 x 1 x 2 9 x 7 x 4 y 2 y 9 x 3 x 2 x 1 x 9 x 8 x 6 x 7 . 3 Cluster expansion formulas for general flips Recall the mo duli spaces A SL n +1 , S of G -lo cal system of type A n . In this section, w e first derive the general expansion formula corresp onding to the cluster mutation sequences asso ciated with a flip of diagonal when S is a quadrilateral. Based on the formula, we will derive the recurrence relation in the case of general n -triangulated m -gon in the next section. As established in Definition 2.18 , the flip of a quadrilateral corresp onds to a sp ecific sequence of m utations in the asso ciated quiv er. W e consider the mo duli space A SL n +1 , S where gluing maps are applied to pairs of triangles such that the diagonal of the resulting quadrilateral is formed by t wo corresp onding colored b oundary interv als with opp osite orien tations [ 10 , 11 ]. W e will consider the unpunctured case as assumed, and give some remarks on the punctured surface cases in Section 3.3 . 3.1 Unpunctured surface case Let S b e a quadrilateral with vertices V 1 , V 2 , V 3 , V 4 lab eled in clo c kwise order, with V 1 V 3 as the initial diagonal. These vertices corresp ond to flags as defined in Section 2.2 . F or conv enience, we treat the quadri- lateral S as a square. The initial cluster seed consists of an ( n 2 + 4 n )-tuple of cluster v ariables, indexed by the b oundary variables y i for i = 1 , 2 , . . . , 4 n , and interior variables x j for j = 1 , 2 , . . . , n 2 assigned to the v ertices of the n -triangulation. The lab eling ( x 1 , ..., x n 2 , y 1 , ..., y 4 n ) of the cluster seed follo ws the con ven tion b elo w: (a) Boundary e dges: • Edge V 4 V 1 : ( y 1 , y 2 , . . . , y n ) • Edge V i V i +1 for i = 1 , 2 , 3: (ordered from V i to V i +1 ) ( y in +1 , y in +2 , . . . , y ( i +1) n ) (b) Diagonal variables: • Initial diagonal V 1 V 3 : (ordered from V 1 to V 3 ) ( x 1 , x 2 , . . . , x n ) • Diagonal V 2 V 4 : (ordered from V 2 to V 4 ) ( ( x n +1 , x n +2 , . . . , x 2 n ) if n is even , ( x n +1 , . . . , x 3 n − 1 2 , x n +1 2 , x 3 n − 1 2 , . . . , x 2 n − 1 ) if n is odd . (16) (c) Interior variables: The remaining x j v ariables are lab eled b y: 14 (i) Considering concen tric lay ers of vertices inside the square (excluding the four edges); (ii) Starting from the outermost la yer con taining n 2 v ertices; (iii) Labeling vertices in clo c kwise order, b eginning from the edge parallel to V 4 V 1 that is closest to V 4 V 1 ; (iv) Proceeding inw ard to subsequent la yers, and rep eating the labeling pro cedure. V 1 V 2 V 3 V 4 V 1 V 2 V 3 V 4 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 y 11 y 12 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 y 11 y 12 y 13 y 14 y 15 y 16 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 x 16 Figure 11: V ertex lab eling examples for n = 3 and n = 4 The vertices are organized in concen tric layers , where the k -th layer (also called layer k for conv enience) con tains ( n − 2 k ) 2 v ertices. After the sequence of mutation corresponding to flipping the diagonal from V 1 V 3 to V 2 V 4 , each vertex in the k -th lay er undergo es exactly k mutations, starting from the outermost lay er ( k = 1) and pro ceeding in ward. Explicitly , consider the cluster realization of A SL n +1 , S o ver the square □ V 1 V 2 V 3 V 4 , i.e. with the asso ciated n -triangulated quiver. Let Υ( V 1 V 2 V 3 V 4 ) b e the n -dimensional vector where each comp onen t is a Laurent p olynomial in Z [ x ± 1 , . . . , x ± n 2 , y 1 , . . . , y 4 n ] represen ting the expansion form ula of the n cluster v ariables labeled along V 2 V 4 with resp ect to the initial v ariables after flipping the diagonal from V 1 V 3 to V 2 V 4 . Then w e hav e: Υ( V 1 V 2 V 3 V 4 ) = ( ( x (1) n +1 , . . . , x ( k ) n + k , x ( k ) n + k +1 , . . . , x (1) 2 n ) if n = 2 k is even , ( x (1) n +1 , . . . , x ( k ) n + k , x ( k +1) k +1 , x ( k ) n + k +1 , . . . , x (1) 2 n − 1 ) if n = 2 k + 1 is o dd (17) where x ( j ) i denotes that v ertex x i is mutated exactly j times in the flip mutation sequence. F or con venience, w e will also call each v ariable x ( j ) i that app ears as a co ordinate in ( 17 ) to b e a vertex of Υ( V 1 V 2 V 3 V 4 ). F or notation conv enience, w e sometimes write x ′ j for x (1) j , x ′′ j for x (2) j , x ′′′ j for x (3) j , etc. The follo wing proposition can be verified directly from the definition b y considering the last tw o m utation steps: Prop osition 3.1. Consider the cluster r e alization of A SL n +1 , S over the squar e □ V 1 V 2 V 3 V 4 with vertic es x i and y j as define d. Define the mutate d variables of the sub-squar e of 1 smal ler r ank to b e: Υ( V 1 y 2 n x n y 1 ) = ( X 1 , 1 , X 1 , 2 , . . . , X 1 ,n − 1 ); Υ( x 1 y 2 n +1 V 3 y 4 n ) = ( X 2 , 1 , X 2 , 2 , . . . , X 2 ,n − 1 ) . (18) Then the mutate d variables along the diagonal V 2 V 4 satisfy: (a) F or n = 1 , we have only one mutable variable x 1 with 4 fr ozen variables y 1 , ..., y 4 . Then Υ( V 1 V 2 V 3 V 4 ) = x ′ 1 := y 1 y 3 + y 2 y 4 x 1 . F or n = 2 , we have 4 mutable variables x 1 , ..., x 4 and 8 fr ozen variables y 1 , ..., y 8 . Then Υ( V 1 V 2 V 3 V 4 ) = ( x ′ 3 , x ′ 4 ) 15 wher e x ′ 3 := y 2 y 5 x 1 + y 4 y 7 x 2 + y 3 y 5 x 4 x 1 x 3 + y 4 y 6 x 4 x 2 x 3 ; and x ′ 4 := y 3 y 8 x 1 + y 1 y 6 x 2 + y 2 y 8 x 3 x 1 x 4 + y 1 y 7 x 3 x 2 x 4 . (b) If n = 2 k ≥ 4 is even, then V 2 V 4 c onsists of vertic es x n +1 , x n +2 , . . . , x 2 n with: Υ( x 1 x n +1 x n x 2 n ) = ( x (1) n +2 , . . . , x ( k − 1) n + k , x ( k − 1) n + k +1 , . . . , x (1) 2 n − 1 ); Υ( V 1 V 2 V 3 V 4 ) = ( x (1) n +1 , . . . , x ( k ) n + k , x ( k ) n + k +1 , . . . , x (1) 2 n ); x ′ n +1 = x (1) n +1 = y 2 n X 2 , 1 + y 2 n +1 X 1 , 1 x n +1 ; x ′ 2 n = x (1) 2 n = y 1 X 2 ,n − 1 + y 4 n X 1 ,n − 1 x 2 n ; and for t = 0 , 1 , . . . , k − 2 : x (2+ t ) n +2+ t = X 1 , 1+ t X 2 , 2+ t + X 1 , 2+ t X 2 , 1+ t x (1+ t ) n +2+ t ; x (2+ t ) 2 n − 1 − t = X 1 ,n − 2 − t X 2 ,n − 1 − t + X 1 ,n − 1 − t X 2 ,n − 2 − t x (1+ t ) 2 n − 1 − t . (19) (c) If n = 2 k + 1 ≥ 3 is o dd, then V 2 V 4 c onsists of vertic es x n +1 , . . . , x n + k , x k +1 , x n + k +1 , . . . , x 2 n − 1 with: Υ( x 1 x n +1 x n x 2 n − 1 ) = ( ( x (1) 2 ) k = 1 ( x (1) n +2 , . . . , x ( k − 1) n + k , x ( k ) k +1 , x ( k − 1) n + k +1 , . . . , x (1) 2 n − 2 ) k > 1 ; Υ( V 1 V 2 V 3 V 4 ) = ( x (1) n +1 ; . . . , x ( k ) n + k , x ( k +1) k +1 , x ( k ) n + k +1 , . . . , x (1) 2 n − 1 ); x ′ n +1 = x (1) n +1 = y 2 n X 2 , 1 + y 2 n +1 X 1 , 1 x n +1 ; x ′ 2 n − 1 = x (1) 2 n − 1 = y 1 X 2 ,n − 1 + y 4 n X 1 ,n − 1 x 2 n − 1 ; x ( k +1) k +1 = X 1 ,k X 2 ,k +1 + X 1 ,k +1 X 2 ,k x ( k ) k +1 ; and if k > 1 , for t = 0 , 1 , . . . , k − 2 : x (2+ t ) n +2+ t = X 1 , 1+ t X 2 , 2+ t + X 1 , 2+ t X 2 , 1+ t x (1+ t ) n +2+ t ; x (2+ t ) 2 n − 2 − t = X 1 ,n − 2 − t X 2 ,n − 1 − t + X 1 ,n − 1 − t X 2 ,n − 2 − t x (1+ t ) 2 n − 2 − t . (20) By the Laurent phenomenon, ev ery co ordinate of Υ is a Laurent p olynomial in the initial v ariables x i and y j . Example 3.2. F or n = 4, we shall compute: Υ( x 1 x 5 x 4 x 8 ) = ( x ′ 6 , x ′ 7 ); x ′ 6 = x 10 x 13 x 2 + x 12 x 15 x 3 + x 7 x 11 x 13 x 2 x 6 + x 7 x 12 x 14 x 3 x 6 ; x ′ 7 = x 11 x 16 x 2 + x 9 x 14 x 3 + x 6 x 10 x 16 x 2 x 7 + x 6 x 9 x 15 x 3 x 7 . Applying Prop osition 3.1 yields: Υ( V 1 y 8 x 4 y 1 ) = ( X 1 , 1 , X 1 , 2 , X 1 , 3 ); 16 X 1 , 1 = x ′ 12 = y 4 x 5 x 1 + y 7 x 15 x 3 + y 5 x 5 x 10 x 1 x 11 + y 7 x 7 x 14 x 3 x 6 + y 6 x 5 x 7 x 2 x 12 + y 7 x 10 x 13 x 2 x 12 + y 7 x 7 x 11 x 13 x 2 x 6 x 12 + y 6 x 5 x 6 x 10 x 2 x 11 x 12 ; X 1 , 2 = x ′′ 2 = y 3 x 13 x 2 + y 6 x 16 x 2 + y 4 x 12 x 16 x 1 x 7 + y 6 x 9 x 14 x 3 x 11 + y 5 x 9 x 13 x 1 x 6 + y 3 x 12 x 15 x 3 x 10 + y 6 x 6 x 10 x 16 x 2 x 7 x 11 + y 3 x 7 x 11 x 13 x 2 x 6 x 10 + y 5 x 10 x 12 x 16 x 1 x 7 x 11 + y 6 x 6 x 9 x 15 x 3 x 7 x 11 + y 4 x 9 x 11 x 13 x 1 x 6 x 10 + y 3 x 7 x 12 x 14 x 3 x 6 x 10 + y 5 x 2 x 9 x 12 x 15 x 1 x 3 x 7 x 11 + y 4 x 2 x 9 x 12 x 15 x 1 x 3 x 7 x 10 + y 4 x 2 x 9 x 12 x 14 x 1 x 3 x 6 x 10 + y 5 x 2 x 9 x 12 x 14 x 1 x 3 x 6 x 11 ; X 1 , 3 = x ′ 9 = y 2 x 14 x 3 + y 5 x 8 x 1 + y 4 x 8 x 11 x 1 x 10 + y 2 x 6 x 15 x 3 x 7 + y 2 x 11 x 16 x 2 x 9 + y 3 x 6 x 8 x 2 x 9 + y 2 x 6 x 10 x 16 x 2 x 7 x 9 + y 3 x 7 x 8 x 11 x 2 x 9 x 10 ; Υ( x 1 y 9 V 3 y 16 ) = ( X 2 , 1 , X 2 , 2 , X 2 , 3 ); X 2 , 1 = x ′ 13 = y 10 x 10 x 2 + y 13 x 5 x 4 + y 10 x 7 x 11 x 2 x 6 + y 12 x 5 x 15 x 4 x 14 + y 10 x 12 x 15 x 3 x 13 + y 11 x 5 x 7 x 3 x 13 + y 11 x 5 x 6 x 15 x 3 x 13 x 14 + y 10 x 7 x 12 x 14 x 3 x 6 x 13 ; X 2 , 2 = x ′′ 3 = y 11 x 9 x 3 + y 14 x 12 x 3 + y 14 x 10 x 13 x 2 x 15 + y 12 x 12 x 16 x 4 x 6 + y 11 x 11 x 16 x 2 x 14 + y 13 x 9 x 13 x 4 x 7 + y 14 x 7 x 12 x 14 x 3 x 6 x 15 + y 11 x 6 x 9 x 15 x 3 x 7 x 14 + y 14 x 7 x 11 x 13 x 2 x 6 x 15 + y 13 x 12 x 14 x 16 x 4 x 6 x 15 + y 11 x 6 x 10 x 16 x 2 x 7 x 14 + y 12 x 9 x 13 x 15 x 4 x 7 x 14 + y 13 x 3 x 11 x 13 x 16 x 2 x 4 x 6 x 15 + y 13 x 3 x 10 x 13 x 16 x 2 x 4 x 7 x 15 + y 12 x 3 x 10 x 13 x 16 x 2 x 4 x 7 x 14 + y 12 x 3 x 11 x 13 x 16 x 2 x 4 x 6 x 14 ; X 2 , 3 = x ′ 16 = y 12 x 8 x 4 + y 15 x 11 x 2 + y 15 x 6 x 10 x 2 x 7 + y 13 x 8 x 14 x 4 x 15 + y 14 x 6 x 8 x 3 x 16 + y 15 x 9 x 14 x 3 x 16 + y 14 x 7 x 8 x 14 x 3 x 15 x 16 + y 15 x 6 x 9 x 15 x 3 x 7 x 16 . Therefore, the expansion formula for n = 4 of the full square □ V 1 V 2 V 3 V 4 is: Υ( V 1 V 2 V 3 V 4 ) = ( x ′ 5 , x ′′ 6 , x ′′ 7 , x ′ 8 ) where x ′ 5 = y 8 X 2 , 1 + y 9 X 1 , 1 x 5 = y 8 y 13 x 4 + y 4 y 9 x 1 + y 5 y 9 x 10 x 1 x 11 + y 6 y 9 x 7 x 2 x 12 + y 8 y 11 x 7 x 3 x 13 + y 8 y 12 x 15 x 4 x 14 + y 7 y 9 x 15 x 3 x 5 + y 8 y 10 x 10 x 2 x 5 + y 6 y 9 x 6 x 10 x 2 x 11 x 12 + y 8 y 11 x 6 x 15 x 3 x 13 x 14 + y 7 y 9 x 7 x 14 x 3 x 5 x 6 + y 7 y 9 x 10 x 13 x 2 x 5 x 12 + y 8 y 10 x 7 x 11 x 2 x 5 x 6 + y 8 y 10 x 12 x 15 x 3 x 5 x 13 + y 7 y 9 x 7 x 11 x 13 x 2 x 5 x 6 x 12 + y 8 y 10 x 7 x 12 x 14 x 3 x 5 x 6 x 13 ; x ′ 8 = y 1 X 2 , 3 + y 16 X 1 , 3 x 8 = y 1 y 12 x 4 + y 6 y 15 x 1 + y 1 y 13 x 14 x 4 x 15 + y 1 y 14 x 6 x 3 x 16 + y 3 y 16 x 7 x 2 x 9 + y 4 y 16 x 11 x 1 x 10 + y 1 y 15 x 11 x 2 x 8 + y 2 y 16 x 14 x 3 x 8 + y 1 y 14 x 7 x 14 x 3 x 15 x 16 + y 3 y 16 x 7 x 11 x 2 x 9 x 10 + y 1 y 15 x 6 x 10 x 2 x 7 x 8 + y 1 y 15 x 9 x 14 x 3 x 8 x 16 + y 2 y 16 x 6 x 15 x 3 x 7 x 8 + y 2 y 16 x 11 x 16 x 2 x 8 x 9 + y 1 y 15 x 6 x 9 x 15 x 3 x 7 x 8 x 16 + y 2 y 16 x 6 x 10 x 16 x 2 x 7 x 8 x 9 ; 17 x ′′ 6 = X 1 , 1 X 2 , 2 + X 1 , 2 X 2 , 1 x ′ 6 = y 3 y 10 x 2 + y 7 y 14 x 3 + y 3 y 13 x 5 x 4 x 10 + y 4 y 14 x 5 x 1 x 15 + y 5 y 10 x 9 x 1 x 6 + y 6 y 10 x 16 x 2 x 13 + y 7 y 11 x 9 x 3 x 12 + y 7 y 12 x 16 x 4 x 6 + y 3 y 10 x 7 x 11 x 2 x 6 x 10 + y 3 y 10 x 12 x 15 x 3 x 10 x 13 + y 3 y 11 x 5 x 7 x 3 x 10 x 13 + y 3 y 12 x 5 x 15 x 4 x 10 x 14 + y 4 y 10 x 9 x 11 x 1 x 6 x 10 + y 4 y 10 x 12 x 16 x 1 x 7 x 13 + y 5 y 14 x 5 x 10 x 1 x 11 x 15 + y 6 y 10 x 9 x 14 x 3 x 11 x 13 + y 6 y 14 x 5 x 7 x 2 x 12 x 15 + y 7 y 11 x 11 x 16 x 2 x 12 x 14 + y 7 y 13 x 9 x 13 x 4 x 7 x 12 + y 7 y 13 x 14 x 16 x 4 x 6 x 15 + y 7 y 14 x 7 x 14 x 3 x 6 x 15 + y 7 y 14 x 10 x 13 x 2 x 12 x 15 + y 3 y 10 x 7 x 12 x 14 x 3 x 6 x 10 x 13 + y 3 y 11 x 5 x 6 x 15 x 3 x 10 x 13 x 14 + y 4 y 11 x 2 x 5 x 9 x 1 x 3 x 10 x 13 + y 4 y 11 x 5 x 6 x 16 x 1 x 7 x 13 x 14 + y 4 y 12 x 3 x 5 x 16 x 1 x 4 x 7 x 14 + y 4 y 13 x 2 x 5 x 9 x 1 x 4 x 7 x 10 + y 4 y 13 x 3 x 5 x 16 x 1 x 4 x 7 x 15 + y 5 y 10 x 10 x 12 x 16 x 1 x 7 x 11 x 13 + y 5 y 11 x 2 x 5 x 9 x 1 x 3 x 11 x 13 + y 5 y 13 x 2 x 5 x 9 x 1 x 4 x 7 x 11 + y 6 y 10 x 6 x 9 x 15 x 3 x 7 x 11 x 13 + y 6 y 10 x 6 x 10 x 16 x 2 x 7 x 11 x 13 + y 6 y 11 x 5 x 6 x 9 x 3 x 11 x 12 x 13 + y 6 y 11 x 5 x 6 x 16 x 2 x 12 x 13 x 14 + y 6 y 12 x 3 x 5 x 16 x 2 x 4 x 12 x 14 + y 6 y 13 x 3 x 5 x 16 x 2 x 4 x 12 x 15 + y 6 y 13 x 5 x 6 x 9 x 4 x 7 x 11 x 12 + y 6 y 14 x 5 x 6 x 10 x 2 x 11 x 12 x 15 + y 7 y 11 x 6 x 9 x 15 x 3 x 7 x 12 x 14 + y 7 y 11 x 6 x 10 x 16 x 2 x 7 x 12 x 14 + y 7 y 12 x 9 x 13 x 15 x 4 x 7 x 12 x 14 + y 7 y 14 x 7 x 11 x 13 x 2 x 6 x 12 x 15 + y 4 y 10 x 2 x 9 x 12 x 14 x 1 x 3 x 6 x 10 x 13 + y 4 y 10 x 2 x 9 x 12 x 15 x 1 x 3 x 7 x 10 x 13 + y 4 y 12 x 2 x 5 x 9 x 15 x 1 x 4 x 7 x 10 x 14 + y 5 y 10 x 2 x 9 x 12 x 14 x 1 x 3 x 6 x 11 x 13 + y 5 y 10 x 2 x 9 x 12 x 15 x 1 x 3 x 7 x 11 x 13 + y 5 y 11 x 5 x 6 x 10 x 16 x 1 x 7 x 11 x 13 x 14 + y 5 y 12 x 2 x 5 x 9 x 15 x 1 x 4 x 7 x 11 x 14 + y 5 y 12 x 3 x 5 x 10 x 16 x 1 x 4 x 7 x 11 x 14 + y 5 y 13 x 3 x 5 x 10 x 16 x 1 x 4 x 7 x 11 x 15 + y 6 y 12 x 5 x 6 x 9 x 15 x 4 x 7 x 11 x 12 x 14 + y 7 y 12 x 3 x 10 x 13 x 16 x 2 x 4 x 7 x 12 x 14 + y 7 y 12 x 3 x 11 x 13 x 16 x 2 x 4 x 6 x 12 x 14 + y 7 y 13 x 3 x 10 x 13 x 16 x 2 x 4 x 7 x 12 x 15 + y 7 y 13 x 3 x 11 x 13 x 16 x 2 x 4 x 6 x 12 x 15 + y 6 y 11 x 2 6 x 5 x 9 x 15 x 3 x 7 x 11 x 12 x 13 x 14 + y 6 y 11 x 2 6 x 5 x 10 x 16 x 2 x 7 x 11 x 12 x 13 x 14 + y 4 y 11 x 2 x 5 x 6 x 9 x 15 x 1 x 3 x 7 x 10 x 13 x 14 + y 5 y 11 x 2 x 5 x 6 x 9 x 15 x 1 x 3 x 7 x 11 x 13 x 14 + y 6 y 12 x 3 x 5 x 6 x 10 x 16 x 2 x 4 x 7 x 11 x 12 x 14 + y 6 y 13 x 3 x 5 x 6 x 10 x 16 x 2 x 4 x 7 x 11 x 12 x 15 ; x ′′ 7 = X 1 , 2 X 2 , 3 + X 1 , 3 X 2 , 2 x ′ 7 = y 2 y 11 x 3 + y 6 y 15 x 2 + y 2 y 13 x 13 x 4 x 7 + y 2 y 14 x 12 x 3 x 9 + y 3 y 15 x 13 x 2 x 16 + y 4 y 15 x 12 x 1 x 7 + y 5 y 11 x 8 x 1 x 14 + y 6 y 12 x 8 x 4 x 11 + y 2 y 11 x 6 x 15 x 3 x 7 x 14 + y 2 y 11 x 11 x 16 x 2 x 9 x 14 + y 2 y 12 x 12 x 16 x 4 x 6 x 9 + y 2 y 12 x 13 x 15 x 4 x 7 x 14 + y 2 y 14 x 10 x 13 x 2 x 9 x 15 + y 3 y 11 x 6 x 8 x 2 x 9 x 14 + y 3 y 15 x 12 x 15 x 3 x 10 x 16 + y 4 y 11 x 8 x 11 x 1 x 10 x 14 + y 5 y 15 x 9 x 13 x 1 x 6 x 16 + y 5 y 15 x 10 x 12 x 1 x 7 x 11 + y 6 y 13 x 8 x 14 x 4 x 11 x 15 + y 6 y 14 x 6 x 8 x 3 x 11 x 16 + y 6 y 15 x 6 x 10 x 2 x 7 x 11 + y 6 y 15 x 9 x 14 x 3 x 11 x 16 + y 2 y 11 x 6 x 10 x 16 x 2 x 7 x 9 x 14 + y 2 y 13 x 12 x 14 x 16 x 4 x 6 x 9 x 15 + y 2 y 14 x 7 x 11 x 13 x 2 x 6 x 9 x 15 + y 2 y 14 x 7 x 12 x 14 x 3 x 6 x 9 x 15 + y 3 y 11 x 7 x 8 x 11 x 2 x 9 x 10 x 14 + y 3 y 12 x 3 x 8 x 13 x 2 x 4 x 9 x 14 + y 3 y 12 x 7 x 8 x 12 x 4 x 6 x 9 x 10 + y 3 y 13 x 3 x 8 x 13 x 2 x 4 x 9 x 15 + y 3 y 14 x 7 x 8 x 12 x 3 x 9 x 10 x 16 + y 3 y 14 x 7 x 8 x 13 x 2 x 9 x 15 x 16 + y 3 y 15 x 7 x 11 x 13 x 2 x 6 x 10 x 16 + y 3 y 15 x 7 x 12 x 14 x 3 x 6 x 10 x 16 + y 4 y 12 x 2 x 8 x 12 x 1 x 4 x 6 x 10 + y 4 y 14 x 2 x 8 x 12 x 1 x 3 x 10 x 16 + y 4 y 15 x 9 x 11 x 13 x 1 x 6 x 10 x 16 + y 5 y 12 x 2 x 8 x 12 x 1 x 4 x 6 x 11 + y 5 y 12 x 3 x 8 x 13 x 1 x 4 x 6 x 14 + y 5 y 13 x 3 x 8 x 13 x 1 x 4 x 6 x 15 + y 5 y 14 x 2 x 8 x 12 x 1 x 3 x 11 x 16 + y 5 y 14 x 7 x 8 x 13 x 1 x 6 x 15 x 16 + y 6 y 14 x 7 x 8 x 14 x 3 x 11 x 15 x 16 + y 6 y 15 x 6 x 9 x 15 x 3 x 7 x 11 x 16 + y 2 y 12 x 3 x 10 x 13 x 16 x 2 x 4 x 7 x 9 x 14 + y 2 y 12 x 3 x 11 x 13 x 16 x 2 x 4 x 6 x 9 x 14 + y 2 y 13 x 3 x 10 x 13 x 16 x 2 x 4 x 7 x 9 x 15 + y 2 y 13 x 3 x 11 x 13 x 16 x 2 x 4 x 6 x 9 x 15 + y 3 y 13 x 7 x 8 x 12 x 14 x 4 x 6 x 9 x 10 x 15 + y 4 y 12 x 3 x 8 x 11 x 13 x 1 x 4 x 6 x 10 x 14 + y 4 y 13 x 2 x 8 x 12 x 14 x 1 x 4 x 6 x 10 x 15 + y 4 y 13 x 3 x 8 x 11 x 13 x 1 x 4 x 6 x 10 x 15 + y 4 y 14 x 7 x 8 x 11 x 13 x 1 x 6 x 10 x 15 x 16 + y 4 y 15 x 2 x 9 x 12 x 14 x 1 x 3 x 6 x 10 x 16 + y 4 y 15 x 2 x 9 x 12 x 15 x 1 x 3 x 7 x 10 x 16 + y 5 y 13 x 2 x 8 x 12 x 14 x 1 x 4 x 6 x 11 x 15 + y 5 y 15 x 2 x 9 x 12 x 14 x 1 x 3 x 6 x 11 x 16 + y 5 y 15 x 2 x 9 x 12 x 15 x 1 x 3 x 7 x 11 x 16 + y 3 y 14 x 2 7 x 8 x 11 x 13 x 2 x 6 x 9 x 10 x 15 x 16 + y 3 y 14 x 2 7 x 8 x 12 x 14 x 3 x 6 x 9 x 10 x 15 x 16 + y 3 y 12 x 3 x 7 x 8 x 11 x 13 x 2 x 4 x 6 x 9 x 10 x 14 + y 3 y 13 x 3 x 7 x 8 x 11 x 13 x 2 x 4 x 6 x 9 x 10 x 15 + y 4 y 14 x 2 x 7 x 8 x 12 x 14 x 1 x 3 x 6 x 10 x 15 x 16 + y 5 y 14 x 2 x 7 x 8 x 12 x 14 x 1 x 3 x 6 x 11 x 15 x 16 . The explicit expressions for these v ariables are computed similarly to the n = 1 case, confirming the general pattern. As in the t yp e A 1 case, i.e. cluster algebra on surfaces, the flip operation is an in volution, since the reverse m utation sequence of a flip pro duces the original configuration. Moreov er, flips from disjoin t quadrilaterals comm ute. 18 The num ber of monomials (counting multiplicities) in the Laurent p olynomial for each co ordinate of Υ( V 1 V 2 V 3 V 4 ) equals: K n := Υ( V 1 V 2 V 3 V 4 ) | x i = y j =1 (21) where n stands for the rank. The follo wing prop osition follows directly from Prop osition 3.1 : Prop osition 3.3. F or the gener al cluster r e alization of A SL n +1 , S on a squar e, the value of the exp ansion formula at unity is: K n = (2 a n, 1 , 2 a n, 2 , . . . , 2 a n,n ) (22) wher e the p ositive inte gers a i,j ( i ≥ j ≥ 1 ) satisfy the r e curr enc e r elations: a n, 1 = a n,n = n ; a n +2 ,k = 1 + a n +1 ,k + a n +1 ,k − 1 − a n,k − 1 for n ≥ 1 , k ≥ 2 . (23) Pr o of. F rom Prop osition 3.1 (a), w e hav e K 1 = (2) and K 2 = (4 , 4) = (2 2 , 2 2 ). W e shall prov e the result b y induction on n . F or n = 1, then K 1 = 2 a 1 , 1 with a 1 , 1 = 1. F or n = 2, then K 2 = (2 a 2 , 1 , 2 a 2 , 2 ) with a 2 , 1 = a 2 , 2 = 2, the claim holds. Supp ose that the claim holds for n − 1 ≥ 2, then consider K n = ( b 1 , b 2 , ..., b n ) in the general cluster realization of A S L n +1 , S . (1) If n = 2 k is ev en , consider Proposition 3.1 (b). Applying the inductive hypothesis, at all v ariables x i = y j = 1, we get: X 1 ,l = X 2 ,l = 2 a n − 1 ,l for l = 1 , 2 , ..., ( n − 1); a n − 1 , 1 = a n − 1 ,n − 1 = n − 1; x (1+ t ) n +2+ t = 2 a n − 2 ,t +1 ; x (1+ t ) 2 n − 1 − t = 2 a n − 2 ,n − 2 − t for t = 0 , 1 , ..., ( k − 2) . Hence, we can compute the following: b 1 = x (1) n +1 = X 2 , 1 + X 1 , 1 = 2 · 2 a n − 1 , 1 = 2 · 2 n − 1 = 2 n ; b n = x (1) 2 n = X 2 ,n − 1 + X 1 ,n − 1 = 2 · 2 a n − 1 ,n − 1 = 2 · 2 n − 1 = 2 n ; b 2+ t = X 1 , 1+ t X 2 , 2+ t + X 1 , 2+ t X 2 , 1+ t x (1+ t ) n +2+ t = 2 · 2 a n − 1 ,t +1 + a n − 1 ,t +2 − a n − 2 ,t +1 = 2 1+ a n − 1 ,t +1 + a n − 1 ,t +2 − a n − 2 ,t +1 ; b n − 1 − t = X 1 ,n − 2 − t X 2 ,n − 1 − t + X 1 ,n − 1 − t X 2 ,n − 2 − t x (1+ t ) 2 n − 1 − t = 2 · 2 a n − 1 ,n − 2 − t + a n − 1 ,n − 1 − t − a n − 2 ,n − 2 − t = 2 1+ a n − 1 ,n − 2 − t + a n − 1 ,n − 1 − t − a n − 2 ,n − 2 − t for all t = 0 , 1 , ..., ( k − 2) whic h implies eac h b l = 2 a n,l , where a n, 1 = a n,n = n and a n,r = 1 + a n − 1 ,r + a n − 1 ,r − 1 − a n − 2 ,r − 1 for all r = 2 , 3 , ..., ( n − 1). This satisfies the claim. (2) If n = 2 k + 1 is o dd , consider Prop osition 3.1 (c). Applying the inductive h yp othesis, at all v ariables x i = y j = 1, we get: X 1 ,l = X 2 ,l = 2 a n − 1 ,l for l = 1 , 2 , ..., ( n − 1); a n − 1 , 1 = a n − 1 ,n − 1 = n − 1; x ( k ) k +1 = 2 a n − 2 ,k ; x (1+ t ) n +2+ t = 2 a n − 2 ,t +1 ; x (1+ t ) 2 n − 2 − t = 2 a n − 2 ,n − 2 − t for t = 0 , 1 , ..., ( k − 2) . 19 Hence, we can compute the following: b 1 = x (1) n +1 = X 2 , 1 + X 1 , 1 = 2 · 2 a n − 1 , 1 = 2 · 2 n − 1 = 2 n ; b n = x (1) 2 n − 1 = X 2 ,n − 1 + X 1 ,n − 1 = 2 · 2 a n − 1 ,n − 1 = 2 · 2 n − 1 = 2 n ; b k +1 = X 1 ,k X 2 ,k +1 + X 1 ,k +1 X 2 ,k x ( k ) k +1 = 2 · 2 a n − 1 ,k +1 + a n − 1 ,k − a n − 2 ,k = 2 1+ a n − 1 ,k +1 + a n − 1 ,k − a n − 2 ,k ; b 2+ t = X 1 , 1+ t X 2 , 2+ t + X 1 , 2+ t X 2 , 1+ t x (1+ t ) n +2+ t = 2 · 2 a n − 1 ,t +1 + a n − 1 ,t +2 − a n − 2 ,t +1 = 2 1+ a n − 1 ,t +1 + a n − 1 ,t +2 − a n − 2 ,t +1 ; b n − 1 − t = X 1 ,n − 2 − t X 2 ,n − 1 − t + X 1 ,n − 1 − t X 2 ,n − 2 − t x (1+ t ) 2 n − 2 − t = 2 · 2 a n − 1 ,n − 2 − t + a n − 1 ,n − 1 − t − a n − 2 ,n − 2 − t = 2 1+ a n − 1 ,n − 2 − t + a n − 1 ,n − 1 − t − a n − 2 ,n − 2 − t for all t = 0 , 1 , ..., ( k − 2) whic h implies eac h b l = 2 a n,l , where a n, 1 = a n,n = n and a n,r = 1 + a n − 1 ,r + a n − 1 ,r − 1 − a n − 2 ,r − 1 for all r = 2 , 3 , ..., ( n − 1). This satisfies the claim. Therefore, the inductive step w orks for all p ositiv e integer n . This concludes the pro of. It turns out the recurrence relations can b e solv ed explicitly , and as a consequence we obtain: Theorem 3.4. The exp onents satisfy a i,j = j ( i − j + 1) for al l i ≥ j ≥ 1 . Ther efor e: K n = (2 n , 2 2 n − 2 , . . . , 2 t ( n − t +1) , . . . , 2 n ) . (24) Pr o of. Firstly , we shall prov e that a i,j = j ( i − j + 1) for all i ≥ j ≥ 1 b y induction on j . Indeed, for j = 1, then a i, 1 = i = 1 · ( i − 1 + 1), satisfied. Supp ose that we already hav e a i,j = j ( i − j + 1) for all i ≥ j for j ≥ 1, then for l ≥ j + 2, w e hav e: a j +1 ,j +1 = j + 1 = ( j + 1) · 1; a l,j +1 = 1 + a l − 1 ,j +1 + a l − 1 ,j − a l − 2 ,j = 1 + a l − 1 ,j +1 + j ( l − j ) − j ( l − j − 1) = 1 + j + a l − 1 ,j +1 . Hence b y induction on l ≥ j + 1, we conclude that a l,j +1 = ( j + 1)( l − j ), and hence the claim also holds for all j , whence a i,j = j ( i − j + 1) for all i ≥ j ≥ 1. As a consequence, as a n,t = t ( n − t + 1), we conclude K n = (2 n , 2 2 n − 2 , . . . , 2 t ( n − t +1) , . . . , 2 n ). The following Prop osition illustrates the relations in case w e set sev eral v ariables to b e equal. Prop osition 3.5. Given the cluster r e alization of A SL n +1 , S for the quadrilater al V 1 V 2 V 3 V 4 with diagonal V 1 V 3 and the c orr esp onding n 2 + 4 n variables x i , y j define d as ab ove, we shal l let p ar ameters x, y , z and then set al l x i = x and y j = y for j = 1 , 2 , ..., 2 n while y j = z for j = 2 n + 1 , 2 n + 2 , ..., 4 n . Then we get the exp ansion formula for diagonal V 2 V 4 of the form ( F 1 , F 2 , ..., F n ) with: F k = 2 k ( n +1 − k ) y z x (25) for k = 1 , . . . , n . As a c onse quenc e, in c ase we set al l variables to x i = x and y j = y (or y = z ), we have: F k = 2 k ( n +1 − k ) y 2 x (26) for k = 1 , . . . , n . Pr o of. W e pro ve the claim b y induction on n . F or n = 1, then F 1 = y 1 y 3 + y 2 y 4 x 1 = 2 y z x . F or n = 2, we ha ve: F 1 = x ′ 3 = y 2 y 5 x 1 + y 4 y 7 x 2 + y 3 y 5 x 4 x 1 x 3 + y 4 y 6 x 4 x 2 x 3 = 4 y z x ; F 2 = x ′ 4 = y 3 y 8 x 1 + y 1 y 6 x 2 + y 2 y 8 x 3 x 1 x 4 + y 1 y 7 x 3 x 2 x 4 = 4 y z x . Supp ose that the claim holds for n − 1 ≥ 2, we shall consider the different cases. 20 (1) If n = 2 k is ev en , consider Proposition 3.1 (b). Applying the inductive hypothesis, at all v ariables x i = y j = 1, we get: X 1 ,l = 2 l ( n − l ) xy x = 2 l ( n − l ) · y ; X 2 ,l = 2 l ( n − l ) xz x = 2 l ( n − l ) · z for l = 1 , 2 , ..., ( n − 1); x (1+ t ) n +2+ t = 2 ( t +1)( n − 2 − t ) x 2 x = 2 ( t +1)( n − 2 − t ) · x ; x (1+ t ) 2 n − 1 − t = 2 ( n − 2 − t )( t +1) x 2 x = 2 ( n − 2 − t )( t +1) · x for t = 0 , 1 , ..., ( k − 2) . Hence, we can compute the following: F 1 = x (1) n +1 = y 2 n X 2 , 1 + y 2 n +1 X 1 , 1 x n +1 = 2 · 2 n − 1 y z x = 2 n y z x ; F n = x (1) 2 n = y 1 X 2 ,n − 1 + y 4 n X 1 ,n − 1 x 2 n = 2 · 2 n − 1 y z x = 2 n y z x ; F 2+ t = X 1 , 1+ t X 2 , 2+ t + X 1 , 2+ t X 2 , 1+ t x (1+ t ) n +2+ t = 2 · 2 ( t +1)( n − 1 − t )+( t +2)( n − 2 − t ) − ( t +1)( n − 2 − t ) y z x = 2 ( t +2)( n − 1 − t ) y z x ; F n − 1 − t = X 1 ,n − 2 − t X 2 ,n − 1 − t + X 1 ,n − 1 − t X 2 ,n − 2 − t x (1+ t ) 2 n − 1 − t = 2 · 2 ( t +1)( n − 1 − t )+( t +2)( n − 2 − t ) − ( t +1)( n − 2 − t ) y z x = 2 ( t +2)( n − 1 − t ) y z x for all t = 0 , 1 , ..., ( k − 2), whic h satisfies the claim. (2) If n = 2 k + 1 is o dd , consider Prop osition 3.1 (c). Applying the inductive h yp othesis, at all v ariables x i = y j = 1, we get: X 1 ,l = 2 l ( n − l ) xy x = 2 l ( n − l ) · y ; X 2 ,l = 2 l ( n − l ) xz x = 2 l ( n − l ) · z for l = 1 , 2 , ..., ( n − 1); x ( k ) k +1 = 2 k 2 x 2 x = 2 k 2 · x ; x (1+ t ) n +2+ t = 2 ( t +1)( n − 2 − t ) x 2 x = 2 ( t +1)( n − 2 − t ) · x ; x (1+ t ) 2 n − 2 − t = 2 ( n − 2 − t )( t +1) x 2 x = 2 ( n − 2 − t )( t +1) · x for t = 0 , 1 , ..., ( k − 2) . Hence, we can compute the following: F 1 = x (1) n +1 = y 2 n X 2 , 1 + y 2 n +1 X 1 , 1 x n +1 = 2 · 2 n − 1 y z x = 2 n y z x ; F n = x (1) 2 n − 1 = y 1 X 2 ,n − 1 + y 4 n X 1 ,n − 1 x 2 n − 1 = 2 · 2 n − 1 y z x = 2 n y z x ; F k +1 = X 1 ,k X 2 ,k +1 + X 1 ,k +1 X 2 ,k x ( k ) k +1 = 2 · 2 k ( k +1)+ k ( k +1) − k 2 y z x = 2 ( k +1) 2 y z x ; F 2+ t = X 1 , 1+ t X 2 , 2+ t + X 1 , 2+ t X 2 , 1+ t x (1+ t ) n +2+ t = 2 · 2 ( t +1)( n − 1 − t )+( t +2)( n − 2 − t ) − ( t +1)( n − 2 − t ) y z x = 2 ( t +2)( n − 1 − t ) y z x ; F n − 1 − t = X 1 ,n − 2 − t X 2 ,n − 1 − t + X 1 ,n − 1 − t X 2 ,n − 2 − t x (1+ t ) 2 n − 2 − t = 2 · 2 ( t +1)( n − 1 − t )+( t +2)( n − 2 − t ) − ( t +1)( n − 2 − t ) y z x = 2 ( t +2)( n − 1 − t ) y z x for all t = 0 , 1 , ..., ( k − 2), whic h satisfies the claim. Therefore, in b oth cases, the claim holds for n . Then the consequence trivially holds by letting y = z . This concludes the pro of. 21 3.2 Com binatorial interpretation via stair paths While the labels x i and y j pro vide a basis for analysis, they are inadequate for deriving a closed-form recursion for Υ( AB C D ) for general n , due to inconsistencies in their indexing under v ariation of n . T o formulate a univ ersal solution, we make a transition to an infinite in teger grid. This setting permits a generalized indexing scheme, obtained by a family of Lauren t p olynomial maps ψ t,k : { x i,j } → Z [ x ± 1 i,j ]. Afterw ards, w e shall introduce the combinatorial constructions n -stair p ath and n -r everse d-stair p ath to help solve in the particular v alues, namely ψ 1 ,k and ψ k,k , then give further connections among these combinatorial ob jects. As a result, note that the new notations in this section are indep enden t of the lab els x i and y j from the previous sections. Consider an infinite integer grid with v ariable x i,j assigned to each point ( i, j ) ∈ Z 2 . F or each t, k ∈ Z ≥ 0 with 0 ≤ t ≤ k + 1, define maps ψ t,k : { x i,j } → Z [ x ± 1 i,j ] by: ψ 0 ,k ( x i,j ) = ψ k +1 ,k ( x i,j ) := x i,j ; ψ 1 , 1 ( x i,j ) := x i − 1 ,j x i +1 ,j + x i,j − 1 x i,j +1 x i,j ; ψ t +1 ,k +2 ( x i,j ) := ψ t +1 ,k +1 ( x i − 1 ,j ) ψ t,k +1 ( x i +1 ,j ) + ψ t,k +1 ( x i,j − 1 ) ψ t +1 ,k +1 ( x i,j +1 ) ψ t,k ( x i,j ) . (27) Using Prop osition 3.1 , ψ t,k ( x i,j ) equals the t -th co ordinate of the index of V 2 V 4 diagonal of the expansion after flipping the square A t,k i,j B t,k i,j C t,k i,j D t,k i,j , where we place the v ertices on the infinite grid as A t,k i,j = ( i + t, j − t + k + 1) , B t,k i,j = ( i + t, j − t ) , C t,k i,j = ( i + t − k − 1 , j − t ) , D t,k i,j = ( i + t − k − 1 , j − t + k + 1) . In particular, b y the Laurent phenomenon, each ψ t,k ( x i,j ) is a Lauren t p olynomial in the v ariables x i,j , hence the maps are well-defined. When all v ariables x i,j = x , applying Prop osition 3.5 yields: ψ t,k ( x ) = 2 t ( k +1 − t ) x. (28) Definition 3.6. Given an infinite in teger grid with v ariables x i,j assigned at each p oin t ( i, j ), an n -stair p ath of ( i, j ) is an alternating sequence of left and up segments with total length n satisfying: (1) The first segmen t contains ( i, j ) as its second point; (2) Consecutiv e segmen ts connect via √ 2-length diagonals: (a) The head of a left segmen t is √ 2 northeast of the tail of the next up segment; (b) The head of an up segmen t is √ 2 southw est of the tail of next left segment; (3) The ending p oint E is one step a wa y from the head of the last segment in the sequence. Equiv alently , E is the p oin t reached by starting at ( i, j ) and following the alternating left/up path according to the connection rules (2), until the sum of segmen t lengths equals n . F or any integer p oin t ( k , l )  = ( i, j ) suc h that k ≤ i and l ≥ j , by a stair p ath fr om ( i, j ) to ( k , l ), we mean an s -stair path of ( i, j ) (where s = ( i − k ) + ( l − j )) such that E = ( k , l ). See Figure 12 for examples. In both examples, the ending p oin t E = ( i − 7 , j + 7) is mark ed in green and eac h example is a stair path from ( i, j ) to E . The left example has segment lengths 3, 4, 4, 3 (sum 14); the righ t has lengths 3, 3, 2, 1, 2, 3 (sum 14). 22 ( i, j ) ( i, j ) E E Figure 12: Examples of 14-stair paths starting from ( i, j ) F or t wo in teger p oin ts a, b with the same horizontal or v ertical coordinate with the corresponding segmen t arro w a → b , regardless of the direction (either left ( ← ), right ( → ), up ( ↑ ), or down ( ↓ )), assign a weight x a x b of Lauren t monomial to that segment. It is clear that for consecutive p oin ts sharing the same direction, the pro duct of weigh ts equals the weigh t of the single segment connecting all p oin ts. Let f n ( x i,j ) := ψ 1 ,n ( x i,j ) for all n ≥ 0, then from the recurrence ( 27 ), we ha ve: f 0 ( x i,j ) = x i,j ; f n +1 ( x i,j ) = f n ( x i − 1 ,j ) x i +1 ,j + x i,j − 1 f n ( x i,j +1 ) x i,j . (29) Define c i,j := x i +1 ,j x i,j and r i,j := x i,j − 1 x i,j , then the recurrence ( 27 ) implies: f 0 ( x i,j ) = x i,j ; f n +1 ( x i,j ) = c i,j f n ( x i − 1 ,j ) + r i,j f n ( x i,j +1 ) . (30) This leads to a combinatorial in terpretation: Prop osition 3.7. The value f n ( x i,j ) = ψ 1 ,n ( x i,j ) e quals the sum of weights of al l p ossible n -stairs p aths starting fr om ( i, j ) , wher e the weight of e ach p ath is the pr o duct of its se gment weights multiplie d by the variable at its ending p oint E . Mor e explicitly, the gener al formula c an b e written as: f n ( x i,j ) = ψ 1 ,n ( x i,j ) = X ( l 1 +1) ,l i ,u i , ( u k +1) ,k ∈ N : P k i =1 ( l i + u i )= n x i − P k s =1 l s ,j + P k t =1 u t k Y v =1 x i +1 − P v − 1 s =1 l s ,j + P v − 1 t =1 u t x i +1 − P v s =1 l s ,j + P v − 1 t =1 u t x i − P v s =1 l s ,j − 1+ P v − 1 t =1 u t x i − P v s =1 l s ,j − 1+ P v t =1 u t ! . (31) Her e and ther e after, the notation for the summation index ( l 1 + 1) ∈ N etc. is e quivalent to adding up al l nonne gative index l 1 ∈ N ∪ { 0 } . Pr o of. W e shall pro ve b y induction on n that: f n ( x i,j ) = X τ 1 ,τ 2 ,...,τ n ∈{ r,c } ( τ 1 ) i,j ( τ 2 ) σ 1 ( i,j ) · · · ( τ n ) ( σ n − 1 ◦···◦ σ 1 )( i,j ) x ( σ n ◦···◦ σ 1 )( i,j ) (32) where σ k ( i ′ , j ′ ) = ( i ′ − 1 , j ′ ) if τ k = c , and σ k ( i ′ , j ′ ) = ( i ′ , j ′ + 1) if τ k = r for 1 ≤ k ≤ n . F or n = 1, then f 1 ( x i,j ) = r i,j x i,j +1 + c i,j x i − 1 ,j , satisfied. 23 Supp ose that the formula holds for n ≥ 1, then consider the ( n + 1) case, from recurrence ( 30 ), we can write: f n +1 ( x i,j ) = c i,j f n ( x i − 1 ,j ) + r i,j f n ( x i,j +1 ) = c i,j . X τ 1 ,τ 2 ,...,τ n ∈{ r,c } ( τ 1 ) i − 1 ,j ( τ 2 ) σ 1 ( i − 1 ,j ) · · · ( τ n ) ( σ n − 1 ◦···◦ σ 1 )( i − 1 ,j ) x ( σ n ◦···◦ σ 1 )( i − 1 ,j ) + r i,j X τ 1 ,τ 2 ,...,τ n ∈{ r,c } ( τ 1 ) i,j +1 ( τ 2 ) σ 1 ( i,j +1) · · · ( τ n ) ( σ n − 1 ◦···◦ σ 1 )( i,j +1) x ( σ n ◦···◦ σ 1 )( i,j +1) = c i,j . X τ 2 ,τ 3 ,...,τ n +1 ∈{ r,c } ( τ 2 ) i − 1 ,j ( τ 3 ) σ 2 ( i − 1 ,j ) · · · ( τ n +1 ) ( σ n ◦···◦ σ 2 )( i − 1 ,j ) x ( σ n +1 ◦···◦ σ 2 )( i − 1 ,j ) + r i,j X τ 2 ,τ 3 ,...,τ n +1 ∈{ r,c } ( τ 2 ) i,j +1 ( τ 3 ) σ 2 ( i,j +1) · · · ( τ n +1 ) ( σ n ◦···◦ σ 2 )( i,j +1) x ( σ n +1 ◦···◦ σ 2 )( i,j +1) = X τ 1 = c ; τ 2 ,...,τ n +1 ∈{ r,c } ( τ 1 ) i,j ( τ 2 ) σ 1 ( i,j ) · · · ( τ n +1 ) ( σ n ◦···◦ σ 1 )( i,j ) x ( σ n +1 ◦···◦ σ 1 )( i,j ) + X τ 1 = r ; τ 2 ,...,τ n +1 ∈{ r,c } ( τ 1 ) i,j ( τ 2 ) σ 1 ( i,j ) · · · ( τ n +1 ) ( σ n ◦···◦ σ 1 )( i,j ) x ( σ n +1 ◦···◦ σ 1 )( i,j ) = X τ 1 ,τ 2 ,...,τ n +1 ∈{ r,c } ( τ 1 ) i,j ( τ 2 ) σ 1 ( i,j ) · · · ( τ n +1 ) ( σ n ◦···◦ σ 1 )( i,j ) x ( σ n +1 ◦···◦ σ 1 )( i,j ) whic h completes the inductiv e pro of. Eac h c i ′ ,j ′ corresp onds to the w eight of the left unit segmen t ( i ′ + 1 , j ′ ) → ( i ′ , j ′ ), while r i ′ ,j ′ corresp onds to the weigh t of the up unit segment ( i ′ , j ′ − 1) → ( i ′ , j ′ ), then for eac h choice of ( τ 1 , τ 2 , ..., τ n ), the term of the summation ab o ve represents an n -stairs path with the v ariable x ( σ n ◦···◦ σ 1 )( i,j ) represen ting the lab el of the ending v ertex E . This establishes the combinatorial in terpretation. The other form ula in the statement is the parametrization for equation ( 32 ), with eac h v alue ( k , l i , u i ) corresp onding to an n -stair path. F or small v alues we compute: f 1 ( x i,j ) = ψ 1 , 1 ( x i,j ) = x i − 1 ,j x i +1 ,j + x i,j − 1 x i,j +1 x i,j , f 2 ( x i,j ) = ψ 1 , 2 ( x i,j ) = x i − 2 ,j x i +1 ,j x i − 1 ,j + x i,j − 1 x i,j +2 x i,j +1 + x i − 1 ,j +1 x i,j x i +1 ,j +1 x i,j x i,j +1 + x i − 1 ,j − 1 x i − 1 ,j +1 x i +1 ,j x i − 1 ,j x i,j . Figure 13 illustrates the calculation of f 3 ( x i,j ) using Prop osition 3.7 , yielding: ψ 1 , 3 ( x i,j ) = x i,j − 1 x i,j +3 x i,j +2 + x i,j − 1 x i +1 ,j +2 x i − 1 ,j +2 x i,j +1 x i,j +2 + x i,j − 1 x i +1 ,j +1 x i − 1 ,j x i − 1 ,j +2 x i,j x i,j +1 x i − 1 ,j +1 + x i,j − 1 x i +1 ,j +1 x i − 2 ,j +1 x i,j x i − 1 ,j +1 + x i +1 ,j x i − 3 ,j x i − 2 ,j + x i +1 ,j x i − 2 ,j − 1 x i − 2 ,j +1 x i − 1 ,j x i − 2 ,j + x i +1 ,j x i − 1 ,j − 1 x i,j +1 x i − 2 ,j +1 x i,j x i − 1 ,j x i − 1 ,j +1 + x i +1 ,j x i − 1 ,j − 1 x i − 1 ,j +2 x i,j x i − 1 ,j +1 . 24 ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E Figure 13: All 2 3 = 8 p ossible 3-stair paths starting from ( i, j ) Similarly , we can also define the rev ersed version: Definition 3.8. An n -r everse d-stair p ath is defined similarly by reflection: an alternating sequence of righ t and down segments with total length n , where: (1) The first segmen t contains ( i, j ) as its second point; (2) Consecutiv e segmen ts connect via √ 2-length diagonals: (a) The head of a righ t segment ends √ 2 southw est of the tail of the next down segmen t; (b) The head of a do wn segment ends √ 2 northeast of the tail of the next righ t segment; (3) The ending p oint E is one step a wa y from the head of the last segment in the sequence. Equiv alently , E is the point reached by starting at ( i, j ) and follo wing the alternating right/do wn path according to the connection rules (2), until the sum of segment lengths equals n . F or an y integer p oint ( k , l )  = ( i, j ) suc h that k ≥ i and l ≤ j , b y a r everse d-stair p ath fr om ( i, j ) to ( k , l ), we mean a r -rev ersed-stair path of ( i, j ) (where r = ( k − i ) + ( j − l )) suc h that E = ( k , l ). The notion allows us to compute the follo wing case: Prop osition 3.9. ψ n,n ( x i,j ) e quals the sum of weights of al l p ossible n -r everse d-stair p aths starting fr om ( i, j ) , wher e the weight of e ach p ath is the pr o duct of its se gment weights multiplie d by the variable at its ending p oint E . Mor e explicitly, the gener al formula c an b e written as: ψ n,n ( x i,j ) = X ( r 1 +1) ,r i ,d i , ( d k +1) ,k ∈ N : P k i =1 ( r i + d i )= n x i + P k s =1 r s ,j − P k t =1 d t k Y v =1 x i − 1+ P v − 1 s =1 r s ,j − P v − 1 t =1 d t x i − 1+ P v s =1 r s ,j − P v − 1 t =1 d t x i + P v s =1 r s ,j +1 − P v − 1 t =1 d t x i + P v s =1 r s ,j +1 − P v t =1 d t ! . (33) Pr o of. Setting t = k + 1, w e obtain the recurrence: ψ k +2 ,k +2 ( x i,j ) = x i − 1 ,j ψ k +1 ,k +1 ( x i +1 ,j ) + ψ k +1 ,k +1 ( x i,j − 1 ) x i,j +1 x i,j , (34) whic h admits a similar combinatorial in terpretation as Prop osition 3.7 . 25 Figure 14 illustrates the calculation of ψ 3 , 3 ( x i,j ): ψ 3 , 3 ( x i,j ) = x i,j +1 x i,j − 3 x i,j − 2 + x i,j +1 x i − 1 ,j − 2 x i +1 ,j − 2 x i,j − 1 x i,j − 2 + x i,j +1 x i − 1 ,j − 1 x i +1 ,j x i +1 ,j − 2 x i,j x i,j − 1 x i +1 ,j − 1 + x i,j +1 x i − 1 ,j − 1 x i +2 ,j − 1 x i,j x i +1 ,j − 1 + x i − 1 ,j x i +3 ,j x i +2 ,j + x i − 1 ,j x i +2 ,j +1 x i +2 ,j − 1 x i +1 ,j x i +2 ,j + x i − 1 ,j x i +1 ,j +1 x i,j − 1 x i +2 ,j − 1 x i,j x i +1 ,j x i +1 ,j − 1 + x i − 1 ,j x i +1 ,j +1 x i +1 ,j − 2 x i,j x i +1 ,j − 1 . ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E ( i, j ) E Figure 14: All 2 3 = 8 p ossible 3-rev ersed-stair paths starting from ( i, j ). F ollowing the idea of the pro of of Proposition 3.7 , we can write the formula in the stair path structure: Prop osition 3.10. The gener al r e curr enc e: ψ t +1 ,k +2 ( x i,j ) = ψ t +1 ,k +1 ( x i − 1 ,j ) ψ t,k +1 ( x i +1 ,j ) + ψ t,k +1 ( x i,j − 1 ) ψ t +1 ,k +1 ( x i,j +1 ) ψ t,k ( x i,j ) (35) le ads to the formula: ψ t,k ( x i,j ) = X ( l 1 +1) ,l i ,u i , ( u g +1) ,g ∈ N : P g i =1 ( l i + u i )= k − t +1 x i − P g s =1 l s ,j + P g h =1 u h g Y v =1 ψ t − 1 ,k − 1 − P v − 1 s =1 l s − P v − 1 h =1 u h ( x i +1 − P v − 1 s =1 l s ,j + P v − 1 h =1 u h ) ψ t − 1 ,k − 1 − P v s =1 l s − P v − 1 h =1 u h ( x i +1 − P v s =1 l s ,j + P v − 1 h =1 u h ) · ψ t − 1 ,k − 1 − P v s =1 l s − P v − 1 h =1 u h ( x i − P v s =1 l s ,j − 1+ P v − 1 h =1 u h ) ψ t − 1 ,k − 1 − P v s =1 l s − P v h =1 u h ( x i − P v s =1 l s ,j − 1+ P v h =1 u h ) ! . (36) Define tw o com binatorial functions: S ( x i,j , x k,l ) := sum of weigh ts of all stair paths from ( i, j ) to ( k , l ) , R ( x i,j , x k,l ) := sum of weigh ts of all reversed-stair paths from ( i, j ) to ( k , l ) . (37) W e can add the notations S ( x i,j , x i,j ) := x i,j and S ( x i,j , x k,l ) := 0 if k > i or l < j , and R ( x i,j , x i,j ) := x i,j and R ( x i,j , x k,l ) := 0 if k < i or l > j . The num b er of stair paths from ( i, j ) to ( k , l ) is  ( i − k )+( l − j ) i − k  when k ≤ i and l ≥ j , and the n umber of reversed-stair paths is  ( k − i )+( j − l ) k − i  when k ≥ i and l ≤ j . Similar to the w ay of parametrizing in the formulas of Proposition 3.7 and 3.9 , we can rewrite b oth functions as: Prop osition 3.11. The two c ombinatorial functions define d ab ove ar e of the form: 26 (1) F or any inte ger p oints ( i, j ) , ( i ′ , j ′ ) such that i ′ ≤ i and j ′ ≥ j , we have: S ( x i,j , x i ′ ,j ′ ) = x i ′ ,j ′ · X ( l 1 +1) ,l i ,u i , ( u k +1) ,k ∈ N : P k s =1 l s = i − i ′ ; P k t =1 u t = j ′ − j k Y v =1 x i +1 − P v − 1 s =1 l s ,j + P v − 1 t =1 u t x i +1 − P v s =1 l s ,j + P v − 1 t =1 u t x i − P v s =1 l s ,j − 1+ P v − 1 t =1 u t x i − P v s =1 l s ,j − 1+ P v t =1 u t . (38) (2) F or any inte ger p oints ( i, j ) , ( i ′ , j ′ ) such that i ′ ≥ i and j ′ ≤ j , we have: R ( x i,j , x i ′ ,j ′ ) = x i ′ ,j ′ · X ( r 1 +1) ,r i ,d i , ( d k +1) ,k ∈ N : P k s =1 r s = i ′ − i ; P k t =1 d t = j − j ′ k Y v =1 x i − 1+ P v − 1 s =1 r s ,j − P v − 1 t =1 d t x i − 1+ P v s =1 r s ,j − P v − 1 t =1 d t x i + P v s =1 r s ,j +1 − P v − 1 t =1 d t x i + P v s =1 r s ,j +1 − P v t =1 d t . (39) Prop osition 3.12. F or al l non-ne gative inte gers k , l and i, j ∈ Z : ψ 1 ,k + l ( x i,j ) = l X t =0 S ( x i,j , x i − t,j + l − t ) ψ 1 ,k ( x i − t,j + l − t ) , ψ k + l,k + l ( x i,j ) = l X t =0 R ( x i,j , x i + t,j − l + t ) ψ k,k ( x i + t,j − l + t ) . (40) Pr o of. F rom the definition ab ov e on the function S and Prop osition 3.7 , we hav e S ( x i,j , x i − t,j + l − t ) is the sum of weigh ts of all stair paths from ( i, j ) to ( i − t, j + l − t ) and ψ 1 ,k ( x i − t,j + l − t ) is the sum of w eights of all k -stair paths of ( i − t, j + l − t ), then as the pro duct of a weigh t of a stair path from ( i, j ) to ( i − t, j + l − t ) and a w eight of an k -stair path of ( i − t, j + l − t ) is precisely a w eight of an ( k + l )-stair path of ( i − t, j + l − t ) that has the p oin t ( i − t, j + l − t ) as the tail of ( l + 1)-th arrow, then taking the sum, w e ha ve P l t =0 S ( x i,j , x i − t,j + l − t ) ψ 1 ,k ( x i − t,j + l − t ) is precisely the sum of w eights of all ( k + l )-stair paths of ( i, j ), whic h is precisely ψ 1 ,k + l ( x i,j ). By similar argument, using the definition abov e on the function R and Prop osition 3.9 implies the remaining formula of the claim. This concludes the pro of. 3.3 Punctured surface case In this section, we giv e some remarks on the punctured surface case, the fundamental difference b eing that self-folded triangles appear during flips. Consider the case of a punctured disk with 2 mark ed p oin ts on the b oundary labeled as follows. P A B Figure 15: A punctured disk with 2 mark ed p oin ts. In an ideal triangulation of such disk, tw o triangles may share 0, 1, or 2 edges. When tw o triangles share 2 edges (indicating punctures), there are tw o possible flips, each pro ducing tw o new triangles with one common edge, one b eing self-folded. The quiv er differs from the unpunctured case: the vertex closest to puncture P on edge AP connects to only 2 vertices instead of 4. Ho wev er, the m utation sequence matches the standard case when treating the 27 quadrilateral as a square with same-colored edges representing identical edges in the original triangulation. F or example, in the case G = SL 4 , we label vertices with v ariables x i , y j ( i = 1 , 2 , . . . , 12; j = 1 , 2 , . . . , 6) as sho wn in Figure 16 . W e can apply the usual m utation sequence for a flip of diagonal, given b y: µ = { x 1 → x 2 → x 3 → x 7 → x 8 → x 10 → x 11 → x 9 → x 12 → x 2 } (41) where the order within groups { x 1 , x 2 , x 3 } , { x 7 , x 8 , x 10 , x 11 } , and { x 9 , x 12 , x 2 } can be p erm uted since the v ertices are disconnected within the same group in the quiver during the mutation procedure. x 1 x 2 x 3 x 4 x 5 x 6 y 1 y 2 y 3 y 6 y 5 y 4 x 9 x 12 x 8 x 11 x 10 x 7 x 1 x 2 x 3 y 3 y 2 y 1 y 4 x 11 x 12 x 6 y 5 x 8 x 10 x 5 y 6 x 9 x 7 x 4 x 6 x 5 x 4 Figure 16: V ertex lab eling for punctured case (left) and its corresponding square lattice (right). The new v ariables are computed to b e: x ′ 1 = y 3 x 8 + y 4 x 11 x 1 ; x ′ 2 = x 7 x 11 + x 8 x 10 x 2 ; x ′ 3 = x 7 + x 10 x 3 ; x ′ 7 = x ′ 3 x 4 x 9 + x ′ 2 x 5 x 7 = x 4 x 9 x 3 + x 4 x 9 x 10 x 3 x 7 + x 5 x 11 x 2 + x 5 x 8 x 10 x 2 x 7 ; x ′ 8 = y 5 x ′ 2 + x 9 x ′ 1 x 8 = x 7 x 11 y 5 x 2 x 8 + x 10 y 5 x 2 + x 9 y 3 x 1 + x 9 x 11 y 4 x 1 x 8 ; x ′ 10 = x ′ 3 x 4 x 12 + x ′ 2 x 5 x 10 = x 4 x 12 x 7 x 3 x 10 + x 4 x 12 x 3 + x 5 x 7 x 11 x 2 x 10 + x 5 x 8 x 2 ; x ′ 11 = y 2 x ′ 2 + x 12 x ′ 1 x 11 = x 7 y 2 x 2 + x 8 x 10 y 2 x 2 x 11 + x 8 x 12 y 3 x 1 x 11 + x 12 y 4 x 1 ; x ′ 9 = x ′ 8 x 6 + x ′ 7 y 6 x 9 = x 4 y 6 x 3 + x 4 x 10 y 6 x 3 x 7 + x 5 x 11 y 6 x 2 x 9 + x 8 x 5 x 10 y 6 x 2 x 7 x 9 + x 7 x 6 x 11 y 5 x 2 x 8 x 9 + x 6 x 10 y 5 x 2 x 9 + x 6 y 3 x 1 + x 6 x 11 y 4 x 1 x 8 ; x ′ 12 = x ′ 11 x 6 + x ′ 10 y 1 x 12 = x 4 x 7 y 1 x 3 x 10 + x 4 y 1 x 3 + x 11 x 5 x 7 y 1 x 2 x 10 x 12 + x 5 x 8 y 1 x 2 x 12 + x 6 x 7 y 2 x 2 x 12 + x 10 x 6 x 8 y 2 x 2 x 11 x 12 + x 6 x 8 y 3 x 1 x 11 + x 6 y 4 x 1 ; x ′′ 2 = x ′ 7 x ′ 11 + x ′ 8 x ′ 10 x ′ 2 = x 7 x 4 x 12 y 5 x 3 x 8 x 10 + x 4 x 9 y 2 x 3 x 11 + x 4 x 12 y 5 x 3 x 8 + x 10 x 4 x 9 y 2 x 3 x 7 x 11 + x 7 x 11 x 5 y 5 x 2 x 8 x 10 + x 5 y 2 x 2 + x 5 y 5 x 2 + x 8 x 10 x 5 y 2 x 2 x 7 x 11 + x 2 x 4 x 9 x 12 y 3 x 3 x 1 x 11 x 10 + x 2 x 4 x 9 x 12 y 4 x 3 x 1 x 8 x 10 + x 2 x 4 x 9 x 12 y 3 x 3 x 7 x 1 x 11 + x 2 x 4 x 9 x 12 y 4 x 3 x 7 x 1 x 8 + x 5 x 9 y 3 x 1 x 10 + x 11 x 5 x 9 y 4 x 1 x 8 x 10 + x 8 x 5 x 12 y 3 x 7 x 1 x 11 + x 5 x 12 y 4 x 7 x 1 . 28 With the exception of x 3 , which follows a mo dified square structure with x 4 replaced by 1, all v ariables adhere to the same pattern as in the unpunctured case. Through analogous computations, we find that all v ariables main tain the unpunctured surface structure up to the flipp ed diagonal vertex nearest the puncture. F urthermore, for the flipp ed diagonal vertex nearest the puncture, the n umber of monomials (up to multi- plicities) of the expansion formula of the v ariable of this vertex remains 2 (in the case of Figure 16 we ha ve x ′ 3 ha ving 2 monomials). Therefore, for the remainder of this pap er, where co efficien t analysis is primary , it suffices to consider only the unpunctured surface case. 3.4 Solving recurrences In this section we attempt to solve (case-by-case) the recurrences ( 27 ) and connect them with famous com binatorial structures ([ 15 ], [ 16 ]) Problem 3.13. Find the general formula of ψ t,k ( x i 0 ,j 0 ) in terms of the v ariables x i,j ? W e present our current progress in deriving a general formula for the expansion structure, b eginning with the ab o ve observ ations (Propositions 3.7 , 3.9 , and ( 36 )). W e shall make connections of the recurrence by defining some auxiliary functions. Prop osition 3.14. Denote the function f : ( Z 2 ≥ 0 \ { (0 , 0) } ) × Z 2 → Z [ x ± i,j ] such that f ( t, k + 1 − t, i, j ) := ψ t,k ( x i,j ) . Then the function f c an b e extende d to Z 2 ≥ 0 × Z 2 , and satisfy the discrete 4D Hirota–Miwa equation fr om [ 16 ], which is the 4-dimensional version of the o ctahe dr on r e curr enc e [ 15 ]. Pr o of. The function f satisfies the recurrence: f (0 , b, i, j ) = f ( a, 0 , i, j ) = x i,j ; f (1 , 1 , i, j ) = x i − 1 ,j x i +1 ,j + x i,j − 1 x i,j +1 x i,j for all a, b ∈ N , i, j ∈ Z ; f ( a + 1 , b + 1 , i, j ) = f ( a + 1 , b, i − 1 , j ) f ( a, b + 1 , i + 1 , j ) + f ( a, b + 1 , i, j − 1) f ( a + 1 , b, i, j + 1) f ( a, b, i, j ) for all ( a, b ) ∈ Z 2 ≥ 0 \ { (0 , 0) } , i, j ∈ Z . F rom the recurrence, we can extend to another v alue f (0 , 0 , i, j ) = x i,j , such that the recurrence b ecomes f : Z 2 ≥ 0 × Z 2 → Z [ x ± i,j ] where: f (0 , b, i, j ) = f ( a, 0 , i, j ) = x i,j for all a, b ∈ Z ≥ 0 , i, j ∈ Z ; f ( a + 1 , b + 1 , i, j ) = f ( a + 1 , b, i − 1 , j ) f ( a, b + 1 , i + 1 , j ) + f ( a, b + 1 , i, j − 1) f ( a + 1 , b, i, j + 1) f ( a, b, i, j ) for all a, b ∈ Z ≥ 0 , i, j ∈ Z . This is exactly the discr ete 4D Hir ota–Miwa e quation . No w define the pro duct of v ariables: X a,b i,j :=    Q p,q ∈ Z ; 1 − a ≤ min { p,q ,p + q }≤ max { p,q ,p + q }≤ b − 1 x i − p,j + q if ( a, b )  = (0 , 0) , x i,j if ( a, b ) = (0 , 0) for all a, b ∈ Z ≥ 0 , i, j ∈ Z where in case there are no such v alues of ( a, b ) satisfying the condition, w e can let the product b e 1. See Figure 17 for an example. 29 x i,j Figure 17: Visualization for all p oin ts app earing in the pro duct X 4 , 6 i,j Then we also ha ve another observ ation: Prop osition 3.15. Consider another function g : Z 2 ≥ 0 × Z 2 → Z [ x ± i,j ] such that g ( a, b, i, j ) := X a,b i,j · f ( a, b, i, j ) . We c an c ompute the b ase c ases: g (0 , 0 , i, j ) = x 2 i,j ; g (0 , b, i, j ) = Y ( k,l ) ∈L (0 ,b ) x i + k,j + l ; g ( a, 0 , i, j ) = Y ( k,l ) ∈L ( a, 0) x i + k,j + l for al l a, b ≥ 1 . Then g satisfies the fol lowing e quation: g ( a + 1 , b + 1 , i, j ) g ( a, b, i, j ) = x i,j − a x i,j + b · g ( a + 1 , b, i − 1 , j ) g ( a, b + 1 , i + 1 , j ) + x i + a,j x i − b,j · g ( a, b + 1 , i, j − 1) g ( a + 1 , b, i, j + 1) for al l a, b ∈ Z ≥ 0 , i, j ∈ Z , with g ( a, b, i, j ) a p olynomial in finitely many variables x i ′ ,j ′ for any a, b ∈ Z ≥ 0 and i, j ∈ Z . Then we c an have a minor observation: g ( a, b, i, j ) = 2 ab · x 1+ ab + ( a + b − 1)( a + b − 2) 2 (42) in c ase al l variables x i ′ ,j ′ = x . The function g satisfies a similar e quation to [ 2 , Cor ol lary 19]. App endix A illustrates further to ols for attempting to derive the general formula for the function g . Additionally , to computationally verify and explore these recurrences, we implemen ted the functions in Maple. The Maple co de in App endix B defines the recurrence relation and allows for the computation of sp ecific cases. This implementation uses memorization to improv e efficiency and handles the base cases explicitly . The recurrence relation matches the discrete 4D Hirota-Miw a equation discussed ab o ve, pro viding a computational to ol to v erify theoretical results and explore new patterns in the structure of ψ t,k . 4 Cluster expansion formulas of n -triangulated m -gon In this section, w e discuss the expansion formula in the general n -triangulated m -gon configuration, i.e., the cluster expansion asso ciated with the mo duli space P SL n +1 , S where S is a disk with m -marked p oin ts with m ≥ 4. 4.1 Notation and lab eling conv en tions W e introduce tw o lab eling sc hemes for the vertices of an n -triangulated p olygon. 30 Definition 4.1 (Grid lab eling) . Consider an n -triangulated quadrilateral as a square with its vertices forming a lattice, assign all rows ( n rows) and columns ( n columns) consisting of all vertices, with ro ws en umerated from top to b ottom and columns from left to right. W e lab el the v ertex at row i and column j by [ i, j ]. W e shall use the following auxiliary construction in the pro of of the well-triangulated theorem below. Definition 4.2. F or a vertex [ i, j ], its envelop e is the smallest square con taining [ i, j ] (excluding b oundaries) whose edges consist of vertices from the initial square and whose main diagonal is con tained in the main diagonal of the initial square. See Figure 18 for the env elop e corresponding to v ertices [2 , 2], [5 , 3], and [2 , 4] with n = 5. 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 [5 , 1] [5 , 1] [5 , 1] [4 , 1] [4 , 1] [4 , 1] [3 , 1] [3 , 1] [3 , 1] [2 , 1] [2 , 1] [2 , 1] [1 , 1] [1 , 1] [1 , 1] [5 , 2] [5 , 2] [5 , 2] [4 , 2] [4 , 2] [4 , 2] [3 , 2] [3 , 2] [3 , 2] [2 , 2] [2 , 2] [2 , 2] [1 , 2] [1 , 2] [1 , 2] [5 , 3] [5 , 3] [5 , 3] [4 , 3] [4 , 3] [4 , 3] [3 , 3] [3 , 3] [3 , 3] [2 , 3] [2 , 3] [2 , 3] [1 , 3] [1 , 3] [1 , 3] [5 , 4] [5 , 4] [5 , 4] [4 , 4] [4 , 4] [4 , 4] [3 , 4] [3 , 4] [3 , 4] [2 , 4] [2 , 4] [2 , 4] [1 , 4] [1 , 4] [1 , 4] [5 , 5] [5 , 5] [5 , 5] [4 , 5] [4 , 5] [4 , 5] [3 , 5] [3 , 5] [3 , 5] [2 , 5] [2 , 5] [2 , 5] [1 , 5] [1 , 5] [1 , 5] Figure 18: The mirror of [2 , 2] (left), [5 , 3] (middle), and [2 , 4] (righ t) in case n = 5 Next we introduce another lab eling for the in ternal cluster v ariables. F or an y m ≥ 2, define Γ m := { ( a, b, c ) ∈ Z 3 ≥ 0 | a + b + c = m } . (43) Definition 4.3 (Γ-labeling) . F or m = 4, consider an n -triangulated quadrilateral AB C D with diagonal AC . Let us assign v ariables x (1) ( a,b,c ) to the vertices of △ AB C and x (2) ( a,b,c ) for △ AC D where ( a, b, c ) ∈ Γ n +1 . The vertices on the diagonal AC are assigned v ariables x (12) ( t, 0 ,n +1 − t ) for t = 1 , . . . , n (Figure 19 sho ws n = 3). Example 4.4. After flipping the diagonal, the new diagonal B D contains v ertices with v ariables x (1) (1 ,n − 1 , 1) , x (1) (2 ,n − 3 , 2) , . . . , x (1) ( k,n +1 − 2 k ,k ) , x (2) ( k,n +1 − 2 k ,k ) , . . . , x (2) (2 ,n − 3 , 2) , x (2) (1 ,n − 1 , 1) (44) where k =  n +1 2  . When n is o dd, the tw o middle v ertices (index k ) coincide, i.e. x (12) ( k,n +1 − 2 k ,k ) = x (1) ( k,n +1 − 2 k ,k ) = x (2) ( k,n +1 − 2 k ,k ) . A B C D A B C D x (2) (1 , 1 , 2) x (2) (1 , 2 , 1) x (2) (1 , 3 , 0) x (2) (2 , 1 , 1) x (2) (2 , 2 , 0) x (2) (3 , 1 , 0) x (2) (0 , 1 , 3) x (2) (0 , 2 , 2) x (2) (0 , 3 , 1) x (12) (1 , 0 , 3) x (12) (2 , 0 , 2) x (12) (3 , 0 , 1) x (1) (1 , 1 , 2) x (1) (1 , 2 , 1) x (1) (1 , 3 , 0) x (1) (2 , 1 , 1) x (1) (2 , 2 , 0) x (1) (3 , 1 , 0) x (1) (0 , 1 , 3) x (1) (0 , 2 , 2) x (1) (0 , 3 , 1) x (2) (0 , 1 , 3) x (2) (0 , 2 , 2) x (2) (0 , 3 , 1) x (1) (0 , 1 , 3) ( x (12) (1 , 0 , 3) ) ′ ( x (2) (1 , 1 , 2) ) ′ ( x (2) (1 , 2 , 1) ) ′ x (2) (1 , 3 , 0) x (1) (0 , 2 , 2) ( x (1) (1 , 1 , 2) ) ′ ( x (12) (2 , 0 , 2) ) ′′ ( x (2) (2 , 1 , 1) ) ′ x (2) (2 , 2 , 0) x (1) (0 , 3 , 1) ( x (1) (1 , 2 , 1) ) ′ ( x (1) (2 , 1 , 1) ) ′ ( x (12) (3 , 0 , 1) ) ′ x (2) (3 , 1 , 0) x (1) (1 , 3 , 0) x (1) (2 , 2 , 0) x (1) (3 , 1 , 0) Figure 19: Lab eling all v ertices of 3-triangulated quadrilateral. 31 Applying results from Prop ositions 3.7 and 3.9 , after a sequence of cluster m utations flipping the diagonal from AC to B D , w e calculate the expansion formula at v ertices x ( r ) (1 ,n − 1 , 1) as follows: ( x (1) (1 ,n − 1 , 1) ) ′ = X ( l 1 +1) ,l i ,u i , ( u k +1) , k ∈ N : P k i =1 ( l i + u i )= n   x (2) ( P k t =1 u t , 1 , P k s =1 l s ) k Y v =1 x (1) (1+ P v − 1 t =1 u t ,n − P v − 1 s =1 l s − P v − 1 t =1 u t , P v − 1 s =1 l s ) x (1) (1+ P v − 1 t =1 u t ,n − P v s =1 l s − P v − 1 t =1 u t , P v s =1 l s ) · x (1) ( P v − 1 t =1 u t ,n − P v s =1 l s − P v − 1 t =1 u t , 1+ P v s =1 l s ) x (1) ( P v t =1 u t ,n − P v s =1 l s − P v t =1 u t , 1+ P v s =1 l s )   ; (45) ( x (2) (1 ,n − 1 , 1) ) ′ = X ( r 1 +1) ,r i ,d i , ( d k +1) , k ∈ N : P k i =1 ( r i + d i )= n   x (1) ( P k t =1 r t , 1 , P k s =1 d s ) k Y v =1 x (2) ( P v − 1 t =1 r t ,n − P v − 1 t =1 r t − P v − 1 s =1 d s , 1+ P v − 1 s =1 d s ) x (2) ( P v t =1 r t ,n − P v t =1 r t − P v − 1 s =1 d s , 1+ P v − 1 s =1 d s ) · x (2) (1+ P v t =1 r t ,n − P v t =1 r t − P v − 1 s =1 d s , P v − 1 s =1 d s ) x (2) (1+ P v t =1 r t ,n − P v t =1 r t − P v s =1 d s , P v s =1 d s )   . (46) The Γ-lab eling can be extended to general m -gon: Example 4.5. Consider m = 5 and the n -triangulated p en tagon AB C D E with tw o main diagonals AC and AD . P oints (except the 5 vertices A, B , C , D , E ) are assigned v ariables x (1) ( a,b,c ) in △ AB C , x (2) ( a,b,c ) in △ AC D , and x (3) ( a,b,c ) in △ AD E for all ( a, b, c ) ∈ Γ n +1 . Returning to the 2-triangulated p en tagon (see Figure 20 ), w e compute B E = ( f 1 , f 3 ), where f 1 = x (1) (1 , 2 , 0) x (3) (0 , 2 , 1) x (23) (1 , 2 , 0) + x (1) (0 , 2 , 1) x (3) (2 , 0 , 1) x (12) (2 , 0 , 1) + x (1) (1 , 2 , 0) x (2) (0 , 1 , 2) x (3) (2 , 0 , 1) x (12) (1 , 0 , 2) x (23) (2 , 1 , 0) + x (1) (0 , 2 , 1) x (1) (2 , 1 , 0) x (3) (1 , 1 , 1) x (1) (1 , 1 , 1) x (23) (2 , 1 , 0) + x (1) (1 , 2 , 0) x (2) (0 , 2 , 1) x (3) (1 , 1 , 1) x (2) (1 , 1 , 1) x (23) (1 , 2 , 0) + x (1) (0 , 2 , 1) x (1) (2 , 1 , 0) x (2) (1 , 1 , 1) x (3) (2 , 0 , 1) x (1) (1 , 1 , 1) x (12) (2 , 0 , 1) x (23) (2 , 1 , 0) + x (1) (1 , 2 , 0) x (2) (0 , 1 , 2) x (12) (2 , 0 , 1) x (3) (1 , 1 , 1) x (12) (1 , 0 , 2) x (2) (1 , 1 , 1) x (23) (2 , 1 , 0) + x (1) (0 , 1 , 2) x (1) (1 , 2 , 0) x (2) (1 , 1 , 1) x (3) (2 , 0 , 1) x (1) (1 , 1 , 1) x (12) (1 , 0 , 2) x (23) (2 , 1 , 0) + x (1) (0 , 1 , 2) x (1) (1 , 2 , 0) x (12) (2 , 0 , 1) x (3) (1 , 1 , 1) x (1) (1 , 1 , 1) x (12) (1 , 0 , 2) x (23) (2 , 1 , 0) ; f 3 = x (1) (0 , 1 , 2) x (3) (1 , 0 , 2) x (12) (1 , 0 , 2) + x (1) (2 , 1 , 0) x (3) (0 , 1 , 2) x (23) (2 , 1 , 0) + x (1) (2 , 1 , 0) x (2) (0 , 2 , 1) x (3) (1 , 0 , 2) x (12) (2 , 0 , 1) x (23) (1 , 2 , 0) + x (1) (1 , 1 , 1) x (3) (0 , 1 , 2) x (3) (2 , 0 , 1) x (12) (2 , 0 , 1) x (3) (1 , 1 , 1) + x (1) (1 , 1 , 1) x (2) (0 , 1 , 2) x (3) (1 , 0 , 2) x (12) (1 , 0 , 2) x (2) (1 , 1 , 1) + x (1) (2 , 1 , 0) x (2) (1 , 1 , 1) x (3) (0 , 1 , 2) x (3) (2 , 0 , 1) x (12) (2 , 0 , 1) x (23) (2 , 1 , 0) x (3) (1 , 1 , 1) + x (1) (1 , 1 , 1) x (2) (0 , 2 , 1) x (23) (2 , 1 , 0) x (3) (1 , 0 , 2) x (12) (2 , 0 , 1) x (2) (1 , 1 , 1) x (23) (1 , 2 , 0) + x (1) (2 , 1 , 0) x (2) (1 , 1 , 1) x (3) (0 , 2 , 1) x (3) (1 , 0 , 2) x (12) (2 , 0 , 1) x (23) (1 , 2 , 0) x (3) (1 , 1 , 1) + x (1) (1 , 1 , 1) x (23) (2 , 1 , 0) x (3) (0 , 2 , 1) x (3) (1 , 0 , 2) x (12) (2 , 0 , 1) x (23) (1 , 2 , 0) x (3) (1 , 1 , 1) . A B E C D x (1) (2 , 1 , 0) x (1) (1 , 2 , 0) x (1) (0 , 2 , 1) x (1) (0 , 1 , 2) x (2) (0 , 1 , 2) x (2) (0 , 2 , 1) x (3) (0 , 2 , 1) x (3) (0 , 1 , 2) x (3) (1 , 0 , 2) x (3) (2 , 0 , 1) x (23) (2 , 1 , 0) x (12) (2 , 0 , 1) x (3) (1 , 1 , 1) x (1) (1 , 1 , 1) x (2) (1 , 1 , 1) x (23) (1 , 2 , 0) x (12) (1 , 0 , 2) Figure 20: Lab eling v ertices in 2-triangulated p entagon. 32 4.2 W ell-triangulated Preserv ation and Num b er of Monomials Theorem In this section, we compute the cluster expansion form ula for the v ariables lying on a non T -diagonal γ of an m -gon with resp ect to the initial triangulation T , meaning that γ intersects eac h diagonal of T exactly once. By the Laurent phenomenon and the Positivit y Theorem (Theorem 2.8 ), their expansions with resp ect to the initial cluster are Lauren t polynomials with all co efficien ts b eing p ositiv e integers. W e study the n umber of monomials, up to multiplicit y , in these Laurent p olynomials. Here, by multiplicit y , w e mean counting the sum of all co efficien ts of those p olynomials. W e introduce the wel l-triangulate d prop erty , and prov e the tw o main theorems on the combinatorics of the cluster expansion, namely the Wel l-triangulate d Pr eservation The or em (Theorem 4.7 ) and the Numb er of Monomials The or em (Theorem 4.8 ). Recall the Γ-lab eling x ( k ) ( a,b,c ) of the cluster v ariables in a triangle. Definition 4.6. Consider an n -triangulated polygon with triangulation T . A triangle T of T together with an assignmen t of scalar parameters to its cluster v ariables is called wel l-triangulate d if there exist parameters x, y , z such that for ev ery ( a, b, c ) ∈ Γ n +1 , the v alue assigned to x ( k ) ( a,b,c ) is x bc y ca z ab (see the left of Figure 21 for an example). The p olygon with v alues assigned to all the cluster v ariables is called wel l-triangulate d if all triangles of T are w ell-triangulated (see the righ t of Figure 21 for an example). z 2 z 2 y 2 x 2 x 2 y 2 xy z r 2 1 r 2 1 r 2 2 r 2 2 r 2 3 r 2 3 r 2 4 r 2 4 r 2 5 r 2 5 r 2 7 r 2 6 r 4 r 5 r 7 r 1 r 2 r 6 r 3 r 6 r 7 r 2 7 r 2 6 Figure 21: A well-triangulated 2-triangulated triangle (left) and a well-triangulated 2-triangulated p entagon where r 1 , r 2 , ..., r 7 are parameters (right). W e shall verify the follo wing theorem: Theorem 4.7 (W ell-triangulated Preserv ation Theorem) . F or a triangulation T of a wel l-triangulate d p oly- gon P with the cluster r e alization of typ e A n , after any se quenc e of flips, the r esulting triangulation is also wel l-triangulate d. In other w ords, any sequence of flips preserves the well-triangulated prop ert y of all triangles of the new triangulation when all triangles of the initial triangulation are well-triangulated. Pr o of. It suffices to chec k for a single flip, then the result follo ws by induction on the num ber of cr ossing p oints of a non T -diagonal, where for a cr ossing p oint of a non T -diagonal γ we mean the intersection of γ and a T -diagonal, note that b y definition, γ intersects ev ery T -diagonal exactly once. Consider the flip of a well-triangulated quadrilateral AB C D with diagonal AC , where x (1) ( i,j,k ) are as- signed to vertices inside △ AB C and x (2) ( i,j,k ) are assigned to vertices inside △ AC D . This means there exists parameters a, b, x for △ AB C and tw o other parameters c, d with x for △ AC D suc h that eac h vertex x (1) ( i,j,k ) has v alue a ij b j k x ik and each vertex x (2) ( i,j,k ) has v alue d ij c j k x ik . Denote l := ac + bd x . 33 Consider AB C D and the inner vertices with the grid lab eling as shown in the Definition 4.2 . Let H [ i,j ] b e the initial v alue of x [ i,j ] at [ i, j ] and H ′ [ i,j ] as the v alue after flip. Then: x [ i,j ] =        x (2) ( j,n +1 − i − j,i ) if i + j ≤ n, x (1) ( n +1 − i,i + j − n − 1 ,n +1 − j ) if i + j ≥ n + 2 , x (12) ( j, 0 ,i ) if i + j = n + 1 . Hence, we compute: H [ i,j ] =      c i ( n +1 − i − j ) d j ( n +1 − i − j ) x ij if i + j ≤ n, a ( n +1 − i )( i + j − n − 1) b ( n +1 − j )( i + j − n − 1) x ( n +1 − i )( n +1 − j ) if i + j ≥ n + 2 , x ij if i + j = n + 1 . After the flip, each vertex [ i, j ] is mutated exactly k times if and only if ( i − k )( j − k )( i − n − 1 + k )( j − n − 1+ k ) = 0 and k ≤ i, j ≤ n + 1 − k . Then: H ′ [1 ,n ] = d n H [2 , n ] + a n H [1 , n − 1] x n = d n bx n − 1 a n − 1 + a n cx n − 1 d n − 1 x n = ld n − 1 a n − 1 ; H ′ [ n, 1] = b n H [ n − 1 , 1] + c n H [ n, 2] x n = b n c n − 1 x n − 1 d + c n b n − 1 x n − 1 a x n = lc n − 1 b n − 1 . Note that for ( l 1 + 1) , l i , u i , ( u k + 1) , k ∈ N such that P k i =1 ( l i + u i ) = n , we hav e the relations: ( l 1 + l 2 + · · · + l k )( u 1 + u 2 + · · · + u k ) − k X v =1  l v  1 + v − 1 X t =1 u t  + u v  1 + v X t =1 l s  = − ( l 1 + l 2 + · · · + l k ) − ( u 1 + u 2 + · · · + u k ) = − n ; k X v =1  l 2 v + l v  v − 1 X s =1 l s  − l v  n − v − 1 X s =1 l s − v − 1 X t =1 u t  + u v  1 + v X t =1 l s  = l 2 1 + l 2 2 + · · · + l 2 k + 2 X 1 ≤ i 0, then the vertex V := V m 1 ,n 1 ,...,m k ,n k is next to B , then end with the remaining vertex V := V m 1 ,n 1 ,...,m k ,n k , 1 next to B . (ii) If n k = 0, then the v ertex V := V m 1 ,n 1 ,...,m k is next to B , then end with the remaining vertex V := V m 1 ,n 1 ,...,m k , 1 next to B . In b oth cases, V is alwa ys on the right of AB . Definition 4.10. A triangulation T of P of t yp e P ( m 1 , n 1 , m 2 , n 2 , ..., m k , n k ) is pro duced by joining v ertices of P in the following w ay: • F or i = 1 , ..., k , join all vertices V m 1 ,n 1 ,m 2 ,n 2 ,...n i − 1 ,x with V m 1 ,n 1 ,m 2 ,n 2 ,...,m i , 1 ,( x = 1 , ..., m i ); • F or i = 1 , ..., k , join all vertices V m 1 ,n 1 ,m 2 ,n 2 ,...,m i ,y with V m 1 ,n 1 ,m 2 ,n 2 ,...,m i ,n i , 1 , ( y = 1 , ..., n i ). By construction the diagonal AB is a non T -diagonal , meaning that AB intersects every triangle of T . See Figure 26 for an example of the setup. Definition 4.11. F or a p olygon P ( p 1 , p 2 , . . . , p K ) with p i > 0, its r efle cte d p olygon is P ( p 1 , p 2 , . . . , p K ) := P ( p K , p K − 1 , . . . , p 1 ) , with the same fixed diagonal AB . By reflecting the sequence, we ma y assume m 1 > 0. F or notation conv enience we ma y omit trailing zeros: e.g., P (3 , 5 , 12 , 0) = P (3 , 5 , 12). F or the rest of the section, we shall fix the triangulation T of an m -gon of type P ( m 1 , n 1 , . . . , m k , n k ) with a designated non T -diagonal γ = AB , where w e assume m 1 , n 1 , m 2 , n 2 , . . . , m k , ( n k + 1) ∈ N such that ( m k , n k ) / ∈ { ( m k , 1) , (1 , 0) } (unless k = 1, then ( m k , n k ) = (1 , 0) is allow ed). Definition 4.12. (Standard sequence of flips) The diagonal γ = AB can b e obtained from T through a sequence of flips, pro ducing diagonals d 1 , d 2 , . . . , d M k + N k . Let r u ∈ { 1 , . . . , m u − 1 } , s u ∈ { 1 , . . . , n u − 1 } . The diagonals are constructed as follows: d 1 = AV 2 , d 2 = AV 3 , . . . , d m 1 − 1 = AV m 1 , d m 1 = AV m 1 ,n 1 , 1 ; d m 1 +1 = AV m 1 , 2 , . . . , d m 1 + n 1 − 1 = AV m 1 ,n 1 , d m 1 + n 1 = AV m 1 ,n 1 ,m 2 , 1 ; . . . d M u − 1 + N u − 1 = AV m 1 ,n 1 ,...,m u − 1 ,n u − 1 ,m u , 1 , d M u − 1 + N u − 1 + r u = AV m 1 ,n 1 ,...,m u − 1 ,n u − 1 ,r u +1 ; d M u + N u − 1 = AV m 1 ,n 1 ,...,m u − 1 ,n u − 1 ,m u ,n u , 1 , d M u + N u − 1 + s u = AV m 1 ,n 1 ,...,m u − 1 ,n u − 1 ,m u ,s u +1 40 and for n k = 0: d M k − 1 + N k − 1 = AV , d M k − 1 + N k − 1 + r k = AV m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,r k +1 ; d M k + N k = AB = γ while for n k > 0: d M k + N k − 1 = AV , d M k + N k − 1 + s k = AV m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ,s k +1 ; d M k + N k = AB = γ . W e shall call the pro cedure ab o ve as the standar d se quenc e of flips of γ . Definition 4.13. (Diagonal and inner vertices notations) Consider an n -triangulated m -gon with triangu- lation T of t yp e P ( m 1 , n 1 , m 2 , n 2 , . . . , m k , n k ) as in Definition 4.10 . Denote the vertices of the diagonal γ = AB by the v ector: ( D [ m 1 ,n 1 ,...,m k ,n k ] 1 , . . . , D [ m 1 ,n 1 ,...,m k ,n k ] n ) , (54) and let the inner vertices of △ AV B b e denoted b y: I [ m 1 ,n 1 ,...,m k ,n k ] ( i,j ) , i, j ∈ N , i + j ≤ n. (55) F or n = 1, define D [ m 1 ,n 1 ,...,m k ,n k ] := D [ m 1 ,n 1 ,...,m k ,n k ] 1 ; for n = 2, define I [ m 1 ,n 1 ,...,m k ,n k ] := I [ m 1 ,n 1 ,...,m k ,n k ] (1 , 1) . See Figure 24 for the case n = 3, where, in case n k > 0, w e denote [ X ] := [ m 1 , n 1 , m 2 , n 2 , ..., m k ] and [ X, ∗∗ ] := [ m 1 , n 1 , m 2 , n 2 , ..., m k , n k ], and in case n k = 0, we denote [ X ] := [ m 1 , n 1 , m 2 , n 2 , ..., n k − 1 ] and [ X, ∗∗ ] := [ m 1 , n 1 , m 2 , n 2 , ..., m k ]. B V A D [ X ] 3 D [ X ] 2 D [ X ] 1 D [ X, ∗∗ ] 3 D [ X, ∗∗ ] 2 D [ X, ∗∗ ] 1 I [ X, ∗∗ ] (1 , 2) I [ X, ∗∗ ] (2 , 1) I [ X, ∗∗ ] (1 , 1) Figure 24: The resulting triangle after the sequence of flips for n = 3 Definition 4.14. Edges in the triangulation are colored (first = red, second = green, third = blue) to assign co ordinates ( a, b, c ) ∈ Γ n +1 . Starting from △ AV m 1 , 1 V 1 as an o dd triangle , all triangles are classified as either o dd or even dep ending on the clockwise or counter-clockwise direction of the colors order red, green, blue on the edges ( v 1 represen ts red, v 2 represen ts green, v 3 represen ts blue), with adjacen t triangles having opposite parit y (see Figure 25 ). 41 v 1 v 2 v 3 v 1 v 3 v 2 Figure 25: Odd triangle (left) and even triangle (righ t) This colouring and parity assignment allow us to iden tify all coordinates in the general n -triangulated m -gon, as illustrated in Figure 26 for n = 2. A B V 1 V 2 V 2 , 1 V 2 , 2 V 2 , 3 V 2 , 3 , 1 V 2 , 3 , 2 V 2 , 3 , 3 V 2 , 3 , 3 , 1 V 2 , 3 , 3 , 1 , 1 V 2 , 3 , 3 , 1 , 2 V 2 , 3 , 3 , 1 , 3 V Figure 26: Co ordinates in the 2-triangulated 15-gon P (2 , 3 , 3 , 1 , 3 , 0) W e are now ready to set up recurrence relations to calculate the general formulas of the functions D [ m 1 ,n 1 ,...,m k ,n k ] l (with l = 1 , 2 , ..., n ) and I [ m 1 ,n 1 ,...,m k ,n k ] ( i,j ) (with i, j ∈ N suc h that i + j ≤ n ) with resp ect to γ = AB based on the standard sequence of flips men tioned in the Definition 4.12 . W e first introduce some short hand notations concerning the parties of m i and n j . Define the partial sums M i := m 1 + m 2 + · · · + m i and N j := n 1 + n 2 + · · · + n j . (56) Definition 4.15. Consider the p erm utation actions υ , τ on Γ m defined by υ · ( w 1 , w 2 , w 3 ) = ( w 3 , w 1 , w 2 ) and τ · ( w 1 , w 2 , w 3 ) = ( w 1 , w 3 , w 2 ) , ( w 1 , w 2 , w 3 ) ∈ Γ m . Clearly , υ 3 = τ 2 = e and υ τ υ = τ . W e define the following sums: σ := ( − 1) m 1 + ( − 1) n 1 + ( − 1) m 2 + ( − 1) n 2 + · · · + ( − 1) m k − 1 + ( − 1) n k − 1 + ( − 1) m k ; (57) σ ′ := ( − 1) m 1 + ( − 1) n 1 + ( − 1) m 1 + ( − 1) n 2 + · · · + ( − 1) m k − 1 + ( − 1) n k − 1 . (58) Finally we let p ∈ { 0 , 1 , 2 } , q ∈ { 0 , 1 } b e suc h that σ ≡ p (mod 3) and M k + N k − 1 ≡ q (mo d 2), and denote ω := υ p τ q . Similarly , let p ′ ∈ { 0 , 1 , 2 } , q ′ ∈ { 0 , 1 } such that σ ′ ≡ p ′ (mo d 3) and M k − 1 + N k − 1 ≡ q ′ (mo d 2), and denote ω ′ := υ p ′ τ q ′ . 42 The color of all edges and T -diagonals of the polygon can be iden tified uniquely (see Figure 26 for an example). Refer to Definition 4.12 again, we start from the last level, and contin ue follo wing the standard sequence of flips recursively . (I) F or n k > 0, w e introduce extra notations V m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k +1 := B and S r := ( m 1 , n 1 , m 2 , n 2 , ..., m r , n r ) for all r ∈ N . When we hav e reac hed the diagonal d M k − 1 + N k , then the color of all edges V S k − 1 ,m k ,j V S k − 1 ,m k ,j +1 for j = 1 , 2 , ..., n k are the same. That common color and the color of the diagonal V V S k − 1 ,m k , 1 is identified uniquely by p and q . Hence, we consider the 6 following cases, where the figure shows ho w the vertices are being lab eled after previous flips. V V V A A A V S k − 1 ,m k , 2 V S k − 1 ,m k ,t +1 V S k − 1 ,m k ,t +1 V S k − 1 ,m k , 1 V S k − 1 ,m k ,t V S k − 1 ,m k ,t t ≥ 2 is ev en t ≥ 3 is odd D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i I [ S k − 1 ,m k ] ( i,j ) I [ S k − 1 ,m k ,t − 1] ( j,i ) I [ S k − 1 ,m k ,t − 1] ( j,i ) D [ S k − 1 ] j D [ S k − 1 ,m k ,t − 1] j D [ S k − 1 ,m k ,t − 1] j Figure 27: Case 1.1 : ( p, q ) = (0 , 0) V V V A A A V S k − 1 ,m k , 2 V S k − 1 ,m k ,t +1 V S k − 1 ,m k ,t +1 V S k − 1 ,m k , 1 V S k − 1 ,m k ,t V S k − 1 ,m k ,t t ≥ 2 is ev en t ≥ 3 is odd D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i I [ S k − 1 ,m k ] ( i,j ) I [ S k − 1 ,m k ,t − 1] ( j,i ) I [ S k − 1 ,m k ,t − 1] ( j,i ) D [ S k − 1 ] j D [ S k − 1 ,m k ,t − 1] j D [ S k − 1 ,m k ,t − 1] j Figure 28: Case 1.2 : ( p, q ) = (0 , 1) V V V A A A V S k − 1 ,m k , 2 V S k − 1 ,m k ,t +1 V S k − 1 ,m k ,t +1 V S k − 1 ,m k , 1 V S k − 1 ,m k ,t V S k − 1 ,m k ,t t ≥ 2 is ev en t ≥ 3 is odd D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i I [ S k − 1 ,m k ] ( i,j ) I [ S k − 1 ,m k ,t − 1] ( j,i ) I [ S k − 1 ,m k ,t − 1] ( j,i ) D [ S k − 1 ] j D [ S k − 1 ,m k ,t − 1] j D [ S k − 1 ,m k ,t − 1] j Figure 29: Case 1.3 : ( p, q ) = (1 , 0) 43 V V V A A A V S k − 1 ,m k , 2 V S k − 1 ,m k ,t +1 V S k − 1 ,m k ,t +1 V S k − 1 ,m k , 1 V S k − 1 ,m k ,t V S k − 1 ,m k ,t t ≥ 2 is ev en t ≥ 3 is odd D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i I [ S k − 1 ,m k ] ( i,j ) I [ S k − 1 ,m k ,t − 1] ( j,i ) I [ S k − 1 ,m k ,t − 1] ( j,i ) D [ S k − 1 ] j D [ S k − 1 ,m k ,t − 1] j D [ S k − 1 ,m k ,t − 1] j Figure 30: Case 1.4 : ( p, q ) = (1 , 1) V V V A A A V S k − 1 ,m k , 2 V S k − 1 ,m k ,t +1 V S k − 1 ,m k ,t +1 V S k − 1 ,m k , 1 V S k − 1 ,m k ,t V S k − 1 ,m k ,t t ≥ 2 is ev en t ≥ 3 is odd D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i I [ S k − 1 ,m k ] ( i,j ) I [ S k − 1 ,m k ,t − 1] ( j,i ) I [ S k − 1 ,m k ,t − 1] ( j,i ) D [ S k − 1 ] j D [ S k − 1 ,m k ,t − 1] j D [ S k − 1 ,m k ,t − 1] j Figure 31: Case 1.5 : ( p, q ) = (2 , 0) V V V A A A V S k − 1 ,m k , 2 V S k − 1 ,m k ,t +1 V S k − 1 ,m k ,t +1 V S k − 1 ,m k , 1 V S k − 1 ,m k ,t V S k − 1 ,m k ,t t ≥ 2 is ev en t ≥ 3 is odd D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i D [ S k − 1 ,m k ] i I [ S k − 1 ,m k ] ( i,j ) I [ S k − 1 ,m k ,t − 1] ( j,i ) I [ S k − 1 ,m k ,t − 1] ( j,i ) D [ S k − 1 ] j D [ S k − 1 ,m k ,t − 1] j D [ S k − 1 ,m k ,t − 1] j Figure 32: Case 1.6 : ( p, q ) = (2 , 1) (I I) F or n k = 0, redefine the notation V m 1 ,n 1 ,m 2 ,n 2 ,...,n k − 1 ,m k +1 := B and keep the notation S r := ( m 1 , n 1 , m 2 , n 2 , ..., m r , n r ) for all r ∈ N . When we hav e reac hed the diagonal d M k − 1 + N k − 1 , then the color of all edges V S k − 1 ,j V S k − 1 ,j +1 for j = 1 , 2 , ..., m k are the same. That common color and the color of the diagonal V V S k − 1 , 1 is iden tified uniquely by p ′ and q ′ . Hence, we consider the 6 following cases, again the figure shows ho w the vertices are b eing labeled after previous flips. 44 V V V A A A V S k − 1 , 2 V S k − 1 ,t ′ +1 V S k − 1 ,t ′ +1 V S k − 1 , 1 V S k − 1 ,t ′ V S k − 1 ,t ′ t ′ ≥ 2 is ev en t ′ ≥ 3 is odd D [ S k − 2 ,m k − 1 ] i D [ S k − 1 ,t ′ − 1] i D [ S k − 1 ,t ′ − 1] i I [ S k − 1 ] ( j,i ) I [ S k − 1 ,t ′ − 1] ( i,j ) I [ S k − 1 ,t ′ − 1] ( i,j ) D [ S k − 1 ] j D [ S k − 1 ] j D [ S k − 1 ] j Figure 33: Case 2.1 : ( p ′ , q ′ ) = (0 , 0) V V V A A A V S k − 1 , 2 V S k − 1 ,t ′ +1 V S k − 1 ,t ′ +1 V S k − 1 , 1 V S k − 1 ,t ′ V S k − 1 ,t ′ t ′ ≥ 2 is ev en t ′ ≥ 3 is odd D [ S k − 2 ,m k − 1 ] i D [ S k − 1 ,t ′ − 1] i D [ S k − 1 ,t ′ − 1] i I [ S k − 1 ] ( j,i ) I [ S k − 1 ,t ′ − 1] ( i,j ) I [ S k − 1 ,t ′ − 1] ( i,j ) D [ S k − 1 ] j D [ S k − 1 ] j D [ S k − 1 ] j Figure 34: Case 2.2 : ( p ′ , q ′ ) = (0 , 1) V V V A A A V S k − 1 , 2 V S k − 1 ,t ′ +1 V S k − 1 ,t ′ +1 V S k − 1 , 1 V S k − 1 ,t ′ V S k − 1 ,t ′ t ′ ≥ 2 is ev en t ′ ≥ 3 is odd D [ S k − 2 ,m k − 1 ] i D [ S k − 1 ,t ′ − 1] i D [ S k − 1 ,t ′ − 1] i I [ S k − 1 ] ( j,i ) I [ S k − 1 ,t ′ − 1] ( i,j ) I [ S k − 1 ,t ′ − 1] ( i,j ) D [ S k − 1 ] j D [ S k − 1 ] j D [ S k − 1 ] j Figure 35: Case 2.3 : ( p ′ , q ′ ) = (1 , 0) 45 V V V A A A V S k − 1 , 2 V S k − 1 ,t ′ +1 V S k − 1 ,t ′ +1 V S k − 1 , 1 V S k − 1 ,t ′ V S k − 1 ,t ′ t ′ ≥ 2 is ev en t ′ ≥ 3 is odd D [ S k − 2 ,m k − 1 ] i D [ S k − 1 ,t ′ − 1] i D [ S k − 1 ,t ′ − 1] i I [ S k − 1 ] ( j,i ) I [ S k − 1 ,t ′ − 1] ( i,j ) I [ S k − 1 ,t ′ − 1] ( i,j ) D [ S k − 1 ] j D [ S k − 1 ] j D [ S k − 1 ] j Figure 36: Case 2.4 : ( p ′ , q ′ ) = (1 , 1) V V V A A A V S k − 1 , 2 V S k − 1 ,t ′ +1 V S k − 1 ,t ′ +1 V S k − 1 , 1 V S k − 1 ,t ′ V S k − 1 ,t ′ t ′ ≥ 2 is ev en t ′ ≥ 3 is odd D [ S k − 2 ,m k − 1 ] i D [ S k − 1 ,t ′ − 1] i D [ S k − 1 ,t ′ − 1] i I [ S k − 1 ] ( j,i ) I [ S k − 1 ,t ′ − 1] ( i,j ) I [ S k − 1 ,t ′ − 1] ( i,j ) D [ S k − 1 ] j D [ S k − 1 ] j D [ S k − 1 ] j Figure 37: Case 2.5 : ( p ′ , q ′ ) = (2 , 0) V V V A A A V S k − 1 , 2 V S k − 1 ,t ′ +1 V S k − 1 ,t ′ +1 V S k − 1 , 1 V S k − 1 ,t ′ V S k − 1 ,t ′ t ′ ≥ 2 is ev en t ′ ≥ 3 is odd D [ S k − 2 ,m k − 1 ] i D [ S k − 1 ,t ′ − 1] i D [ S k − 1 ,t ′ − 1] i I [ S k − 1 ] ( j,i ) I [ S k − 1 ,t ′ − 1] ( i,j ) I [ S k − 1 ,t ′ − 1] ( i,j ) D [ S k − 1 ] j D [ S k − 1 ] j D [ S k − 1 ] j Figure 38: Case 2.6 : ( p ′ , q ′ ) = (2 , 1) Definition 4.11 simplifies calculating expansion formulas b y reducing the n umber of co ordinates to chec k. The case of 1 -triangulated m -gon After the previous reduction, our problem b ecomes just solving the general formula D [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] . Actually , this is just an application of p erfect matc hings on snake graph [ 25 , 28 ]. This part will elab orate the general formula in the recurrence form ula format whic h is differen t from the previous combinatorial method, but maintain the same idea ab out the standard sequence of flips. By the calculation from each figure, we can rewrite all recurrences in the statement b elo w: 46 Theorem 4.16 (Recurrence formula for a general 1-triangulated p olygon) . Consider a gener al 1 -triangulate d m -gon with T -triangulation of typ e P ( m 1 , n 1 , . . . , m k , n k ) . L et D [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] b e the exp ansion for- mula with r esp e ct to the non- T diagonal γ = AB . Then the functions satisfy the fol lowing r e curr enc e r elations: I. (T yp e P ( m 1 ) .) F or any k ∈ N , D [2 k − 1] = x (1) (1 , 1 , 0) · x (2 k )(2 k +1) (1 , 1 , 0) · k X j =1 x (2 j ) (0 , 1 , 1) x (2 j )(2 j +1) (1 , 1 , 0) x (2 j − 1)(2 j ) (1 , 0 , 1) + x (2 j − 1) (0 , 1 , 1) x (2 j − 1)(2 j ) (1 , 0 , 1) x (2 j − 2)(2 j − 1) (1 , 1 , 0) ! ; D [2 k ] = x (1) (1 , 1 , 0) · x (2 k +1)(2 k +2) (1 , 0 , 1) · " x (2 k +1) (0 , 1 , 1) x (2 k +1)(2 k +2) (1 , 0 , 1) x (2 k )(2 k +1) (1 , 1 , 0) + k X j =1 x (2 j ) (0 , 1 , 1) x (2 j )(2 j +1) (1 , 1 , 0) x (2 j − 1)(2 j ) (1 , 0 , 1) + x (2 j − 1) (0 , 1 , 1) x (2 j − 1)(2 j ) (1 , 0 , 1) x (2 j − 2)(2 j − 1) (1 , 1 , 0) !# . I I. (T yp e P ( m 1 , n 1 ) .) F or any k , l ∈ N , then: D [2 k − 1 , 2 l − 1] = x ((2 k +2 l − 1)(2 k +2 l )) (1 , 0 , 1) x ((2 k )(2 k +1)) (0 , 1 , 1) · x (1) (1 , 1 , 0) + x ((2 k +2 l − 1)(2 k +2 l )) (1 , 0 , 1) · " x (2 k +2 l − 1) (1 , 1 , 0) x ((2 k +2 l − 1)(2 k +2 l )) (1 , 0 , 1) x ((2 k +2 l − 2)(2 k +2 l − 1)) (0 , 1 , 1) + l − 1 X j =1 x (2 k +2 j ) (1 , 1 , 0) x ((2 k +2 j )(2 k +2 j +1)) (0 , 1 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 1) + x (2 k +2 j − 1) (1 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 1) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 1) !# · " x (1) (1 , 1 , 0) · x (2 k )(2 k +1) (1 , 1 , 0) · k X j =1 x (2 j ) (0 , 1 , 1) x (2 j )(2 j +1) (1 , 1 , 0) x (2 j − 1)(2 j ) (1 , 0 , 1) + x (2 j − 1) (0 , 1 , 1) x (2 j − 1)(2 j ) (1 , 0 , 1) x (2 j − 2)(2 j − 1) (1 , 1 , 0) !# ; D [2 k − 1 , 2 l ] = x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 1) x ((2 k )(2 k +1)) (0 , 1 , 1) · x (1) (1 , 1 , 0) + x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 1) · x (1) (1 , 1 , 0) · " l X j =1 x (2 k +2 j ) (1 , 1 , 0) x ((2 k +2 j )(2 k +2 j +1)) (0 , 1 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 1) + x (2 k +2 j − 1) (1 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 1) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 1) !# · " x (2 k )(2 k +1) (1 , 1 , 0) · k X j =1 x (2 j ) (0 , 1 , 1) x (2 j )(2 j +1) (1 , 1 , 0) x (2 j − 1)(2 j ) (1 , 0 , 1) + x (2 j − 1) (0 , 1 , 1) x (2 j − 1)(2 j ) (1 , 0 , 1) x (2 j − 2)(2 j − 1) (1 , 1 , 0) !# ; D [2 k, 2 l − 1] = x ((2 k +2 l )(2 k +2 l +1)) (1 , 1 , 0) x ((2 k +1)(2 k +2)) (0 , 1 , 1) · x (1) (1 , 1 , 0) + x ((2 k +2 l )(2 k +2 l +1)) (1 , 1 , 0) · " x (2 k +2 l ) (1 , 0 , 1) x ((2 k +2 l )(2 k +2 l +1)) (1 , 1 , 0) x ((2 k +2 l − 1)(2 k +2 l )) (0 , 1 , 1) + l − 1 X j =1 x (2 k +2 j +1) (1 , 0 , 1) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 1 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 1 , 0) + x (2 k +2 j ) (1 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 1 , 1) !# · " x (1) (1 , 1 , 0) · x (2 k +1)(2 k +2) (1 , 0 , 1) · " x (2 k +1) (0 , 1 , 1) x (2 k +1)(2 k +2) (1 , 0 , 1) x (2 k )(2 k +1) (1 , 1 , 0) + k X j =1 x (2 j ) (0 , 1 , 1) x (2 j )(2 j +1) (1 , 1 , 0) x (2 j − 1)(2 j ) (1 , 0 , 1) + x (2 j − 1) (0 , 1 , 1) x (2 j − 1)(2 j ) (1 , 0 , 1) x (2 j − 2)(2 j − 1) (1 , 1 , 0) !## ; D [2 k, 2 l ] = x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 1 , 1) x ((2 k +1)(2 k +2)) (0 , 1 , 1) · x (1) (1 , 1 , 0) + x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 1 , 1) · " l X j =1 x (2 k +2 j +1) (1 , 0 , 1) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 1 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 1 , 0) + x (2 k +2 j ) (1 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 0 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 1 , 1) !# · " x (1) (1 , 1 , 0) · x (2 k +1)(2 k +2) (1 , 0 , 1) · " x (2 k +1) (0 , 1 , 1) x (2 k +1)(2 k +2) (1 , 0 , 1) x (2 k )(2 k +1) (1 , 1 , 0) + k X j =1 x (2 j ) (0 , 1 , 1) x (2 j )(2 j +1) (1 , 1 , 0) x (2 j − 1)(2 j ) (1 , 0 , 1) + x (2 j − 1) (0 , 1 , 1) x (2 j − 1)(2 j ) (1 , 0 , 1) x (2 j − 2)(2 j − 1) (1 , 1 , 0) !## . I II-1. (F or n k ≥ 1 ) F or any k , m 1 , n 1 , m 2 , n 2 , . . . , m k , ( n k − 1) ∈ N , the function satisfies the fol lowing 47 r elations: for any n k = 2 l ≥ 2 even: D [ m 1 ,n 1 ,...,m k ,n k ] = x (( M k + N k − 1 +2 l +1)( M k + N k − 1 +2 l +2)) ω · (1 , 1 , 0) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω · (1 , 1 , 0) D [ m 1 ,n 1 ,...,n k − 1 ] + x (( M k + N k − 1 +2 l +1)( M k + N k − 1 +2 l +2)) ω · (1 , 1 , 0) · · " l X j =1 x ( M k + N k − 1 +2 j ) ω · (0 , 1 , 1) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 1) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (1 , 1 , 0) + x ( M k + N k − 1 +2 j +1) ω · (0 , 1 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (1 , 1 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 1) !# · D [ m 1 ,n 1 ,...,m k ] . F or any n k = 2 l + 1 ≥ 1 o dd: D [ m 1 ,n 1 ,...,m k ,n k ] = x (( M k + N k − 1 +2 l +2)( M k + N k − 1 +2 l +3)) ω · (1 , 0 , 1) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω · (1 , 1 , 0) D [ m 1 ,n 1 ,...,n k − 1 ] + x (( M k + N k − 1 +2 l +2)( M k + N k − 1 +2 l +3)) ω · (1 , 0 , 1) · " x ( M k + N k − 1 +2 l +2) ω · (0 , 1 , 1) x (( M k + N k − 1 +2 l +2)( M k + N k − 1 +2 l +3)) ω · (1 , 0 , 1) x (( M k + N k − 1 +2 l +1)( M k + N k − 1 +2 l +2)) ω · (1 , 1 , 0) + l X j =1 x ( M k + N k − 1 +2 j ) ω · (0 , 1 , 1) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 1) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (1 , 1 , 0) + x ( M k + N k − 1 +2 j +1) ω · (0 , 1 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (1 , 1 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 1) !# · D [ m 1 ,n 1 ,...,m k ] . I II-2. (F or n k = 0 ) F or any ( k − 1) , m 1 , n 1 , m 2 , n 2 , . . . , m k − 1 , n k − 1 , ( m k − 1) ∈ N , the function satisfies the fol lowing r elations: F or any m k = 2 l ≥ 2 even: D [ m 1 ,n 1 ,...,m k ] = x (( M k − 1 + N k − 1 +2 l +1)( M k − 1 + N k − 1 +2 l +2)) ω ′ · (1 , 0 , 1) x (( M k − 1 + N k − 1 +1)( M k − 1 + N k − 1 +2)) ω ′ · (1 , 0 , 1) D [ m 1 ,n 1 ,...,m k − 1 ] + x (( M k − 1 + N k − 1 +2 l +1)( M k − 1 + N k − 1 +2 l +2)) ω ′ · (1 , 0 , 1) · " l X j =1 x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 1 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 1 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (1 , 0 , 1) + x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 1 , 1) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (1 , 0 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 1 , 0) !# · D [ m 1 ,n 1 ,...,n k − 1 ] . F or any m k = 2 l + 1 ≥ 1 o dd: D [ m 1 ,n 1 ,...,m k ] = x (( M k − 1 + N k − 1 +2 l +2)( M k − 1 + N k − 1 +2 l +3)) ω ′ · (1 , 1 , 0) x (( M k − 1 + N k − 1 +1)( M k − 1 + N k − 1 +2)) ω ′ · (1 , 0 , 1) D [ m 1 ,n 1 ,...,m k − 1 ] + x (( M k − 1 + N k − 1 +2 l +2)( M k − 1 + N k − 1 +2 l +3)) ω ′ · (1 , 1 , 0) · " x ( M k − 1 + N k − 1 +2 l +2) ω ′ · (0 , 1 , 1) x (( M k − 1 + N k − 1 +2 l +2)( M k − 1 + N k − 1 +2 l +3)) ω ′ · (1 , 1 , 0) x (( M k − 1 + N k − 1 +2 l +1)( M k − 1 + N k − 1 +2 l +2)) ω ′ · (1 , 0 , 1) + l X j =1 x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 1 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 1 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (1 , 0 , 1) + x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 1 , 1) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (1 , 0 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 1 , 0) !# · D [ m 1 ,n 1 ,...,n k − 1 ] . The case of 2 -triangulated m -gon In this case our problem b ecomes solving the general formula D [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] i (for i = 1 , 2) and the inner v ertex I [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] = I [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] (1 , 1) . By doing part-by-part after eac h flip, we can dra w quadrilaterals to calculate the desired expansion formula. 48 Theorem 4.17 (Recurrence formula for a general 2-triangulated p olygon) . In the gener al 2 -triangulate d m -gon with T -triangulation in typ e P ( m 1 , n 1 , ..., m k , n k ) , denote the fol lowing functions: (i) ( D [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] 1 , D [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k ,n k ] 2 ) b e the exp ansion formula of the non T -diagonal d = AB ; (ii) I [ m 1 ,n 1 ,...,m k ,n k ] b e the exp ansion formula of the inner vertex of △ AV B forme d by taking the flips in the right-to-left or der (i.e. fr om A to B ). Then the functions satisfy the fol lowing r e curr enc e r elations: I. ( P ( m 1 ) ) F or any k ∈ N , then: D [1] 2 = x (1) (0 , 2 , 1) x (2) (2 , 1 , 0) x (12) (2 , 0 , 1) + x (1) (1 , 2 , 0) x (2) (0 , 1 , 2) x (12) (1 , 0 , 2) + x (1) (1 , 1 , 1) x (2) (0 , 1 , 2) x (2) (1 , 2 , 0) x (12) (1 , 0 , 2) x (2) (1 , 1 , 1) + x (1) (1 , 1 , 1) x (2) (0 , 2 , 1) x (2) (2 , 1 , 0) x (12) (2 , 0 , 1) x (2) (1 , 1 , 1) ; D [2 k ] 2 = x (2 k +1) (0 , 1 , 2) x (1) (2 , 1 , 0) x ((2 k )(2 k +1)) (2 , 1 , 0) + " x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k ) (2 , 1 , 0) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k ) (2 , 1 , 0) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) !# · x (2 k +1) (0 , 1 , 2) x (2 k +1) (1 , 1 , 1) · x (2 k +1) (2 , 0 , 1) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (2 k +1) (1 , 0 , 2) x (2 k +1) (1 , 1 , 1) · x (2 k +1) (0 , 2 , 1) x ((2 k )(2 k +1)) (1 , 2 , 0) ! x (2 k +1) (1 , 0 , 2) x ((2 k )(2 k +1)) (1 , 2 , 0) · D [2 k − 1] 2 = x (2 k +1) (1 , 0 , 2) x (2 k ) (0 , 2 , 1) x (1) (2 , 1 , 0) x ((2 k )(2 k +1)) (1 , 2 , 0) x ((2 k − 1)(2 k )) (2 , 0 , 1) + x (2 k +1) (0 , 1 , 2) x (1) (2 , 1 , 0) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (2 k +1) (1 , 0 , 2) x (1) (2 , 1 , 0) x (2 k − 1) (1 , 1 , 1) x ((2 k )(2 k +1)) (1 , 2 , 0) x ((2 k − 2)(2 k − 1)) (2 , 1 , 0) · x (2 k ) (0 , 2 , 1) x (2 k ) (1 , 1 , 1) · x (2 k ) (2 , 1 , 0) x ((2 k − 1)(2 k )) (2 , 0 , 1) + x (2 k ) (1 , 2 , 0) x (2 k ) (1 , 1 , 1) · x (2 k ) (0 , 1 , 2) x ((2 k − 1)(2 k )) (1 , 0 , 2) ! + " x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k ) (2 , 1 , 0) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k ) (2 , 1 , 0) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) !# · x (2 k +1) (0 , 1 , 2) x (2 k +1) (1 , 1 , 1) · x (2 k +1) (2 , 0 , 1) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (2 k +1) (1 , 0 , 2) x (2 k +1) (1 , 1 , 1) · x (2 k +1) (0 , 2 , 1) x ((2 k )(2 k +1)) (1 , 2 , 0) ! + x (2 k +1) (1 , 0 , 2) x ((2 k )(2 k +1)) (1 , 2 , 0) · " x (1) (2 , 1 , 0) · k − 2 X j =0 x (2 k − 1) (2 , 0 , 1) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k − 1) (2 , 0 , 1) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) !# · x (2 k ) (0 , 2 , 1) x (2 k ) (1 , 1 , 1) · x (2 k ) (2 , 1 , 0) x ((2 k − 1)(2 k )) (2 , 0 , 1) + x (2 k ) (1 , 2 , 0) x (2 k ) (1 , 1 , 1) · x (2 k ) (0 , 1 , 2) x ((2 k − 1)(2 k )) (1 , 0 , 2) ! + x (2 k +1) (1 , 0 , 2) x ((2 k − 1)(2 k )) (1 , 0 , 2) · D [2 k − 2] 2 ; 49 D [2 k +1] 2 = x (2 k +2) (0 , 2 , 1) x (1) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (2 , 0 , 1) + " x (1) (2 , 1 , 0) x (2 k +1) (1 , 1 , 1) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k +1) (2 , 0 , 1) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k +1) (2 , 0 , 1) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) !# · x (2 k +2) (0 , 2 , 1) x (2 k +2) (1 , 1 , 1) · x (2 k +2) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (2 , 0 , 1) + x (2 k +2) (1 , 2 , 0) x (2 k +2) (1 , 1 , 1) · x (2 k +2) (0 , 1 , 2) x ((2 k +1)(2 k +2)) (1 , 0 , 2) ! + x (2 k +2) (1 , 2 , 0) x ((2 k +1)(2 k +2)) (1 , 0 , 2) · D [2 k ] 2 = x (2 k +2) (1 , 2 , 0) x (2 k +1) (0 , 1 , 2) x (1) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (1 , 0 , 2) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (2 k +2) (0 , 2 , 1) x (1) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (2 , 0 , 1) + x (1) (2 , 1 , 0) x (2 k +1) (1 , 1 , 1) x ((2 k )(2 k +1)) (2 , 1 , 0) · x (2 k +2) (0 , 2 , 1) x (2 k +2) (1 , 1 , 1) · x (2 k +2) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (2 , 0 , 1) + x (2 k +2) (1 , 2 , 0) x (2 k +2) (1 , 1 , 1) · x (2 k +2) (0 , 1 , 2) x ((2 k +1)(2 k +2)) (1 , 0 , 2) ! + " x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k +1) (2 , 0 , 1) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k +1) (2 , 0 , 1) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) !# · x (2 k +2) (0 , 2 , 1) x (2 k +2) (1 , 1 , 1) · x (2 k +2) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (2 , 0 , 1) + x (2 k +2) (1 , 2 , 0) x (2 k +2) (1 , 1 , 1) · x (2 k +2) (0 , 1 , 2) x ((2 k +1)(2 k +2)) (1 , 0 , 2) ! + x (2 k +2) (1 , 2 , 0) x ((2 k +1)(2 k +2)) (1 , 0 , 2) · " x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k ) (2 , 1 , 0) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k ) (2 , 1 , 0) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) !# · x (2 k +1) (0 , 1 , 2) x (2 k +1) (1 , 1 , 1) · x (2 k +1) (2 , 0 , 1) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (2 k +1) (1 , 0 , 2) x (2 k +1) (1 , 1 , 1) · x (2 k +1) (0 , 2 , 1) x ((2 k )(2 k +1)) (1 , 2 , 0) ! + x (2 k +2) (1 , 2 , 0) x ((2 k )(2 k +1)) (1 , 2 , 0) · D [2 k − 1] 2 ; I [1] = x (1) (1 , 1 , 1) x (2) (2 , 1 , 0) x (12) (2 , 0 , 1) + x (1) (2 , 1 , 0) x (2) (1 , 1 , 1) x (12) (2 , 0 , 1) ; I [2] = x (1) (2 , 1 , 0) x (3) (1 , 1 , 1) x (23) (2 , 1 , 0) + x (1) (1 , 1 , 1) x (3) (2 , 0 , 1) x (12) (2 , 0 , 1) + x (1) (2 , 1 , 0) x (2) (1 , 1 , 1) x (3) (2 , 0 , 1) x (12) (2 , 0 , 1) x (23) (2 , 1 , 0) ; I [2 k − 1] = x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k ) (2 , 1 , 0) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k ) (2 , 1 , 0) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) ! ; I [2 k ] = x (1) (2 , 1 , 0) x (2 k +1) (1 , 1 , 1) x ((2 k )(2 k +1)) (2 , 1 , 0) + x (1) (2 , 1 , 0) · k − 1 X j =0 x (2 k +1) (2 , 0 , 1) x (2 j +2) (1 , 1 , 1) x ((2 j +2)(2 j +3)) (2 , 1 , 0) x ((2 j +1)(2 j +2)) (2 , 0 , 1) + x (2 k +1) (2 , 0 , 1) x (2 j +1) (1 , 1 , 1) x ((2 j +1)(2 j +2)) (2 , 0 , 1) x ((2 j )(2 j +1)) (2 , 1 , 0) ! . I I. ( P ( m 1 , n 1 ) ) F or any k , ( l + 1) ∈ N , then: D [2 k − 1 , 2 l ] 2 = x ((2 k +2 l )(2 k +2 l +1)) (0 , 2 , 1) x (1) (2 , 1 , 0) x ((2 k )(2 k +1)) (0 , 2 , 1) + x ((2 k +2 l )(2 k +2 l +1)) (0 , 2 , 1) · · " l X j =1 x (2 k +2 j ) (1 , 2 , 0) x ((2 k +2 j )(2 k +2 j +1)) (0 , 2 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) + x (2 k +2 j − 1) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (2 , 0 , 1) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 2) !# · D [2 k − 1] 2 + x ((2 k +2 l )(2 k +2 l +1)) (0 , 2 , 1) · l X j =1 " 1 x (2 k +2 j ) (1 , 1 , 1) · x (2 k +2 j ) (1 , 2 , 0) x ((2 k +2 j )(2 k +2 j +1)) (0 , 1 , 2) x ((2 k +2 j )(2 k +2 j +1)) (0 , 2 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) + x (2 k +2 j ) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (2 , 0 , 1) ! · I [2 k − 1 , 2 j − 1] + 1 x (2 k +2 j − 1) (1 , 1 , 1) · x (2 k +2 j − 1) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) x ((2 k +2 j − 1)(2 k +2 j )) (2 , 0 , 1) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 2) + x (2 k +2 j − 1) (1 , 2 , 0) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 2 , 1) ! · I [2 k − 1 , 2 j − 2] # ; 50 D [2 k − 1 , 2 l +1] 2 = x ((2 k +2 l +1)(2 k +2 l +2)) (2 , 0 , 1) x (1) (2 , 1 , 0) x ((2 k )(2 k +1)) (0 , 2 , 1) + x ((2 k +2 l +1)(2 k +2 l +2)) (2 , 0 , 1) · " x (2 k +2 l +1) (2 , 1 , 0) x ((2 k +2 l +1)(2 k +2 l +2)) (2 , 0 , 1) x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 2) + l X j =1 x (2 k +2 j ) (1 , 2 , 0) x ((2 k +2 j )(2 k +2 j +1)) (0 , 2 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) + x (2 k +2 j − 1) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (2 , 0 , 1) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 2) !# · D [2 k − 1] 2 + x ((2 k +2 l +1)(2 k +2 l +2)) (2 , 0 , 1) · " 1 x (2 k +2 l +1) (1 , 1 , 1) · x (2 k +2 l +1) (2 , 1 , 0) x ((2 k +2 l +1)(2 k +2 l +2)) (1 , 0 , 2) x ((2 k +2 l +1)(2 k +2 l +2)) (2 , 0 , 1) x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 2) + x (2 k +2 l +1) (1 , 2 , 0) x ((2 k +2 l )(2 k +2 l +1)) (0 , 2 , 1) ! · I [2 k − 1 , 2 l ] + l X j =1 " 1 x (2 k +2 j ) (1 , 1 , 1) · x (2 k +2 j ) (1 , 2 , 0) x ((2 k +2 j )(2 k +2 j +1)) (0 , 1 , 2) x ((2 k +2 j )(2 k +2 j +1)) (0 , 2 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) + x (2 k +2 j ) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (2 , 0 , 1) ! · I [2 k − 1 , 2 j − 1] + 1 x (2 k +2 j − 1) (1 , 1 , 1) · x (2 k +2 j − 1) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) x ((2 k +2 j − 1)(2 k +2 j )) (2 , 0 , 1) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 2) + x (2 k +2 j − 1) (1 , 2 , 0) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 2 , 1) ! · I [2 k − 1 , 2 j − 2] ## ; D [2 k, 2 l ] 2 = x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 1 , 2) x (1) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (0 , 1 , 2) + x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 1 , 2) · " l X j =1 x (2 k +2 j +1) (1 , 0 , 2) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 1 , 2) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) + x (2 k +2 j ) (2 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 2 , 1) !# · D [2 k ] 2 + x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 1 , 2) · l X j =1 " 1 x (2 k +2 j +1) (1 , 1 , 1) · x (2 k +2 j +1) (1 , 0 , 2) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 2 , 1) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 1 , 2) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) + x (2 k +2 j +1) (2 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (2 , 1 , 0) ! · I [2 k, 2 j − 1] + 1 x (2 k +2 j ) (1 , 1 , 1) · x (2 k +2 j ) (2 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) x ((2 k +2 j )(2 k +2 j +1)) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 2 , 1) + x (2 k +2 j ) (1 , 0 , 2) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 1 , 2) ! · I [2 k, 2 j − 2] # ; D [2 k, 2 l +1] 2 = x ((2 k +2 l +2)(2 k +2 l +3)) (2 , 1 , 0) x (1) (2 , 1 , 0) x ((2 k +1)(2 k +2)) (0 , 1 , 2) + x ((2 k +2 l +2)(2 k +2 l +3)) (2 , 1 , 0) · " x (2 k +2 l +2) (2 , 0 , 1) x ((2 k +2 l +2)(2 k +2 l +3)) (2 , 1 , 0) x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 2 , 1) + l X j =1 x (2 k +2 j +1) (1 , 0 , 2) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 1 , 2) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) + x (2 k +2 j ) (2 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 2 , 1) !# · D [2 k ] 2 + x ((2 k +2 l +2)(2 k +2 l +3)) (2 , 1 , 0) · " 1 x (2 k +2 l +2) (1 , 1 , 1) · x (2 k +2 l +2) (2 , 0 , 1) x ((2 k +2 l +2)(2 k +2 l +3)) (1 , 2 , 0) x ((2 k +2 l +2)(2 k +2 l +3)) (2 , 1 , 0) x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 2 , 1) + x (2 k +2 l +2) (1 , 0 , 2) x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 1 , 2) ! · I [2 k, 2 l ] + l X j =1 " 1 x (2 k +2 j +1) (1 , 1 , 1) · x (2 k +2 j +1) (1 , 0 , 2) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 2 , 1) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 1 , 2) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) + x (2 k +2 j +1) (2 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (2 , 1 , 0) ! · I [2 k, 2 j − 1] + 1 x (2 k +2 j ) (1 , 1 , 1) · x (2 k +2 j ) (2 , 0 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) x ((2 k +2 j )(2 k +2 j +1)) (2 , 1 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 2 , 1) + x (2 k +2 j ) (1 , 0 , 2) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 1 , 2) ! · I [2 k, 2 j − 2] ## ; 51 I [2 k − 1 , 2 l ] = x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 2) x ((2 k )(2 k +1)) (0 , 1 , 2) · I [2 k − 1] + x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 2) · " l X j =1 x (2 k +2 j ) (1 , 1 , 1) x ((2 k +2 j )(2 k +2 j +1)) (0 , 1 , 2) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) + x (2 k +2 j − 1) (1 , 1 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 2) !# · D [2 k − 1] 2 ; I [2 k − 1 , 2 l +1] = x ((2 k +2 l +1)(2 k +2 l +2)) (1 , 0 , 2) x ((2 k )(2 k +1)) (0 , 1 , 2) · I [2 k − 1] + x ((2 k +2 l +1)(2 k +2 l +2)) (1 , 0 , 2) · " x (2 k +2 l +1) (1 , 1 , 1) x ((2 k +2 l +1)(2 k +2 l +2)) (1 , 0 , 2) x ((2 k +2 l )(2 k +2 l +1)) (0 , 1 , 2) + l X j =1 x (2 k +2 j ) (1 , 1 , 1) x ((2 k +2 j )(2 k +2 j +1)) (0 , 1 , 2) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) + x (2 k +2 j − 1) (1 , 1 , 1) x ((2 k +2 j − 1)(2 k +2 j )) (1 , 0 , 2) x ((2 k +2 j − 2)(2 k +2 j − 1)) (0 , 1 , 2) !# · D [2 k − 1] 2 ; I [2 k, 2 l ] = x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 2 , 1) x ((2 k +1)(2 k +2)) (0 , 2 , 1) · I [2 k ] + x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 2 , 1) · " l X j =1 x (2 k +2 j +1) (1 , 1 , 1) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 2 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) + x (2 k +2 j ) (1 , 1 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 2 , 1) !# · D [2 k ] 2 ; I [2 k, 2 l +1] = x ((2 k +2 l +2)(2 k +2 l +3)) (1 , 2 , 0) x ((2 k +1)(2 k +2)) (0 , 2 , 1) · I [2 k ] + x ((2 k +2 l +2)(2 k +2 l +3)) (1 , 2 , 0) · " x (2 k +2 l +2) (1 , 1 , 1) x ((2 k +2 l +2)(2 k +2 l +3)) (1 , 2 , 0) x ((2 k +2 l +1)(2 k +2 l +2)) (0 , 2 , 1) + l X j =1 x (2 k +2 j +1) (1 , 1 , 1) x ((2 k +2 j +1)(2 k +2 j +2)) (0 , 2 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) + x (2 k +2 j ) (1 , 1 , 1) x ((2 k +2 j )(2 k +2 j +1)) (1 , 2 , 0) x ((2 k +2 j − 1)(2 k +2 j )) (0 , 2 , 1) !# · D [2 k ] 2 . I II-1. (F or n k ≥ 1 ) F or any k , m 1 , n 1 , m 2 , n 2 , ..., m k , ( n k − 1) ∈ N , then the functions satisfy the fol lowing r elations: If n k = 2 l ≥ 2 is even: I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ,n k ] = x (( M k + N k +1)( M k + N k +2)) ω · (2 , 1 , 0) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω · (2 , 1 , 0) · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] + x (( M k + N k +1)( M k + N k +2)) ω · (2 , 1 , 0) · " l X j =1 x ( M k + N k − 1 +2 j +1) (1 , 1 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (2 , 1 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) + x ( M k + N k − 1 +2 j ) (1 , 1 , 1) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (2 , 1 , 0) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] 2 ; 52 D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ,n k ] 2 = x (( M k + N k +1)( M k + N k +2)) ω · (1 , 2 , 0) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω · (1 , 2 , 0) · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ] 2 + x (( M k + N k +1)( M k + N k +2)) ω · (1 , 2 , 0) · " l X j =1 x ( M k + N k − 1 +2 j +1) ω · (0 , 2 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (1 , 2 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) + x ( M k + N k − 1 +2 j ) ω · (0 , 1 , 2) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 2) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (2 , 1 , 0) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] 2 + x (( M k + N k +1)( M k + N k +2)) ω · (1 , 2 , 0) · l X j =1 " 1 x ( M k + N k − 1 +2 j +1) (1 , 1 , 1) · x ( M k + N k − 1 +2 j +1) ω · (0 , 2 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (2 , 1 , 0) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (1 , 2 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) + x ( M k + N k − 1 +2 j +1) ω · (0 , 1 , 2) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 2) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k , 2 j − 1] + 1 x ( M k + N k − 1 +2 j ) (1 , 1 , 1) · x ( M k + N k − 1 +2 j ) ω · (0 , 1 , 2) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 2) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (2 , 1 , 0) + x ( M k + N k − 1 +2 j ) ω · (0 , 2 , 1) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (1 , 2 , 0) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k , 2 j − 2] # . If n k = 2 l + 1 ≥ 1 is o dd: I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ,n k ] = x (( M k + N k +1)( M k + N k +2)) ω · (2 , 0 , 1) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω · (2 , 1 , 0) · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] + x (( M k + N k +1)( M k + N k +2)) ω · (2 , 0 , 1) · " x ( M k + N k +1) (1 , 1 , 1) x (( M k + N k +1)( M k + N k +2)) ω · (2 , 0 , 1) x (( M k + N k )( M k + N k +1)) ω · (2 , 1 , 0) + l X j =1 x ( M k + N k − 1 +2 j +1) (1 , 1 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (2 , 1 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) + x ( M k + N k − 1 +2 j ) (1 , 1 , 1) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (2 , 1 , 0) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] 2 ; 53 D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ,n k ] 2 = x (( M k + N k +1)( M k + N k +2)) ω · (1 , 0 , 2) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω · (1 , 2 , 0) · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ] 2 + x (( M k + N k +1)( M k + N k +2)) ω · (1 , 0 , 2) · " x ( M k + N k +1) ω · (0 , 1 , 2) x (( M k + N k +1)( M k + N k +2)) ω · (1 , 0 , 2) x (( M k + N k )( M k + N k +1)) ω · (2 , 1 , 0) + l X j =1 x ( M k + N k − 1 +2 j +1) ω · (0 , 2 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (1 , 2 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) + x ( M k + N k − 1 +2 j ) ω · (0 , 1 , 2) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 2) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (2 , 1 , 0) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] 2 + x (( M k + N k +1)( M k + N k +2)) ω · (1 , 0 , 2) · " 1 x ( M k + N k +1) (1 , 1 , 1) · x ( M k + N k +1) ω · (0 , 1 , 2) x (( M k + N k +1)( M k + N k +2)) ω · (2 , 0 , 1) x (( M k + N k +1)( M k + N k +2)) ω · (1 , 0 , 2) x (( M k + N k )( M k + N k +1)) ω · (2 , 1 , 0) + x ( M k + N k +1) ω · (0 , 2 , 1) x (( M k + N k )( M k + N k +1)) ω · (1 , 2 , 0) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k , 2 l ] + l X j =1 " 1 x ( M k + N k − 1 +2 j +1) (1 , 1 , 1) · x ( M k + N k − 1 +2 j +1) ω · (0 , 2 , 1) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (2 , 1 , 0) x (( M k + N k − 1 +2 j +1)( M k + N k − 1 +2 j +2)) ω · (1 , 2 , 0) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) + x ( M k + N k − 1 +2 j +1) ω · (0 , 1 , 2) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 2) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k , 2 j − 1] + 1 x ( M k + N k − 1 +2 j ) (1 , 1 , 1) · x ( M k + N k − 1 +2 j ) ω · (0 , 1 , 2) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (2 , 0 , 1) x (( M k + N k − 1 +2 j )( M k + N k − 1 +2 j +1)) ω · (1 , 0 , 2) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (2 , 1 , 0) + x ( M k + N k − 1 +2 j ) ω · (0 , 2 , 1) x (( M k + N k − 1 +2 j − 1)( M k + N k − 1 +2 j )) ω · (1 , 2 , 0) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k , 2 j − 2] ## . I II-2. (F or n k = 0 ) F or any ( k − 1) , m 1 , n 1 , m 2 , n 2 , ..., m k − 1 , n k − 1 , ( m k − 1) ∈ N , then the functions satisfy the fol lowing r elations: If m k = 2 l ≥ 2 is even: I [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k − 1 ,n k − 1 ,m k ] = x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (2 , 0 , 1) x (( M k − 1 + N k − 1 +1)( M k − 1 + N k − 1 +2)) ω ′ · (2 , 0 , 1) · I [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k − 1 ,n k − 1 ] + x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (2 , 0 , 1) · " l X j =1 x ( M k − 1 + N k − 1 +2 j +1) (1 , 1 , 1) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (2 , 0 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) + x ( M k − 1 + N k − 1 +2 j ) (1 , 1 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (2 , 0 , 1) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ] 2 ; 54 D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] 2 = x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 0 , 2) x (( M k − 1 + N k − 1 +1)( M k − 1 + N k − 1 +2)) ω ′ · (1 , 0 , 2) · D [ m 1 ,n 1 ,...,m k − 2 ,n k − 2 ,m k − 1 ] 2 + x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 0 , 2) · " l X j =1 x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 1 , 2) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (1 , 0 , 2) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) + x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 2 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 2 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (2 , 0 , 1) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ] 2 + x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 0 , 2) · l X j =1 " 1 x ( M k − 1 + N k − 1 +2 j +1) (1 , 1 , 1) · x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 1 , 2) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (2 , 0 , 1) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (1 , 0 , 2) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) + x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 2 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 2 , 0) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 , 2 j − 1] + 1 x ( M k − 1 + N k − 1 +2 j ) (1 , 1 , 1) · x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 2 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 2 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (2 , 0 , 1) + x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 1 , 2) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (1 , 0 , 2) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 , 2 j − 2] # . If m k = 2 l + 1 ≥ 1 is o dd: I [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k − 1 ,n k − 1 ,m k ] = x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (2 , 1 , 0) x (( M k − 1 + N k − 1 +1)( M k − 1 + N k − 1 +2)) ω ′ · (2 , 0 , 1) · I [ m 1 ,n 1 ,m 2 ,n 2 ,...,m k − 1 ,n k − 1 ] + x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (2 , 1 , 0) · " x ( M k + N k − 1 +1) (1 , 1 , 1) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (2 , 1 , 0) x (( M k + N k − 1 )( M k + N k − 1 +1)) ω ′ · (2 , 0 , 1) + l X j =1 x ( M k − 1 + N k − 1 +2 j +1) (1 , 1 , 1) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (2 , 0 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) + x ( M k − 1 + N k − 1 +2 j ) (1 , 1 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (2 , 0 , 1) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ] 2 ; 55 D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ,m k ] 2 = x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 2 , 0) x (( M k − 1 + N k − 1 +1)( M k − 1 + N k − 1 +2)) ω ′ · (1 , 0 , 2) · D [ m 1 ,n 1 ,...,m k − 2 ,n k − 2 ,m k − 1 ] 2 + x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 2 , 0) · " x ( M k + N k − 1 +1) ω ′ · (0 , 2 , 1) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 2 , 0) x (( M k + N k − 1 )( M k + N k − 1 +1)) ω ′ · (2 , 0 , 1) + l X j =1 x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 1 , 2) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (1 , 0 , 2) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) + x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 2 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 2 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (2 , 0 , 1) !# · D [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 ] 2 + x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 2 , 0) · " 1 x ( M k + N k − 1 +1) (1 , 1 , 1) · x ( M k + N k − 1 +1) ω ′ · (0 , 2 , 1) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (2 , 1 , 0) x (( M k + N k − 1 +1)( M k + N k − 1 +2)) ω ′ · (1 , 2 , 0) x (( M k + N k − 1 )( M k + N k − 1 +1)) ω ′ · (2 , 0 , 1) + x ( M k + N k − 1 +1) ω ′ · (0 , 1 , 2) x (( M k + N k − 1 )( M k + N k − 1 +1)) ω ′ · (1 , 0 , 2) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 , 2 l ] + l X j =1 " 1 x ( M k − 1 + N k − 1 +2 j +1) (1 , 1 , 1) · x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 1 , 2) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (2 , 0 , 1) x (( M k − 1 + N k − 1 +2 j +1)( M k − 1 + N k − 1 +2 j +2)) ω ′ · (1 , 0 , 2) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) + x ( M k − 1 + N k − 1 +2 j +1) ω ′ · (0 , 2 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 2 , 0) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 , 2 j − 1] + 1 x ( M k − 1 + N k − 1 +2 j ) (1 , 1 , 1) · x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 2 , 1) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (2 , 1 , 0) x (( M k − 1 + N k − 1 +2 j )( M k − 1 + N k − 1 +2 j +1)) ω ′ · (1 , 2 , 0) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (2 , 0 , 1) + x ( M k − 1 + N k − 1 +2 j ) ω ′ · (0 , 1 , 2) x (( M k − 1 + N k − 1 +2 j − 1)( M k − 1 + N k − 1 +2 j )) ω ′ · (1 , 0 , 2) ! · I [ m 1 ,n 1 ,...,m k − 1 ,n k − 1 , 2 j − 2] ## . 4.4 General n umber of terms prop ert y and examples Consider the n -triangulated T -triangulation of p olygon of t yp e P ( p 1 , p 2 , . . . , p N ) for p i > 0. Let a l ( p 1 , p 2 , . . . , p N ) b e the v alue of D [ p 1 ,p 2 ,...,p N ] l when all v ariables x ( r ) ( w 1 ,w 2 ,w 3 ) = 1. In other w ords a l ( p 1 , p 2 , . . . , p N ) is also the n umber of terms (coun ting multiplicities) in the Laurent expansion of D [ p 1 ,p 2 ,...,p N ] l . F or n = 1, w e simply write a ( p 1 , p 2 , . . . , p N ) := a 1 ( p 1 , p 2 , . . . , p N ). Let a ( ∅ ) := 1 (i.e. in case N = 0). Based on all p ossible cases ab o v e, we deduce the following recurrence: a ( p 1 ) = p 1 + 1; a ( p 1 , p 2 ) = p 1 p 2 + p 2 + 1; a ( p 1 , p 2 , . . . , p N +2 ) = p N +2 · a ( p 1 , p 2 , . . . , p N +1 ) + a ( p 1 , p 2 , . . . , p N ) for all N , p 1 , p 2 , . . . , p N +2 ∈ N . By Prop osition 2.10 , we conclude that a ( p 1 , p 2 , . . . , p N ) is precisely the nu- merator of the lo west form of the contin ued fraction [1; p 1 , p 2 , . . . , p N ]. F or general n , apply Theorem 4.8 , then thinking of a ( p 1 , p 2 , . . . , p N ) as the numerator of the low est form of the contin ued fraction [1; p 1 , p 2 , . . . , p N ], w e can deduce that a l ( p 1 , p 2 , . . . , p N ) = a ( p 1 , p 2 , . . . , p N ) l ( n +1 − l ) . Therefore, counting multiplicities when coun ting terms, w e obtain the follo wing corollary: Corollary 4.18. F or any n -triangulate d p olygon P ( p 1 , p 2 , . . . , p N ) , the gener al formula D [ p 1 ,p 2 ,...,p N ] l ( l = 1 , 2 , . . . , n ) of the diagonal has pr e cisely a ( p 1 , p 2 , . . . , p N ) l ( n +1 − l ) terms, wher e a ( p 1 , p 2 , . . . , p N ) is the nu- mer ator of the lowest form of the c ontinue d fr action [1; p 1 , p 2 , . . . , p N ] . 56 F urthermore, applying Theorem 4.7 , as any triangle in Figure 24 is w ell-triangulated, we get the follo wing consequence: Corollary 4.19. F or any n -triangulate d p olygon P ( p 1 , p 2 , . . . , p N ) , the gener al formula I [ p 1 ,p 2 ,...,p N ] ( i,j ) ( i, j ∈ N , i + j ≤ n ) has pr e cisely a ( p 1 , p 2 , . . . , p N ) i ( n +1 − i − j ) a ( p 1 , p 2 , . . . , p N − 1 ) j ( n +1 − i − j ) terms, wher e a ( p 1 , p 2 , . . . , p N ) is the numer ator of the lowest form of the c ontinue d fr action [1; p 1 , p 2 , . . . , p N ] . Returning to Theorem 4.16 and Theorem 4.17 , we see that the functions can b e computed inductively b y the linear com bination of the functions that are computed for smaller cases. Example 4.20. W e shall return to the example in Figure 26 . Recall the p olygon P (2 , 3 , 3 , 1 , 3 , 0), or equiv alen tly , P (2 , 3 , 3 , 1 , 3), the target is to calculate the sp ecific v alues D [2 , 3 , 3 , 1 , 3] i with i = 1 , 2 , . . . , n for b oth cases n = 1 and n = 2. W e shall apply Theorem 4.16 and Theorem 4.17 ab o ve. Indeed: (1) F or n = 1, w e only need to compute D [2 , 3 , 3 , 1 , 3] . Applying Theorem 4.16 , we get: D [2] = x (1) (1 , 1 , 0) x (3) (0 , 1 , 1) x (2)(3) (1 , 1 , 0) + x (1) (1 , 1 , 0) x (2) (0 , 1 , 1) x (3)(4) (1 , 0 , 1) x (1)(2) (1 , 0 , 1) x (2)(3) (1 , 1 , 0) + x (1) (0 , 1 , 1) x (3)(4) (1 , 0 , 1) x (1)(2) (1 , 0 , 1) ; D [2 , 3] = x (1) (1 , 1 , 0) x (6)(7) (1 , 1 , 0) x (3)(4) (0 , 1 , 1) + " x (6) (1 , 0 , 1) x (5)(6) (1 , 1 , 0) + x (5) (1 , 0 , 1) x (6)(7) (1 , 1 , 0) x (4)(5) (1 , 1 , 0) x (5)(6) (0 , 1 , 1) + x (4) (1 , 0 , 1) x (6)(7) (1 , 1 , 0) x (3)(4) (0 , 1 , 1) x (4)(5) (1 , 1 , 0) # · D [2] ; D [2 , 3 , 3] = x (9)(10) (1 , 0 , 1) x (6)(7) (1 , 1 , 0) · D [2] + x (9)(10) (1 , 0 , 1) · " x (9) (0 , 1 , 1) x (9)(10) (1 , 0 , 1) x (8)(9) (1 , 1 , 0) + x (8) (0 , 1 , 1) x (8)(9) (1 , 1 , 0) x (7)(8) (1 , 0 , 1) + x (7) (0 , 1 , 1) x (7)(8) (1 , 0 , 1) x (6)(7) (1 , 1 , 0) # · D [2 , 3] ; D [2 , 3 , 3 , 1] = x (10)(11) (0 , 1 , 1) x (9)(10) (1 , 0 , 1) · D [2 , 3] + x (10) (1 , 1 , 0) x (9)(10) (1 , 0 , 1) · D [2 , 3 , 3] ; D [2 , 3 , 3 , 1 , 3] = x (13)(14) (1 , 1 , 0) x (10)(11) (0 , 1 , 1) · D [2 , 3 , 3] + x (13)(14) (1 , 1 , 0) · " x (13) (1 , 0 , 1) x (13)(14) (1 , 1 , 0) x (12)(13) (0 , 1 , 1) + x (12) (1 , 0 , 1) x (12)(13) (0 , 1 , 1) x (11)(12) (1 , 1 , 0) + x (11) (1 , 0 , 1) x (11)(12) (1 , 1 , 0) x (10)(11) (0 , 1 , 1) # · D [2 , 3 , 3 , 1] whence we can directly imply the formula. As the formula is long, we shall omit the detail result. F or example D [2 , 3 , 3 , 1 , 3] consists of 162 Laurent monomials (up to multiplicities) with ev ery co efficient b eing 1. (2) F or n = 2, we need to compute D [2 , 3 , 3 , 1 , 3] 1 , D [2 , 3 , 3 , 1 , 3] 2 , and I [2 , 3 , 3 , 1 , 3] . W e shall only discuss how to apply Theorem 4.17 since the directed computation is v ery large and inappropriate to present in details. The theorem expresses D [2 , 3 , 3 , 1 , 3] 2 and I [2 , 3 , 3 , 1 , 3] as linear com binations of D [2 , 3 , 3] 2 , D [2 , 3 , 3 , 1] 2 , and all I [2 , 3 , 3 , 1 ,t ] for t = 0 , 1 , 2 (note I [2 , 3 , 3 , 1 , 0] = I [2 , 3 , 3 , 1] ). This application is applied b y the congruences ( − 1) 2 + ( − 1) 3 + ( − 1) 3 + ( − 1) 1 ≡ 1 (mo d 3) and 2 + 3 + 3 + 1 = 9 ≡ 1 (mo d 2). Iterating this pro cess, we express these quantities in terms of D [2 , 3] 2 , D [2 , 3 , 3] 2 , I [2 , 3 , 3] , and I [2 , 3 , 3 , 1] . Con tinuing recursiv ely (noting that the applicable cases must b e v erified at eac h step) even tually yields closed forms for D [2 , 3 , 3 , 1 , 3] 2 and I [2 , 3 , 3 , 1 , 3] . F or D [2 , 3 , 3 , 1 , 3] 1 , w e consider the reflected p olygon P (3 , 1 , 3 , 3 , 2) = P (2 , 3 , 3 , 1 , 3). After re-enumerating v ertices for the reflected polygon, w e apply Theorem 4.17 to D [3 , 1 , 3 , 3 , 2] 2 , and recov er the original v ariable assignmen ts to obtain the closed form for D [2 , 3 , 3 , 1 , 3] 1 . Each D [2 , 3 , 3 , 1 , 3] 1 and D [2 , 3 , 3 , 1 , 3] 2 consists of 162 2 = 26 , 244 Laurent monomials (up to multiplicities) with ev ery co efficien t b eing 1, while the same holds for I [2 , 3 , 3 , 1 , 3] whic h consists of 6 , 966 Laurent monomials (up to multiplicities). Example 4.21. Consider the following 3-triangulated 5-gon: 57 D C B A E y 3 y 1 y 2 x 6 x 4 x 5 x 11 x 1 x 2 x 3 y 14 y 13 x 9 y 15 x 12 x 10 y 5 y 4 y 6 x 7 y 12 x 8 y 11 y 10 x 13 x 14 y 9 x 15 y 7 y 8 Figure 39: m = 5, n = 3 In order to reach the diagonal B E , we can use an y flip sequence from Figure 8 . Then we can compute: V 2 V 4 = ( x ′ 10 , x ′′ 5 , x ′ 14 ); V 3 V 5 = ( x ′ 15 , x ′′ 2 , x ′ 7 ); V 2 V 5 = ( X 1 2 , 5 , X 2 2 , 5 , X 3 2 , 5 ); x ′ 10 = y 6 y 10 x 6 + y 7 x 1 x 4 + y 4 y 7 x 13 x 4 x 11 + y 6 y 9 x 15 x 6 x 12 + y 5 y 7 x 15 x 5 x 10 + y 6 y 8 x 13 x 5 x 10 + y 6 y 8 x 11 x 15 x 5 x 10 x 12 + y 5 y 7 x 12 x 13 x 5 x 10 x 11 ; x ′′ 5 = y 5 y 11 x 5 + y 8 x 2 x 5 + y 5 y 9 x 14 x 6 x 11 + y 4 y 8 x 14 x 4 x 12 + y 11 x 1 x 10 x 4 x 15 + y 10 x 2 x 10 x 6 x 13 + y 5 y 11 x 12 x 13 x 5 x 11 x 15 + y 4 y 11 x 10 x 13 x 4 x 11 x 15 + y 5 y 10 x 12 x 14 x 6 x 11 x 15 + y 8 x 2 x 11 x 15 x 5 x 12 x 13 + y 8 x 1 x 11 x 14 x 4 x 12 x 13 + y 9 x 2 x 10 x 15 x 6 x 12 x 13 + y 4 y 10 x 5 x 10 x 14 x 4 x 6 x 11 x 15 + y 4 y 9 x 5 x 10 x 14 x 4 x 6 x 11 x 12 + y 10 x 1 x 5 x 10 x 14 x 4 x 6 x 13 x 15 + y 9 x 1 x 5 x 10 x 14 x 4 x 6 x 12 x 13 ; x ′ 14 = y 4 y 12 x 4 + y 9 x 3 x 6 + y 12 x 1 x 11 x 4 x 13 + y 10 x 3 x 12 x 6 x 15 + y 11 x 3 x 11 x 5 x 14 + y 12 x 2 x 12 x 5 x 14 + y 11 x 3 x 12 x 13 x 5 x 14 x 15 + y 12 x 2 x 11 x 15 x 5 x 13 x 14 ; x ′ 15 = y 3 y 10 x 1 + y 13 x 6 x 3 + y 10 x 4 x 9 x 1 x 13 + y 12 x 6 x 8 x 3 x 14 + y 10 x 5 x 8 x 2 x 15 + y 11 x 6 x 9 x 2 x 15 + y 11 x 6 x 8 x 13 x 2 x 14 x 15 + y 10 x 5 x 9 x 14 x 2 x 13 x 15 ; x ′′ 2 = y 2 y 11 x 2 + y 14 x 5 x 2 + y 3 y 14 x 15 x 1 x 8 + y 2 y 13 x 15 x 3 x 9 + y 12 x 5 x 7 x 3 x 13 + y 11 x 4 x 7 x 1 x 14 + y 2 y 11 x 8 x 13 x 2 x 9 x 14 + y 3 y 11 x 7 x 13 x 1 x 9 x 14 + y 2 y 12 x 8 x 15 x 3 x 9 x 14 + y 14 x 5 x 9 x 14 x 2 x 8 x 13 + y 14 x 4 x 9 x 15 x 1 x 8 x 13 + y 13 x 5 x 7 x 14 x 3 x 8 x 13 + y 3 y 13 x 2 x 7 x 15 x 1 x 3 x 8 x 9 + y 3 y 12 x 2 x 7 x 15 x 1 x 3 x 9 x 14 + y 13 x 2 x 4 x 7 x 15 x 1 x 3 x 8 x 13 + y 12 x 2 x 4 x 7 x 15 x 1 x 3 x 13 x 14 ; x ′ 7 = y 1 y 12 x 3 + y 15 x 4 x 1 + y 3 y 15 x 13 x 1 x 9 + y 1 y 13 x 14 x 3 x 8 + y 1 y 14 x 13 x 2 x 7 + y 2 y 15 x 14 x 2 x 7 + y 1 y 14 x 9 x 14 x 2 x 7 x 8 + y 2 y 15 x 8 x 13 x 2 x 7 x 9 ; X 1 2 , 5 = y 3 y 7 x 4 + y 6 y 13 x 3 + y 3 y 6 y 10 x 1 x 6 + y 6 y 12 x 8 x 3 x 14 + y 6 y 11 x 9 x 2 x 15 + y 4 y 7 x 9 x 1 x 11 + y 5 y 7 x 8 x 2 x 10 + y 3 y 4 y 7 x 13 x 1 x 4 x 11 + y 3 y 6 y 9 x 15 x 1 x 6 x 12 + y 3 y 6 y 8 x 13 x 1 x 5 x 10 + y 6 y 10 x 4 x 9 x 1 x 6 x 13 + y 6 y 10 x 5 x 8 x 2 x 6 x 15 + y 6 y 11 x 8 x 13 x 2 x 14 x 15 + y 6 y 9 x 5 x 8 x 2 x 6 x 12 + y 5 y 7 x 9 x 14 x 2 x 10 x 13 + y 6 y 8 x 4 x 9 x 1 x 5 x 10 + y 3 y 5 y 7 x 15 x 1 x 5 x 10 + y 6 y 8 x 8 x 11 x 2 x 10 x 12 + y 3 y 6 y 8 x 11 x 15 x 1 x 5 x 10 x 12 + y 3 y 5 y 7 x 12 x 13 x 1 x 5 x 10 x 11 + y 6 y 10 x 5 x 9 x 14 x 2 x 6 x 13 x 15 + y 6 y 9 x 5 x 9 x 14 x 2 x 6 x 12 x 13 + y 6 y 9 x 4 x 9 x 15 x 1 x 6 x 12 x 13 + y 5 y 7 x 4 x 9 x 15 x 1 x 5 x 10 x 13 + y 6 y 8 x 9 x 11 x 14 x 2 x 10 x 12 x 13 + y 5 y 7 x 4 x 9 x 12 x 1 x 5 x 10 x 11 + y 6 y 8 x 4 x 9 x 11 x 15 x 1 x 5 x 10 x 12 x 13 ; 58 X 2 2 , 5 = y 5 y 14 x 2 + y 2 y 8 x 5 + y 2 y 5 y 11 x 2 x 5 + y 5 y 12 x 7 x 3 x 13 + y 2 y 10 x 10 x 6 x 13 + y 3 y 14 x 10 x 4 x 8 + y 4 y 8 x 7 x 1 x 12 + y 2 y 5 y 9 x 14 x 2 x 6 x 11 + y 2 y 5 y 13 x 15 x 3 x 5 x 9 + y 2 y 4 y 8 x 14 x 2 x 4 x 12 + y 3 y 5 y 14 x 15 x 1 x 5 x 8 + y 2 y 11 x 1 x 10 x 2 x 4 x 15 + y 5 y 11 x 4 x 7 x 1 x 5 x 14 + y 2 y 13 x 1 x 10 x 3 x 4 x 9 + y 5 y 9 x 4 x 7 x 1 x 6 x 11 + y 2 y 9 x 10 x 15 x 6 x 12 x 13 + y 5 y 13 x 7 x 14 x 3 x 8 x 13 + y 2 y 8 x 11 x 15 x 5 x 12 x 13 + y 5 y 14 x 9 x 14 x 2 x 8 x 13 + y 3 y 8 x 7 x 11 x 4 x 9 x 12 + y 4 y 14 x 9 x 10 x 1 x 8 x 11 + y 2 y 5 y 10 x 12 x 14 x 2 x 6 x 11 x 15 + y 2 y 5 y 12 x 8 x 15 x 3 x 5 x 9 x 14 + y 2 y 5 y 11 x 8 x 13 x 2 x 5 x 9 x 14 + y 2 y 5 y 11 x 12 x 13 x 2 x 5 x 11 x 15 + y 2 y 5 y 9 x 8 x 13 x 2 x 6 x 9 x 11 + y 2 y 5 y 13 x 12 x 13 x 3 x 5 x 9 x 11 + y 2 y 4 y 8 x 8 x 13 x 2 x 4 x 9 x 12 + y 3 y 5 y 14 x 12 x 13 x 1 x 5 x 8 x 11 + y 2 y 4 y 11 x 10 x 13 x 2 x 4 x 11 x 15 + y 3 y 5 y 11 x 7 x 13 x 1 x 5 x 9 x 14 + y 2 y 4 y 13 x 10 x 13 x 3 x 4 x 9 x 11 + y 3 y 5 y 9 x 7 x 13 x 1 x 6 x 9 x 11 + y 3 y 4 y 8 x 7 x 13 x 1 x 4 x 9 x 12 + y 3 y 4 y 14 x 10 x 13 x 1 x 4 x 8 x 11 + y 2 y 8 x 1 x 8 x 11 x 2 x 4 x 9 x 12 + y 5 y 14 x 4 x 9 x 12 x 1 x 5 x 8 x 11 + y 2 y 12 x 1 x 8 x 10 x 3 x 4 x 9 x 14 + y 5 y 10 x 4 x 7 x 12 x 1 x 6 x 11 x 15 + y 2 y 8 x 1 x 11 x 14 x 2 x 4 x 12 x 13 + y 5 y 14 x 4 x 9 x 15 x 1 x 5 x 8 x 13 + y 3 y 9 x 5 x 7 x 10 x 4 x 6 x 9 x 12 + y 4 y 13 x 2 x 7 x 10 x 1 x 3 x 8 x 11 + y 3 y 10 x 5 x 7 x 10 x 4 x 6 x 9 x 15 + y 4 y 12 x 2 x 7 x 10 x 1 x 3 x 11 x 14 + y 3 y 11 x 7 x 10 x 13 x 4 x 9 x 14 x 15 + y 4 y 11 x 7 x 10 x 13 x 1 x 11 x 14 x 15 + y 3 y 12 x 2 x 7 x 10 x 3 x 4 x 9 x 14 + y 4 y 10 x 5 x 7 x 10 x 1 x 6 x 11 x 15 + y 3 y 13 x 2 x 7 x 10 x 3 x 4 x 8 x 9 + y 4 y 9 x 5 x 7 x 10 x 1 x 6 x 11 x 12 + y 2 y 5 y 10 x 8 x 12 x 13 x 2 x 6 x 9 x 11 x 15 + y 2 y 5 y 12 x 8 x 12 x 13 x 3 x 5 x 9 x 11 x 14 + y 2 y 4 y 9 x 5 x 10 x 14 x 2 x 4 x 6 x 11 x 12 + y 3 y 5 y 13 x 2 x 7 x 15 x 1 x 3 x 5 x 8 x 9 + y 2 y 4 y 12 x 8 x 10 x 13 x 3 x 4 x 9 x 11 x 14 + y 3 y 5 y 10 x 7 x 12 x 13 x 1 x 6 x 9 x 11 x 15 + y 2 y 4 y 10 x 5 x 10 x 14 x 2 x 4 x 6 x 11 x 15 + y 3 y 5 y 12 x 2 x 7 x 15 x 1 x 3 x 5 x 9 x 14 + y 2 y 9 x 1 x 5 x 8 x 10 x 2 x 4 x 6 x 9 x 12 + y 5 y 13 x 2 x 4 x 7 x 12 x 1 x 3 x 5 x 8 x 11 + y 2 y 9 x 1 x 5 x 10 x 14 x 2 x 4 x 6 x 12 x 13 + y 5 y 13 x 2 x 4 x 7 x 15 x 1 x 3 x 5 x 8 x 13 + y 2 y 10 x 1 x 5 x 8 x 10 x 2 x 4 x 6 x 9 x 15 + y 5 y 12 x 2 x 4 x 7 x 12 x 1 x 3 x 5 x 11 x 14 + y 2 y 10 x 1 x 5 x 10 x 14 x 2 x 4 x 6 x 13 x 15 + y 5 y 12 x 2 x 4 x 7 x 15 x 1 x 3 x 5 x 13 x 14 + y 2 y 11 x 1 x 8 x 10 x 13 x 2 x 4 x 9 x 14 x 15 + y 5 y 11 x 4 x 7 x 12 x 13 x 1 x 5 x 11 x 14 x 15 + y 2 y 4 y 9 x 5 x 8 x 10 x 13 x 2 x 4 x 6 x 9 x 11 x 12 + y 3 y 5 y 13 x 2 x 7 x 12 x 13 x 1 x 3 x 5 x 8 x 9 x 11 + y 2 y 4 y 10 x 5 x 8 x 10 x 13 x 2 x 4 x 6 x 9 x 11 x 15 + y 3 y 5 y 12 x 2 x 7 x 12 x 13 x 1 x 3 x 5 x 9 x 11 x 14 + y 3 y 4 y 9 x 5 x 7 x 10 x 13 x 1 x 4 x 6 x 9 x 11 x 12 + y 3 y 4 y 13 x 2 x 7 x 10 x 13 x 1 x 3 x 4 x 8 x 9 x 11 + y 3 y 4 y 10 x 5 x 7 x 10 x 13 x 1 x 4 x 6 x 9 x 11 x 15 + y 3 y 4 y 12 x 2 x 7 x 10 x 13 x 1 x 3 x 4 x 9 x 11 x 14 + y 2 y 5 y 11 x 8 x 12 x 2 13 x 2 x 5 x 9 x 11 x 14 x 15 + y 2 y 4 y 11 x 8 x 10 x 2 13 x 2 x 4 x 9 x 11 x 14 x 15 + y 3 y 5 y 11 x 7 x 12 x 2 13 x 1 x 5 x 9 x 11 x 14 x 15 + y 3 y 4 y 11 x 7 x 10 x 2 13 x 1 x 4 x 9 x 11 x 14 x 15 ; X 3 2 , 5 = y 4 y 15 x 1 + y 1 y 9 x 6 + y 1 y 4 y 12 x 3 x 4 + y 1 y 10 x 12 x 6 x 15 + y 1 y 11 x 11 x 5 x 14 + y 3 y 15 x 11 x 4 x 9 + y 2 y 15 x 12 x 5 x 7 + y 3 y 4 y 15 x 13 x 1 x 4 x 9 + y 1 y 4 y 13 x 14 x 3 x 4 x 8 + y 1 y 4 y 14 x 13 x 2 x 4 x 7 + y 2 y 4 y 15 x 14 x 2 x 4 x 7 + y 1 y 12 x 1 x 11 x 3 x 4 x 13 + y 1 y 12 x 2 x 12 x 3 x 5 x 14 + y 1 y 11 x 12 x 13 x 5 x 14 x 15 + y 1 y 13 x 2 x 12 x 3 x 5 x 8 + y 2 y 15 x 11 x 15 x 5 x 7 x 13 + y 1 y 14 x 1 x 11 x 2 x 4 x 7 + y 1 y 14 x 9 x 12 x 5 x 7 x 8 + y 1 y 4 y 14 x 9 x 14 x 2 x 4 x 7 x 8 + y 2 y 4 y 15 x 8 x 13 x 2 x 4 x 7 x 9 + y 1 y 12 x 2 x 11 x 15 x 3 x 5 x 13 x 14 + y 1 y 13 x 2 x 11 x 15 x 3 x 5 x 8 x 13 + y 1 y 13 x 1 x 11 x 14 x 3 x 4 x 8 x 13 + y 2 y 15 x 1 x 11 x 14 x 2 x 4 x 7 x 13 + y 1 y 14 x 9 x 11 x 15 x 5 x 7 x 8 x 13 + y 2 y 15 x 1 x 8 x 11 x 2 x 4 x 7 x 9 + y 1 y 14 x 1 x 9 x 11 x 14 x 2 x 4 x 7 x 8 x 13 . This result verifies that the num b er of monomials in the Laurent p olynomial in each co ordinate of the expansion formula of diagonal V 2 V 5 is (27 , 81 , 27) = (3 3 , 3 4 , 3 3 ), whic h satisfies the Number of Monomials Theorem 4.8 since in the base case (cluster realization of type A 1 ), the num b er of monomials (counting m ultiplicities) of diagonal B E is 3. Clearly , eac h co efficien t of every component is 1. W e shall end the section with a conjecture and the hop e ab out finding general recursive form ulas for general n . Based on the observ ation ab o ve, w e hav e that the conjecture holds for n = 1 , 2. Conjecture 4.22 (Multiplicity free) . Fix a non- T diagonal γ = AB in a gener al n -triangulate d m -gon with T -triangulation of typ e P ( m 1 , n 1 , . . . , m k , n k ) . Then for l = 1 , 2 , ..., n and i, j ∈ N such that i + j ≤ n , every function D [ m 1 ,n 1 ,...,m k ,n k ] l and I [ m 1 ,n 1 ,...,m k ,n k ] ( i,j ) is a L aur ent p olynomial of variables x ( r ) ( a,b,c ) wher e r = 1 , 2 , ..., ( m − 2) and ( a, b, c ) ∈ Γ n +1 with e ach c o efficient b eing 1 . 5 General expansion formulas of cluster realization of t yp e G 2 in 4 -gon In this section, we are going to illustrate ho w the basic quiver is asso ciated with a surface in the cluster realization where G is of t yp e G 2 [ 18 ], and compute the cluster expansion formula related to the flip of the diagonal ov er a quadrilateral. 59 5.1 Cluster realization of t yp e G 2 lo cal systems Definition 5.1. F or any triangle of a triangulation of a surface S , the typ e G 2 b asic quiver can be dra wn as either one of the tw o wa ys b elow (Figure 40 and Figure 41 ), corresp onding to the longest reduced expression b eing w 0 = s 2 s 1 s 2 s 1 s 2 s 1 or w 0 = s 1 s 2 s 1 s 2 s 1 s 2 resp ectiv ely , where the index 1 belongs to the long ro ot while 2 b elongs to the short ro ot. L 1 L 2 L 3 L 4 L 5 S 1 S 2 S 3 S 4 S 5 S 5 L 5 S 1 S 2 S 3 S 4 L 1 L 2 L 3 L 4 Figure 40: The type G 2 quiv er on eac h triangle for reduced wor d w 0 = s 2 s 1 s 2 s 1 s 2 s 1 L 1 L 2 L 3 L 4 L 5 S 1 S 2 S 3 S 4 S 5 S 1 L 1 L 4 S 3 L 3 S 5 S 4 S 2 L 2 L 5 Figure 41: The type G 2 quiv er on eac h triangle for reduced wor d w 0 = s 1 s 2 s 1 s 2 s 1 s 2 Here for the arrows: (i) Red dotted arro w (dashed, thick) refers to half of an undashed, thick arro w; (ii) Undashed, thin arro w refers to 1 3 of an undashed, thick arro w. Moreo ver, for the vertices (ignoring the color on the figure ab o ve): (i) Blac k v ertex refers to the vertex corresponding to the long root; (ii) Blue v ertex refers to the vertex corresponding to the short ro ot. Definition 5.2. The flip se quenc e of typ e G 2 (i.e. the sequence of mutations to get the quiver after the flip) can b e applied in tw o w ays, where the sequence can b e applied in the following order (b elo w, b y ... → a → b → ... we mean m utating at vertex a first, then b , and so on): (1) The flips sequence µ 1 : µ 1 = { x 1 → x 7 → x 8 → x 9 → x 7 → x 2 → x 9 → x 7 → x 2 → x 8 → x 7 → x 10 → x 6 → x 7 → x 8 → x 2 → x 7 → x 5 → x 2 → x 7 → x 5 → x 8 → x 7 → x 9 → x 3 → x 7 → x 8 → x 5 → x 7 → x 4 → x 5 → x 7 → x 4 → x 8 → x 7 → x 2 } 60 where the long roots (satisfying d i = 1) are { x 1 , x 3 , x 6 , x 7 , x 8 , y 2 , y 3 , y 5 , y 8 } and short roots (satisfying d i = 1 3 ) are { x 2 , x 4 , x 5 , x 9 , x 10 , y 1 , y 4 , y 6 , y 7 } . µ 1 y 1 y 2 y 7 x 10 x 9 x 2 y 8 x 7 x 8 x 1 y 5 x 4 x 3 y 4 y 6 x 5 x 6 y 3 A B C D y 1 y 2 x (5) 2 x (3) 9 x (1) 10 y 4 x (1) 3 x (1) 6 x (1) 1 y 3 y 7 y 8 y 5 x (4) 5 x (12) 7 y 6 x (2) 4 x (6) 8 A B C D Figure 42: Mutation sequence µ 1 for type G 2 (2) The flips sequence µ 2 : µ 2 = { x 1 → x 2 → x 7 → x 8 → x 10 → x 7 → x 9 → x 10 → x 7 → x 9 → x 8 → x 7 → x 6 → x 5 → x 7 → x 8 → x 9 → x 7 → x 2 → x 9 → x 7 → x 2 → x 8 → x 7 → x 3 → x 4 → x 7 → x 8 → x 2 → x 7 → x 5 → x 2 → x 7 → x 5 → x 8 → x 7 } where the long roots (satisfying d i = 1) are { x 1 , x 3 , x 6 , x 7 , x 8 , y 2 , y 3 , y 5 , y 7 } and short roots (satisfying d i = 1 3 ) are { x 2 , x 4 , x 5 , x 9 , x 10 , y 1 , y 4 , y 6 , y 8 } . µ 2 y 1 y 2 y 7 x 10 x 9 x 2 y 8 x 7 x 8 x 1 y 5 x 4 x 3 y 4 y 6 x 5 x 6 y 3 A B C D y 1 y 2 x (5) 2 x (4) 9 x (1) 10 y 4 x (1) 3 x (1) 6 x (1) 1 y 3 y 7 y 8 y 5 x (3) 5 x (12) 7 y 6 x (2) 4 x (6) 8 A B C D Figure 43: Mutation sequence µ 2 for type G 2 By drawing the triangulation using equilateral triangles as ab o ve, an imp ortan t observ ation from [ 18 ] is that after applying the m utation sequence, the resulting quiver o ver eac h triangle is still the same as one of the tw o basic ones from Figure 40 and 41 , but up to a rotation. 5.2 Detailed calculations Prop osition 5.3. In the quiver for typ e G 2 lo c al system, we have the fol lowing pr op erties: (i) F or any arr ow of weight w ij c onne cting vertic es i and j (of c ourse w ij ≥ 0 ), then the c o efficients satisfy b ij = w ij d i , b j i = − w ij d j . 61 (ii) If the arr ow is undashe d and thick, then: b ij = ( 1 if the vertex i is black 3 if the vertex i is blue ; b j i = ( − 1 if the vertex j is black − 3 if the vertex j is blue . F or the flip sequence µ 1 , we get the new expansion formulas of the v ertices of the form: ( y 1 , y 2 , ..., y 8 , x 1 , x 2 , ..., x 10 ) → ( y 1 , y 2 , ..., y 8 , x (1) 1 , x (5) 2 , x (1) 3 , x (2) 4 , x (4) 5 , x (1) 6 , x (12) 7 , x (6) 8 , x (3) 9 , x (1) 10 ) . W e shall omit the results of remaining vertices whereas the form ulas are too long. In the expressions b elow, eac h fraction is written in a canonical form with all y -v ariables (in increasing index) preceding all x -v ariables (also increasing index) in the numerator, and the terms are ordered by the num b er of distinct v ariables follo wed b y their exp onen ts. W e finally get the diagonal expansion form ula after flip: B D = ( x (1) 3 , x (5) 2 ) where: x (5) 2 = y 1 y 6 x 2 + y 4 y 7 x 2 + 2 y 1 y 4 x 4 x 10 x 5 x 9 + y 1 y 4 y 8 x 5 x 6 x 10 + y 1 y 4 y 5 x 9 x 4 x 8 + y 1 y 4 x 3 x 2 10 x 2 5 x 7 + y 1 y 4 x 2 4 x 7 x 3 x 2 9 + y 4 y 7 x 5 x 7 x 6 x 9 x 10 + 2 y 4 y 7 x 4 x 9 x 2 x 5 x 10 + y 4 y 7 x 5 x 8 x 2 x 6 x 9 + y 1 y 6 x 6 x 9 x 2 x 5 x 8 + 2 y 1 y 6 x 5 x 10 x 2 x 4 x 9 + y 1 y 6 x 3 x 9 x 4 x 5 x 8 + y 2 y 6 y 7 x 6 x 1 x 5 x 10 + y 3 y 6 y 7 x 8 x 1 x 5 x 10 + y 2 y 6 y 7 x 6 x 1 x 4 x 9 + y 3 y 6 y 7 x 8 x 1 x 4 x 9 + 2 y 6 y 7 x 5 x 9 x 2 2 x 4 x 10 + y 1 y 6 x 2 5 x 7 x 4 x 6 x 2 9 + y 1 y 6 x 6 x 2 10 x 2 2 x 4 x 7 + y 4 y 7 x 2 4 x 8 x 2 2 x 3 x 10 + y 4 y 7 x 3 x 2 9 x 2 5 x 8 x 10 + y 3 y 6 y 7 x 2 x 7 x 1 x 4 x 9 x 10 + y 1 y 3 y 6 y 8 x 2 x 1 x 4 x 10 + y 2 y 4 y 5 y 7 x 2 x 1 x 4 x 10 + y 6 y 7 x 6 x 2 9 x 2 2 x 4 x 8 x 10 + y 6 y 7 x 2 5 x 8 x 2 2 x 4 x 6 x 10 + 2 y 1 y 4 x 4 x 6 x 2 10 x 2 x 2 5 x 7 + 2 y 1 y 4 x 2 4 x 8 x 10 x 2 x 3 x 2 9 + y 2 y 4 y 7 x 2 2 x 3 x 1 x 2 5 x 9 + y 1 y 3 y 6 x 2 2 x 7 x 1 x 5 x 2 9 + y 2 y 6 y 7 x 2 x 3 x 1 x 4 x 5 x 10 + 2 y 4 y 7 x 4 x 6 x 2 9 x 2 x 2 5 x 8 x 10 + 2 y 1 y 6 x 2 5 x 8 x 10 x 2 x 4 x 6 x 2 9 + 2 y 1 y 6 x 5 x 8 x 2 10 x 2 2 x 4 x 7 x 9 + 2 y 4 y 7 x 2 4 x 6 x 9 x 2 2 x 3 x 5 x 10 + 2 y 1 y 4 x 4 x 8 x 2 10 x 2 x 5 x 7 x 9 + 2 y 1 y 4 x 2 4 x 6 x 10 x 2 x 3 x 5 x 9 + y 1 y 4 x 2 4 x 2 8 x 2 10 x 2 2 x 3 x 7 x 2 9 + y 1 y 4 x 2 4 x 2 6 x 2 10 x 2 2 x 3 x 2 5 x 7 + 2 y 2 y 4 y 7 x 2 x 4 x 6 x 1 x 2 5 x 9 + 2 y 3 y 4 y 7 x 2 x 4 x 8 x 1 x 2 5 x 9 + 2 y 1 y 2 y 6 x 2 x 6 x 10 x 1 x 5 x 2 9 + 2 y 1 y 3 y 6 x 2 x 8 x 10 x 1 x 5 x 2 9 + y 1 y 3 y 6 x 2 8 x 2 10 x 1 x 5 x 7 x 2 9 + y 2 y 4 y 7 x 2 4 x 2 6 x 1 x 3 x 2 5 x 9 + 2 y 1 y 6 y 8 x 5 x 2 9 x 2 2 x 4 x 7 x 10 + 2 y 4 y 5 y 7 x 2 5 x 9 x 2 2 x 3 x 4 x 10 + 2 y 1 y 4 y 8 x 4 x 2 9 x 2 x 5 x 7 x 10 + 2 y 1 y 4 y 5 x 2 5 x 10 x 2 x 3 x 4 x 9 + 2 y 1 y 3 y 4 y 8 x 2 2 x 4 x 1 x 2 5 x 10 + 2 y 1 y 2 y 4 y 5 x 2 2 x 10 x 1 x 4 x 2 9 + y 1 y 4 y 8 x 3 x 3 9 x 2 5 x 7 x 8 x 10 + y 1 y 4 y 5 x 3 5 x 7 x 3 x 4 x 6 x 2 9 + y 3 y 4 y 5 y 7 x 5 x 8 x 1 x 3 x 4 x 9 + y 1 y 2 y 6 y 8 x 2 x 6 x 1 x 4 x 8 x 10 + y 3 y 4 y 5 y 7 x 2 x 8 x 1 x 4 x 6 x 10 + y 1 y 2 y 6 y 8 x 6 x 9 x 1 x 5 x 7 x 10 + y 1 y 3 y 6 y 8 x 8 x 9 x 1 x 5 x 7 x 10 + y 2 y 4 y 5 y 7 x 5 x 6 x 1 x 3 x 4 x 9 + y 1 y 6 x 2 5 x 2 8 x 2 10 x 2 2 x 4 x 6 x 7 x 2 9 + y 4 y 7 x 2 4 x 2 6 x 2 9 x 2 2 x 3 x 2 5 x 8 x 10 + y 1 y 2 y 6 x 6 x 8 x 2 10 x 1 x 5 x 7 x 2 9 + y 3 y 4 y 7 x 2 4 x 6 x 8 x 1 x 3 x 2 5 x 9 + y 1 y 4 y 5 x 5 x 6 x 2 10 x 2 2 x 3 x 4 x 7 + y 3 y 4 y 7 x 2 2 x 3 x 8 x 1 x 2 5 x 6 x 9 + 2 y 1 y 2 y 6 x 2 2 x 3 x 10 x 1 x 4 x 5 x 2 9 + 2 y 3 y 4 y 7 x 2 2 x 4 x 7 x 1 x 2 5 x 9 x 10 + y 1 y 2 y 6 x 2 2 x 6 x 7 x 1 x 5 x 8 x 2 9 + y 1 y 4 y 8 x 2 4 x 8 x 9 x 2 2 x 3 x 7 x 10 + y 1 y 6 y 8 x 6 x 3 9 x 2 2 x 4 x 7 x 8 x 10 + y 4 y 5 y 7 x 3 5 x 8 x 2 2 x 3 x 4 x 6 x 10 + y 1 y 2 y 4 y 8 x 3 2 x 3 x 1 x 2 5 x 8 x 10 + y 1 y 3 y 4 y 8 x 3 2 x 3 x 1 x 2 5 x 6 x 10 + y 1 y 2 y 4 y 5 x 3 2 x 7 x 1 x 4 x 8 x 2 9 + y 1 y 3 y 4 y 5 x 3 2 x 7 x 1 x 4 x 6 x 2 9 + y 3 y 6 y 7 x 2 x 3 x 8 x 1 x 4 x 5 x 6 x 10 + y 2 y 6 y 7 x 2 x 6 x 7 x 1 x 4 x 8 x 9 x 10 + y 1 y 2 y 4 y 5 y 8 x 2 x 9 x 1 x 4 x 7 x 10 + y 1 y 3 y 4 y 5 y 8 x 2 x 5 x 1 x 3 x 4 x 10 + 2 y 1 y 4 x 2 4 x 6 x 8 x 2 10 x 2 2 x 3 x 5 x 7 x 9 + y 1 y 6 y 8 x 2 5 x 8 x 9 x 2 2 x 4 x 6 x 7 x 10 + y 4 y 5 y 7 x 5 x 6 x 2 9 x 2 2 x 3 x 4 x 8 x 10 + 2 y 1 y 4 y 5 x 2 5 x 8 x 2 10 x 2 2 x 3 x 4 x 7 x 9 + 2 y 1 y 4 y 8 x 2 4 x 6 x 2 9 x 2 2 x 3 x 5 x 7 x 10 + 2 y 1 y 2 y 4 y 8 x 2 2 x 4 x 6 x 1 x 2 5 x 8 x 10 + 2 y 1 y 3 y 4 y 5 x 2 2 x 8 x 10 x 1 x 4 x 6 x 2 9 + y 1 y 2 y 4 y 5 x 2 x 8 x 2 10 x 1 x 4 x 7 x 2 9 + y 1 y 3 y 4 y 8 x 2 x 2 4 x 6 x 1 x 3 x 2 5 x 10 + 2 y 1 y 4 y 5 y 8 x 2 5 x 2 9 x 2 2 x 3 x 4 x 7 x 10 + y 2 y 4 y 7 x 3 2 x 3 x 7 x 1 x 2 5 x 8 x 9 x 10 + y 3 y 4 y 7 x 3 2 x 3 x 7 x 1 x 2 5 x 6 x 9 x 10 + y 1 y 2 y 6 x 3 2 x 3 x 7 x 1 x 4 x 5 x 8 x 2 9 + y 1 y 3 y 6 x 3 2 x 3 x 7 x 1 x 4 x 5 x 6 x 2 9 + 2 y 1 y 4 y 5 x 3 5 x 8 x 10 x 2 x 3 x 4 x 6 x 2 9 + 2 y 1 y 4 y 8 x 4 x 6 x 3 9 x 2 x 2 5 x 7 x 8 x 10 + y 1 y 2 y 6 y 8 x 2 x 3 x 9 x 1 x 4 x 5 x 7 x 10 + y 3 y 4 y 5 y 7 x 2 x 5 x 7 x 1 x 3 x 4 x 9 x 10 + y 3 y 4 y 7 x 2 x 2 4 x 6 x 7 x 1 x 3 x 2 5 x 9 x 10 + 2 y 1 y 3 y 6 x 2 2 x 3 x 8 x 10 x 1 x 4 x 5 x 6 x 2 9 + 2 y 2 y 4 y 7 x 2 2 x 4 x 6 x 7 x 1 x 2 5 x 8 x 9 x 10 + y 1 y 2 y 6 x 2 x 3 x 8 x 2 10 x 1 x 4 x 5 x 7 x 2 9 + y 1 y 2 y 4 y 8 x 2 x 2 4 x 2 6 x 1 x 3 x 2 5 x 8 x 10 + y 1 y 3 y 4 y 5 x 2 x 2 8 x 2 10 x 1 x 4 x 6 x 7 x 2 9 + y 1 y 4 y 8 x 2 4 x 2 6 x 3 9 x 2 2 x 3 x 2 5 x 7 x 8 x 10 + y 1 y 4 y 5 x 3 5 x 2 8 x 2 10 x 2 2 x 3 x 4 x 6 x 7 x 2 9 + y 1 y 3 y 4 y 5 y 8 x 2 x 8 x 9 x 1 x 4 x 6 x 7 x 10 + y 1 y 2 y 4 y 5 y 8 x 2 x 5 x 6 x 1 x 3 x 4 x 8 x 10 + y 2 y 4 y 7 x 2 x 2 4 x 2 6 x 7 x 1 x 3 x 2 5 x 8 x 9 x 10 + y 1 y 3 y 6 x 2 x 3 x 2 8 x 2 10 x 1 x 4 x 5 x 6 x 7 x 2 9 + y 1 y 4 y 5 y 8 x 5 x 6 x 3 9 x 2 2 x 3 x 4 x 7 x 8 x 10 + y 1 y 4 y 5 y 8 x 3 5 x 8 x 9 x 2 2 x 3 x 4 x 6 x 7 x 10 + y 1 y 3 y 6 y 8 x 2 x 3 x 8 x 9 x 1 x 4 x 5 x 6 x 7 x 10 + y 2 y 4 y 5 y 7 x 2 x 5 x 6 x 7 x 1 x 3 x 4 x 8 x 9 x 10 ; x (1) 3 = y 2 y 5 x 1 + y 3 y 8 x 1 + 6 x 4 x 10 x 2 2 + 2 x 3 x 7 x 6 x 8 + 3 x 3 x 10 x 2 x 6 + 3 x 4 x 7 x 2 x 8 + 3 x 2 4 x 9 x 2 2 x 5 + 3 x 5 x 2 10 x 2 2 x 9 + x 2 3 x 3 9 x 3 5 x 2 8 + x 3 5 x 2 7 x 2 6 x 3 9 + x 6 x 3 10 x 3 2 x 7 + x 3 4 x 8 x 3 2 x 3 + y 2 y 8 x 6 x 1 x 8 62 + y 3 y 5 x 8 x 1 x 6 + 3 y 5 x 2 5 x 9 x 3 2 x 3 + 3 y 8 x 5 x 2 9 x 3 2 x 7 + 3 x 3 x 9 x 10 x 2 x 5 x 8 + 3 x 4 x 5 x 7 x 2 x 6 x 9 + 3 x 3 x 4 x 2 9 x 2 x 2 5 x 8 + 3 x 2 5 x 7 x 10 x 2 x 6 x 2 9 + 6 x 2 4 x 6 x 2 9 x 2 2 x 2 5 x 8 + 6 x 2 5 x 8 x 2 10 x 2 2 x 6 x 2 9 + 3 x 2 4 x 2 6 x 3 9 x 2 2 x 3 5 x 2 8 + 3 x 3 5 x 2 8 x 2 10 x 2 2 x 2 6 x 3 9 + 3 x 5 x 8 x 3 10 x 3 2 x 7 x 9 + 3 x 3 4 x 6 x 9 x 3 2 x 3 x 5 + y 5 x 3 5 x 8 x 3 2 x 3 x 6 + y 5 x 2 6 x 3 9 x 3 2 x 3 x 2 8 + y 8 x 3 5 x 2 8 x 3 2 x 2 6 x 7 + y 8 x 6 x 3 9 x 3 2 x 7 x 8 + 9 y 3 x 2 4 x 6 x 10 x 1 x 3 5 + 9 y 2 x 4 x 8 x 2 10 x 1 x 3 9 + 3 y 3 y 5 x 6 x 10 x 1 x 2 x 3 + 2 y 2 y 8 x 3 x 8 x 1 x 6 x 7 + 3 y 2 y 8 x 3 x 9 x 1 x 5 x 7 + 3 y 3 y 5 x 5 x 7 x 1 x 3 x 9 + 2 y 3 y 5 x 6 x 7 x 1 x 3 x 8 + 3 y 2 y 8 x 4 x 8 x 1 x 2 x 7 + 3 x 4 x 5 x 8 x 10 x 2 2 x 6 x 9 + 3 x 4 x 6 x 9 x 10 x 2 2 x 5 x 8 + 6 y 3 x 2 x 2 4 x 7 x 1 x 2 5 x 9 + 6 y 2 x 2 x 3 x 2 10 x 1 x 5 x 2 9 + 6 y 2 x 2 4 x 6 x 10 x 1 x 2 5 x 9 + 6 y 3 x 2 4 x 8 x 10 x 1 x 2 5 x 9 + 6 y 2 x 4 x 6 x 2 10 x 1 x 5 x 2 9 + 6 y 3 x 4 x 8 x 2 10 x 1 x 5 x 2 9 + 2 y 3 y 8 x 3 x 2 8 x 1 x 2 6 x 7 + 2 y 2 y 5 x 2 6 x 7 x 1 x 3 x 2 8 + y 2 2 y 5 y 8 x 2 6 x 2 1 x 3 x 7 + y 2 3 y 5 y 8 x 2 8 x 2 1 x 3 x 7 + 3 x 3 4 x 2 6 x 2 9 x 3 2 x 3 x 2 5 x 8 + x 3 4 x 3 6 x 3 9 x 3 2 x 3 x 3 5 x 2 8 + 3 x 3 x 4 x 6 x 3 9 x 2 x 3 5 x 2 8 + 3 x 3 5 x 7 x 8 x 10 x 2 x 2 6 x 3 9 + x 3 5 x 3 8 x 3 10 x 3 2 x 2 6 x 7 x 3 9 + 3 x 2 5 x 2 8 x 3 10 x 3 2 x 6 x 7 x 2 9 + 3 y 5 x 5 x 6 x 2 9 x 3 2 x 3 x 8 + 2 y 3 x 3 2 x 3 x 2 7 x 1 x 2 6 x 3 9 + 3 y 8 x 2 5 x 8 x 9 x 3 2 x 6 x 7 + 2 y 2 x 3 2 x 2 3 x 7 x 1 x 3 5 x 2 8 + 3 y 2 x 2 2 x 2 3 x 10 x 1 x 3 5 x 8 + 3 y 3 x 2 2 x 2 3 x 10 x 1 x 3 5 x 6 + 3 y 2 x 2 2 x 4 x 2 7 x 1 x 8 x 3 9 + 3 y 3 x 2 2 x 4 x 2 7 x 1 x 6 x 3 9 + 9 y 3 x 2 x 3 x 4 x 10 x 1 x 3 5 + 9 y 2 x 2 x 4 x 7 x 10 x 1 x 3 9 + 9 y 2 x 2 4 x 2 6 x 10 x 1 x 3 5 x 8 + 9 y 3 x 4 x 2 8 x 2 10 x 1 x 6 x 3 9 + y 2 2 y 8 x 3 2 x 2 3 x 2 1 x 3 5 x 7 + y 2 3 y 5 x 3 2 x 2 7 x 2 1 x 3 x 3 9 + 3 y 2 2 x 4 2 x 2 3 x 2 10 x 2 1 x 3 5 x 3 9 + 3 y 2 3 x 4 2 x 2 4 x 2 7 x 2 1 x 3 5 x 3 9 + 3 y 2 x 2 x 3 x 4 x 10 x 1 x 2 5 x 9 + 3 y 3 x 2 x 4 x 7 x 10 x 1 x 5 x 2 9 + 3 y 2 y 5 x 2 6 x 10 x 1 x 2 x 3 x 8 + 3 y 3 y 8 x 4 x 2 8 x 1 x 2 x 6 x 7 + 3 y 3 x 3 4 x 2 6 x 10 x 1 x 2 x 3 x 3 5 + 2 y 2 x 3 2 x 3 x 2 7 x 1 x 6 x 8 x 3 9 + 2 y 3 x 3 2 x 2 3 x 7 x 1 x 3 5 x 6 x 8 + 6 y 2 x 2 x 2 4 x 2 6 x 7 x 1 x 3 5 x 2 8 + 6 y 3 x 2 x 2 4 x 6 x 7 x 1 x 3 5 x 8 + 2 y 2 x 3 x 2 8 x 3 10 x 1 x 6 x 7 x 3 9 + 2 y 3 x 3 x 3 8 x 3 10 x 1 x 2 6 x 7 x 3 9 + 3 y 2 x 3 x 8 x 3 10 x 1 x 5 x 7 x 2 9 + 3 y 3 x 3 4 x 6 x 7 x 1 x 3 x 2 5 x 9 + 2 y 2 x 3 4 x 3 6 x 7 x 1 x 3 x 3 5 x 2 8 + 2 y 3 x 3 4 x 2 6 x 7 x 1 x 3 x 3 5 x 8 + 6 y 3 x 2 2 x 3 x 4 x 7 x 1 x 3 5 x 8 + 6 y 2 x 2 2 x 3 x 7 x 10 x 1 x 6 x 3 9 + 6 y 2 x 2 x 3 x 8 x 2 10 x 1 x 6 x 3 9 + 6 y 3 x 2 x 3 x 2 8 x 2 10 x 1 x 2 6 x 3 9 + 3 y 2 x 4 x 2 8 x 3 10 x 1 x 2 x 7 x 3 9 + 9 y 2 2 x 2 2 x 2 4 x 2 6 x 2 10 x 2 1 x 3 5 x 3 9 + 9 y 2 3 x 2 2 x 2 4 x 2 8 x 2 10 x 2 1 x 3 5 x 3 9 + 3 y 2 2 y 8 x 2 x 2 4 x 2 6 x 2 1 x 3 5 x 7 + 3 y 2 3 y 8 x 2 x 2 4 x 2 8 x 2 1 x 3 5 x 7 + 3 y 2 2 y 5 x 2 x 2 6 x 2 10 x 2 1 x 3 x 3 9 + 3 y 2 3 y 5 x 2 x 2 8 x 2 10 x 2 1 x 3 x 3 9 + y 2 2 y 8 x 3 4 x 3 6 x 2 1 x 3 x 3 5 x 7 + y 2 3 y 5 x 3 8 x 3 10 x 2 1 x 3 x 7 x 3 9 + y 2 2 x 6 2 x 2 3 x 2 7 x 2 1 x 3 5 x 2 8 x 3 9 + y 2 3 x 6 2 x 2 3 x 2 7 x 2 1 x 3 5 x 2 6 x 3 9 + 3 y 3 y 8 x 4 x 8 x 9 x 1 x 2 x 5 x 7 + 3 y 2 y 5 x 5 x 6 x 10 x 1 x 2 x 3 x 9 + 3 y 3 y 5 x 5 x 8 x 10 x 1 x 2 x 3 x 9 + 3 y 3 y 8 x 3 x 8 x 9 x 1 x 5 x 6 x 7 + 3 y 2 y 5 x 5 x 6 x 7 x 1 x 3 x 8 x 9 + 3 y 2 y 8 x 4 x 6 x 9 x 1 x 2 x 5 x 7 + 6 y 2 x 2 x 2 4 x 6 x 7 x 1 x 2 5 x 8 x 9 + 3 y 2 x 2 2 x 3 x 4 x 7 x 1 x 2 5 x 8 x 9 + 3 y 3 x 2 2 x 3 x 4 x 7 x 1 x 2 5 x 6 x 9 + 3 y 2 x 2 2 x 3 x 7 x 10 x 1 x 5 x 8 x 2 9 + 3 y 3 x 2 2 x 3 x 7 x 10 x 1 x 5 x 6 x 2 9 + 6 y 3 x 2 x 3 x 8 x 2 10 x 1 x 5 x 6 x 2 9 + 2 y 2 y 3 y 5 y 8 x 6 x 8 x 2 1 x 3 x 7 + 3 y 3 x 4 x 2 8 x 3 10 x 1 x 2 x 5 x 7 x 2 9 + 3 y 2 x 3 4 x 3 6 x 10 x 1 x 2 x 3 x 3 5 x 8 + 3 y 2 x 3 4 x 2 6 x 10 x 1 x 2 x 3 x 2 5 x 9 + 3 y 3 x 3 x 2 8 x 3 10 x 1 x 5 x 6 x 7 x 2 9 + 3 y 2 x 3 4 x 2 6 x 7 x 1 x 3 x 2 5 x 8 x 9 + 6 y 2 x 2 2 x 3 x 4 x 6 x 7 x 1 x 3 5 x 2 8 + 6 y 3 x 2 2 x 3 x 7 x 8 x 10 x 1 x 2 6 x 3 9 + 9 y 2 x 2 x 3 x 4 x 6 x 10 x 1 x 3 5 x 8 + 9 y 3 x 2 x 4 x 7 x 8 x 10 x 1 x 6 x 3 9 + 3 y 3 x 4 x 3 8 x 3 10 x 1 x 2 x 6 x 7 x 3 9 + y 2 2 x 3 2 x 2 3 x 8 x 3 10 x 2 1 x 3 5 x 7 x 3 9 + y 2 3 x 3 2 x 3 4 x 6 x 2 7 x 2 1 x 3 x 3 5 x 3 9 + 3 y 2 3 x 2 x 2 4 x 3 8 x 3 10 x 2 1 x 3 5 x 7 x 3 9 + 3 y 2 2 x 2 x 3 4 x 3 6 x 2 10 x 2 1 x 3 x 3 5 x 3 9 + 9 y 2 2 x 3 2 x 3 x 4 x 6 x 2 10 x 2 1 x 3 5 x 3 9 + 9 y 2 3 x 3 2 x 2 4 x 7 x 8 x 10 x 2 1 x 3 5 x 3 9 + y 2 3 y 8 x 3 2 x 2 3 x 2 8 x 2 1 x 3 5 x 2 6 x 7 + y 2 2 y 5 x 3 2 x 2 6 x 2 7 x 2 1 x 3 x 2 8 x 3 9 + 3 y 2 3 y 5 x 2 2 x 7 x 8 x 10 x 2 1 x 3 x 3 9 + y 2 3 y 8 x 3 4 x 6 x 2 8 x 2 1 x 3 x 3 5 x 7 + y 2 2 y 5 x 2 6 x 8 x 3 10 x 2 1 x 3 x 7 x 3 9 + 3 y 2 2 y 8 x 2 2 x 3 x 4 x 6 x 2 1 x 3 5 x 7 + 3 y 2 3 x 4 2 x 2 3 x 2 8 x 2 10 x 2 1 x 3 5 x 2 6 x 3 9 + 3 y 2 2 x 4 2 x 2 4 x 2 6 x 2 7 x 2 1 x 3 5 x 2 8 x 3 9 + 3 y 2 2 x 5 2 x 2 3 x 7 x 10 x 2 1 x 3 5 x 8 x 3 9 + 3 y 2 3 x 5 2 x 3 x 4 x 2 7 x 2 1 x 3 5 x 6 x 3 9 + 3 y 3 x 2 x 3 x 4 x 8 x 10 x 1 x 2 5 x 6 x 9 + 3 y 2 x 2 x 4 x 6 x 7 x 10 x 1 x 5 x 8 x 2 9 + 3 y 3 x 3 4 x 6 x 8 x 10 x 1 x 2 x 3 x 2 5 x 9 + 3 y 2 x 4 x 6 x 8 x 3 10 x 1 x 2 x 5 x 7 x 2 9 + y 2 3 x 3 2 x 3 8 x 2 3 x 3 10 x 2 1 x 3 5 x 2 6 x 7 x 3 9 + y 2 2 x 3 2 x 3 6 x 3 4 x 2 7 x 2 1 x 3 x 3 5 x 2 8 x 3 9 + 3 y 2 2 x 2 x 2 4 x 2 6 x 8 x 3 10 x 2 1 x 3 5 x 7 x 3 9 + 3 y 2 3 x 2 x 3 4 x 6 x 2 8 x 2 10 x 2 1 x 3 x 3 5 x 3 9 + y 2 2 x 3 4 x 3 6 x 8 x 3 10 x 2 1 x 3 x 3 5 x 7 x 3 9 + y 2 3 x 3 4 x 6 x 3 8 x 3 10 x 2 1 x 3 x 3 5 x 7 x 3 9 + 9 y 2 3 x 3 2 x 3 x 4 x 2 8 x 2 10 x 2 1 x 3 5 x 6 x 3 9 + 9 y 2 2 x 3 2 x 2 4 x 2 6 x 7 x 10 x 2 1 x 3 5 x 8 x 3 9 + 3 y 2 3 x 2 2 x 3 x 4 x 3 8 x 3 10 x 2 1 x 3 5 x 6 x 7 x 3 9 + 18 y 2 y 3 x 3 2 x 3 x 4 x 8 x 2 10 x 2 1 x 3 5 x 3 9 + 18 y 2 y 3 x 3 2 x 2 4 x 6 x 7 x 10 x 2 1 x 3 5 x 3 9 + 18 y 2 y 3 x 2 2 x 2 4 x 6 x 8 x 2 10 x 2 1 x 3 5 x 3 9 + 3 y 2 2 y 5 x 2 2 x 2 6 x 7 x 10 x 2 1 x 3 x 8 x 3 9 + 3 y 2 3 y 8 x 2 2 x 3 x 4 x 2 8 x 2 1 x 3 5 x 6 x 7 + 6 y 2 y 3 y 8 x 2 2 x 3 x 4 x 8 x 2 1 x 3 5 x 7 + 6 y 2 y 3 y 5 x 2 2 x 6 x 7 x 10 x 2 1 x 3 x 3 9 + 6 y 2 y 3 y 8 x 2 x 2 4 x 6 x 8 x 2 1 x 3 5 x 7 + 6 y 2 y 3 y 5 x 2 x 6 x 8 x 2 10 x 2 1 x 3 x 3 9 + 2 y 2 y 3 y 8 x 3 4 x 2 6 x 8 x 2 1 x 3 x 3 5 x 7 + 2 y 2 y 3 y 5 x 6 x 2 8 x 3 10 x 2 1 x 3 x 7 x 3 9 + 2 y 2 y 3 y 8 x 3 2 x 2 3 x 8 x 2 1 x 3 5 x 6 x 7 + 2 y 2 y 3 y 5 x 3 2 x 6 x 2 7 x 2 1 x 3 x 8 x 3 9 + 18 y 2 y 3 x 4 2 x 3 x 4 x 7 x 10 x 2 1 x 3 5 x 3 9 + 6 y 2 y 3 x 4 2 x 2 3 x 8 x 2 10 x 2 1 x 3 5 x 6 x 3 9 + 6 y 2 y 3 x 4 2 x 2 4 x 6 x 2 7 x 2 1 x 3 5 x 8 x 3 9 + 3 y 2 3 x 5 2 x 2 3 x 7 x 8 x 10 x 2 1 x 3 5 x 2 6 x 3 9 + 3 y 2 2 x 5 2 x 3 x 4 x 6 x 2 7 x 2 1 x 3 5 x 2 8 x 3 9 + 6 y 2 y 3 x 5 2 x 2 3 x 7 x 10 x 2 1 x 3 5 x 6 x 3 9 + 6 y 2 y 3 x 5 2 x 3 x 4 x 2 7 x 2 1 x 3 5 x 8 x 3 9 + 2 y 2 y 3 x 6 2 x 2 3 x 2 7 x 2 1 x 3 5 x 6 x 8 x 3 9 + 3 y 2 2 x 2 2 x 3 4 x 3 6 x 7 x 10 x 2 1 x 3 x 3 5 x 8 x 3 9 + 3 y 2 3 x 2 2 x 3 4 x 6 x 7 x 8 x 10 x 2 1 x 3 x 3 5 x 3 9 + 3 y 2 2 x 2 2 x 3 x 4 x 6 x 8 x 3 10 x 2 1 x 3 5 x 7 x 3 9 + 6 y 2 y 3 x 2 2 x 3 x 4 x 2 8 x 3 10 x 2 1 x 3 5 x 6 x 7 x 3 9 + 6 y 2 y 3 x 2 2 x 3 4 x 2 6 x 7 x 10 x 2 1 x 3 x 3 5 x 3 9 + 6 y 2 y 3 x 2 x 2 4 x 6 x 2 8 x 3 10 x 2 1 x 3 5 x 7 x 3 9 + 6 y 2 y 3 x 2 x 3 4 x 2 6 x 8 x 2 10 x 2 1 x 3 x 3 5 x 3 9 + 2 y 2 y 3 x 3 4 x 2 6 x 2 8 x 3 10 x 2 1 x 3 x 3 5 x 7 x 3 9 + 2 y 2 y 3 x 3 2 x 2 3 x 2 8 x 3 10 x 2 1 x 3 5 x 6 x 7 x 3 9 + 2 y 2 y 3 x 3 2 x 3 4 x 2 6 x 2 7 x 2 1 x 3 x 3 5 x 8 x 3 9 + 9 y 2 2 x 4 2 x 3 x 4 x 6 x 7 x 10 x 2 1 x 3 5 x 8 x 3 9 + 9 y 2 3 x 4 2 x 3 x 4 x 7 x 8 x 10 x 2 1 x 3 5 x 6 x 3 9 . 63 F or letting all v ariables x i = x and y j = y , we can calculate the desired expansion form ula for diagonal γ ′ = ( x (1) 3 , x (5) 2 ): x (1) 3 = 80 + 256 y + 256 y 2 + 16 y + 48 y 2 + 64 y 3 x + 4 y 4 x 2 ; x (5) 2 = 16 y 2 + 24 y 2 + 52 y 3 + 16 y 4 x + 4 y 2 + 16 y 3 + 16 y 4 + 4 y 5 x 2 . In general, for all vertices: x (1) 1 = 2 y ; x (5) 2 = 16 y 2 + 24 y 2 + 52 y 3 + 16 y 4 x + 4 y 2 + 16 y 3 + 16 y 4 + 4 y 5 x 2 ; x (1) 3 = 80 + 256 y + 256 y 2 + 16 y + 48 y 2 + 64 y 3 x + 4 y 4 x 2 ; x (2) 4 = 96 y 2 + 248 y 3 + 96 y 4 + 64 xy 2 + 56 y 2 + 272 y 3 + 432 y 4 + 220 y 5 + 32 y 6 x + 8 y 3 + 36 y 4 + 56 y 5 + 28 y 6 + 4 y 7 x 2 ; x (4) 5 = 12 y + 36 y 2 + 16 y 3 + 16 xy + 4 y 2 + 10 y 3 + 4 y 4 x ; x (1) 6 = 8 + 24 y + 32 y 2 + 4 y 3 x ; x (12) 7 = 9216 y 3 + 23552 y 4 + 8192 y 5 + 4096 xy 3 + 6912 y 3 + 36288 y 4 + 58784 y 5 + 30208 y 6 + 4608 y 7 x + 1728 y 3 + 14688 y 4 + 44352 y 5 + 58112 y 6 + 32256 y 7 + 7296 y 8 + 512 y 9 x 2 + 640 y 4 + 4576 y 5 + 12128 y 6 + 14528 y 7 + 7792 y 8 + 1768 y 9 + 128 y 10 x 3 + 64 y 5 + 416 y 6 + 1008 y 7 + 1112 y 8 + 560 y 9 + 120 y 10 + 8 y 11 x 4 ; x (6) 8 = 64 y 4 + 144 y 4 + 312 y 5 + 96 y 6 x + 80 y 3 + 424 y 4 + 904 y 5 + 808 y 6 + 284 y 7 + 32 y 8 x 2 + 8 y 4 + 48 y 5 + 116 y 6 + 104 y 7 + 36 y 8 + 4 y 9 x 3 ; x (3) 9 = 12 y + 16 y 2 + 4 y + 12 y 2 + 6 y 3 x ; x (1) 10 = 2 y + 8 y 2 + 2 y 2 + 2 y 3 x . F or the flip sequence µ 2 , we get the new expansion formulas of the v ertices of the form: ( y 1 , y 2 , ..., y 8 , x 1 , x 2 , ..., x 10 ) → ( y 1 , y 2 , ..., y 8 , x (1) 1 , x (5) 2 , x (1) 3 , x (1) 4 , x (3) 5 , x (1) 6 , x (12) 7 , x (6) 8 , x (4) 9 , x (2) 10 ) . W e shall omit the results of the remaining vertices as the form ulas are to o long. W e finally get the diagonal expansion formula after the flip: B D = ( x (1) 3 , x (5) 2 ) 64 where: x (5) 2 = y 1 y 6 x 2 + y 4 y 8 x 2 + y 4 x 5 x 9 x 2 x 6 + y 3 y 6 x 10 x 1 x 5 + y 3 y 6 x 9 x 1 x 4 + y 4 x 2 4 x 10 x 2 2 x 3 + 2 y 1 y 4 x 4 x 9 x 5 x 10 + 3 y 1 y 6 y 7 x 6 x 4 x 8 + y 1 y 4 y 5 y 8 x 4 x 9 + y 6 x 2 5 x 10 x 2 2 x 4 x 6 + y 4 y 2 8 x 3 x 10 x 2 5 x 2 9 + 2 y 2 1 y 6 x 6 x 9 x 5 x 2 10 + 3 y 2 1 y 6 x 6 x 2 9 x 4 x 3 10 + 2 y 4 y 8 x 4 x 10 x 2 x 5 x 9 + 2 y 1 y 6 x 5 x 9 x 2 x 4 x 10 + y 2 y 6 x 6 x 10 x 1 x 5 x 7 + y 2 y 6 x 6 x 9 x 1 x 4 x 7 + y 1 y 6 y 8 x 3 x 4 x 5 x 9 + y 1 y 6 y 8 x 6 x 2 x 5 x 9 + 2 y 1 y 6 y 8 x 6 x 2 x 4 x 10 + 2 y 4 x 4 x 7 x 2 9 x 5 x 8 x 10 + 2 y 3 y 4 x 2 x 4 x 9 x 1 x 2 5 + 2 y 4 y 7 x 4 x 2 10 x 5 x 8 x 9 + 2 y 6 y 7 x 6 x 7 x 9 x 4 x 2 8 + 2 y 6 y 8 x 5 x 10 x 2 2 x 4 x 9 + y 6 y 2 8 x 6 x 10 x 2 2 x 4 x 2 9 + 3 y 1 y 4 y 7 x 2 2 x 3 x 2 5 x 8 + 2 y 1 y 4 y 8 x 2 x 3 x 2 5 x 10 + 4 y 1 y 4 y 8 x 4 x 6 x 2 5 x 10 + 2 y 2 1 y 4 y 5 x 2 x 9 x 4 x 2 10 + 3 y 2 1 y 4 x 2 2 x 3 x 2 9 x 2 5 x 3 10 + y 6 y 2 7 x 6 x 3 10 x 4 x 2 8 x 2 9 + y 3 1 y 6 x 6 x 8 x 5 x 7 x 2 10 + y 6 x 6 x 2 7 x 4 9 x 4 x 2 8 x 3 10 + y 1 y 6 y 7 x 6 x 10 x 5 x 8 x 9 + y 2 y 4 y 5 x 2 x 10 x 1 x 4 x 7 + y 3 y 4 y 5 x 2 x 10 x 1 x 4 x 6 + y 3 y 4 y 5 x 5 x 9 x 1 x 3 x 4 + 2 y 6 x 5 x 7 x 2 9 x 2 x 4 x 8 x 10 + y 2 y 4 x 2 2 x 3 x 9 x 1 x 2 5 x 7 + y 3 y 4 x 2 2 x 3 x 9 x 1 x 2 5 x 6 + y 3 y 4 x 2 4 x 6 x 9 x 1 x 3 x 2 5 + 2 y 6 y 7 x 5 x 2 10 x 2 x 4 x 8 x 9 + 2 y 4 y 7 x 2 2 x 3 x 7 x 9 x 2 5 x 2 8 + y 1 y 6 x 6 x 7 x 2 9 x 5 x 8 x 2 10 + 2 y 4 y 2 8 x 4 x 6 x 10 x 2 x 2 5 x 2 9 + 2 y 2 1 y 6 x 2 x 3 x 9 x 4 x 5 x 2 10 + 6 y 1 y 4 y 7 x 2 x 4 x 6 x 2 5 x 8 + 3 y 1 y 4 y 7 x 2 4 x 2 6 x 3 x 2 5 x 8 + 3 y 1 y 6 x 6 x 7 x 3 9 x 4 x 8 x 3 10 + y 4 y 5 x 3 5 x 10 x 2 2 x 3 x 4 x 6 + y 4 y 2 7 x 2 2 x 3 x 3 10 x 2 5 x 2 8 x 2 9 + 6 y 2 1 y 4 x 2 x 4 x 6 x 2 9 x 2 5 x 3 10 + 3 y 2 1 y 4 x 2 4 x 2 6 x 2 9 x 3 x 2 5 x 3 10 + y 3 1 y 6 x 6 x 8 x 9 x 4 x 7 x 3 10 + y 3 1 y 4 y 5 x 2 x 8 x 4 x 7 x 2 10 + y 4 x 2 2 x 3 x 2 7 x 4 9 x 2 5 x 2 8 x 3 10 + 2 y 6 y 8 x 6 x 7 x 9 x 2 x 4 x 8 x 10 + y 2 y 6 x 2 x 3 x 10 x 1 x 4 x 5 x 7 + y 3 y 6 x 2 x 3 x 10 x 1 x 4 x 5 x 6 + y 1 y 4 y 5 y 7 x 2 x 10 x 4 x 8 x 9 + 3 y 1 y 4 y 5 y 7 x 5 x 6 x 3 x 4 x 8 + 2 y 1 y 4 x 2 4 x 6 x 9 x 2 x 3 x 5 x 10 + 2 y 2 y 4 x 2 x 4 x 6 x 9 x 1 x 2 5 x 7 + y 2 y 4 x 2 4 x 2 6 x 9 x 1 x 3 x 2 5 x 7 + 2 y 4 y 8 x 2 4 x 6 x 10 x 2 2 x 3 x 5 x 9 + 4 y 4 y 7 x 2 x 4 x 6 x 7 x 9 x 2 5 x 2 8 + 2 y 4 y 7 x 2 4 x 2 6 x 7 x 9 x 3 x 2 5 x 2 8 + 2 y 4 y 8 x 2 x 3 x 7 x 9 x 2 5 x 8 x 10 + 4 y 4 y 8 x 4 x 6 x 7 x 9 x 2 5 x 8 x 10 + y 4 y 2 8 x 2 4 x 2 6 x 10 x 2 2 x 3 x 2 5 x 2 9 + y 1 y 4 y 5 x 2 x 7 x 2 9 x 4 x 8 x 2 10 + 2 y 6 y 7 y 8 x 6 x 2 10 x 2 x 4 x 8 x 2 9 + 2 y 4 y 5 y 8 x 2 5 x 10 x 2 2 x 3 x 4 x 9 + 2 y 4 y 7 y 8 x 2 x 3 x 2 10 x 2 5 x 8 x 2 9 + 4 y 4 y 7 y 8 x 4 x 6 x 2 10 x 2 5 x 8 x 2 9 + 2 y 1 y 4 y 8 x 2 4 x 2 6 x 2 x 3 x 2 5 x 10 + 2 y 1 y 4 y 5 x 2 5 x 9 x 2 x 3 x 4 x 10 + 3 y 1 y 4 x 2 2 x 3 x 7 x 3 9 x 2 5 x 8 x 3 10 + 3 y 1 y 4 x 2 4 x 2 6 x 7 x 3 9 x 3 x 2 5 x 8 x 3 10 + 2 y 4 y 2 7 x 2 x 4 x 6 x 3 10 x 2 5 x 2 8 x 2 9 + y 4 y 2 7 x 2 4 x 2 6 x 3 10 x 3 x 2 5 x 2 8 x 2 9 + y 3 1 y 4 x 2 2 x 3 x 8 x 9 x 2 5 x 7 x 3 10 + y 3 1 y 6 x 2 x 3 x 8 x 4 x 5 x 7 x 2 10 + 3 y 2 1 y 4 y 5 x 5 x 6 x 2 9 x 3 x 4 x 3 10 + 2 y 4 x 2 x 4 x 6 x 2 7 x 4 9 x 2 5 x 2 8 x 3 10 + y 4 x 2 4 x 2 6 x 2 7 x 4 9 x 3 x 2 5 x 2 8 x 3 10 + y 1 y 6 y 7 x 2 x 3 x 10 x 4 x 5 x 8 x 9 + y 2 y 4 y 5 x 5 x 6 x 9 x 1 x 3 x 4 x 7 + 2 y 1 y 4 y 5 y 8 x 5 x 6 x 2 x 3 x 4 x 10 + 2 y 4 x 2 4 x 6 x 7 x 2 9 x 2 x 3 x 5 x 8 x 10 + 2 y 4 y 7 x 2 4 x 6 x 2 10 x 2 x 3 x 5 x 8 x 9 + 2 y 4 y 5 x 2 5 x 7 x 2 9 x 2 x 3 x 4 x 8 x 10 + y 1 y 6 x 2 x 3 x 7 x 2 9 x 4 x 5 x 8 x 2 10 + 2 y 4 y 5 y 7 x 5 x 6 x 7 x 9 x 3 x 4 x 2 8 + 2 y 4 y 5 y 7 x 2 5 x 2 10 x 2 x 3 x 4 x 8 x 9 + y 4 y 5 y 2 8 x 5 x 6 x 10 x 2 2 x 3 x 4 x 2 9 + 6 y 1 y 4 x 2 x 4 x 6 x 7 x 3 9 x 2 5 x 8 x 3 10 + 2 y 3 1 y 4 x 2 x 4 x 6 x 8 x 9 x 2 5 x 7 x 3 10 + y 3 1 y 4 x 2 4 x 2 6 x 8 x 9 x 3 x 2 5 x 7 x 3 10 + y 4 y 5 y 2 7 x 5 x 6 x 3 10 x 3 x 4 x 2 8 x 2 9 + y 4 y 5 x 5 x 6 x 2 7 x 4 9 x 3 x 4 x 2 8 x 3 10 + 2 y 4 y 8 x 2 4 x 2 6 x 7 x 9 x 2 x 3 x 2 5 x 8 x 10 + 2 y 4 y 7 y 8 x 2 4 x 2 6 x 2 10 x 2 x 3 x 2 5 x 8 x 2 9 + 3 y 1 y 4 y 5 x 5 x 6 x 7 x 3 9 x 3 x 4 x 8 x 3 10 + y 3 1 y 4 y 5 x 5 x 6 x 8 x 9 x 3 x 4 x 7 x 3 10 + 2 y 4 y 5 y 8 x 5 x 6 x 7 x 9 x 2 x 3 x 4 x 8 x 10 + 2 y 4 y 5 y 7 y 8 x 5 x 6 x 2 10 x 2 x 3 x 4 x 8 x 2 9 ; x (1) 3 = y 2 y 5 x 1 + y 3 y 7 x 1 + 2 y 2 7 x 6 x 7 x 2 8 + x 3 5 x 8 x 3 2 x 2 6 + x 3 4 x 7 x 3 2 x 3 + 2 y 7 x 3 x 7 x 6 x 8 + 3 y 7 x 4 x 7 x 2 x 8 + 6 y 1 y 7 x 4 x 2 x 9 + y 3 y 5 x 7 x 1 x 6 + y 2 y 7 x 6 x 1 x 7 + 3 y 1 y 7 x 5 x 2 x 10 + 3 y 1 y 7 x 3 x 6 x 9 + 2 y 3 x 3 x 8 x 1 x 2 6 + 6 y 8 x 4 x 7 x 2 2 x 10 + 3 y 1 y 2 7 x 6 x 8 x 9 + y 2 3 y 5 x 8 x 2 1 x 3 + 3 y 2 8 x 5 x 8 x 3 2 x 2 9 + y 3 7 x 6 x 3 10 x 2 8 x 3 9 + y 3 8 x 2 3 x 7 x 3 5 x 3 9 + y 3 8 x 6 x 8 x 3 2 x 3 9 + 3 y 2 1 y 7 x 6 x 9 x 3 10 + 3 y 2 x 4 x 8 x 1 x 2 x 7 + 3 y 3 x 4 x 8 x 1 x 2 x 6 + 3 y 8 x 3 x 7 x 2 x 6 x 10 + 2 y 2 x 3 x 8 x 1 x 6 x 7 + 3 y 8 x 2 4 x 7 x 2 2 x 5 x 9 + 3 y 7 x 2 5 x 10 x 2 2 x 6 x 9 + 3 y 2 7 x 5 x 2 10 x 2 x 8 x 2 9 + 3 y 2 1 x 2 4 x 8 x 2 x 5 x 2 10 + 6 y 7 y 8 x 5 x 10 x 2 2 x 2 9 + 6 y 2 1 y 7 x 4 x 6 x 5 x 2 10 + 6 y 2 1 y 7 x 2 x 3 x 5 x 2 10 + 6 y 1 y 7 y 8 x 3 x 5 x 2 9 + 2 x 3 x 2 7 x 3 9 x 6 x 8 x 3 10 + 3 x 4 x 2 7 x 3 9 x 2 x 8 x 3 10 + 6 y 1 x 3 x 7 x 2 9 x 6 x 3 10 + y 5 x 3 5 x 7 x 3 2 x 3 x 6 + y 7 x 6 x 2 7 x 3 9 x 2 8 x 3 10 + 9 y 1 x 4 x 7 x 2 9 x 2 x 3 10 + 3 y 8 x 2 5 x 8 x 3 2 x 6 x 9 + 3 y 2 7 x 3 2 x 2 3 x 2 7 x 3 5 x 3 8 + 6 y 2 1 x 3 x 8 x 9 x 6 x 3 10 + 9 y 2 1 x 4 x 8 x 9 x 2 x 3 10 + 3 y 2 3 x 2 x 2 4 x 8 x 2 1 x 3 5 + 2 y 3 1 x 3 x 2 8 x 6 x 7 x 3 10 + 3 y 3 1 x 4 x 2 8 x 2 x 7 x 3 10 + 9 y 2 1 y 8 x 3 x 8 x 5 x 3 10 + 3 y 7 y 2 8 x 6 x 10 x 2 2 x 3 9 + 3 y 5 y 2 7 x 2 6 x 2 7 x 3 x 3 8 + y 3 1 y 7 x 6 x 8 x 7 x 3 10 + 12 y 3 1 y 7 x 3 2 x 2 3 x 3 5 x 3 10 + 12 y 3 1 y 5 y 7 x 2 6 x 3 x 3 10 + 3 y 7 x 5 x 7 x 9 x 2 x 8 x 10 + 3 y 7 y 8 x 3 x 7 x 5 x 8 x 9 + 3 y 7 y 8 x 6 x 7 x 2 x 8 x 10 + 3 y 3 y 7 x 4 x 10 x 1 x 5 x 9 + 3 y 1 y 7 y 8 x 6 x 2 x 9 x 10 + 3 x 2 4 x 2 7 x 2 9 x 2 x 5 x 8 x 2 10 + 3 x 4 x 5 x 7 x 9 x 2 2 x 6 x 10 + 6 y 2 x 2 4 x 6 x 9 x 1 x 2 5 x 10 + 6 y 3 x 2 4 x 7 x 9 x 1 x 2 5 x 10 + 6 y 1 x 2 4 x 7 x 9 x 2 x 5 x 2 10 + 3 y 1 x 4 x 5 x 8 x 2 2 x 6 x 10 + 3 y 2 8 x 3 x 4 x 7 x 2 x 2 5 x 2 9 + 3 y 2 8 x 3 x 7 x 2 x 5 x 9 x 10 + 6 y 2 8 x 2 4 x 6 x 7 x 2 2 x 2 5 x 2 9 + 6 y 1 y 3 x 2 4 x 8 x 1 x 2 5 x 10 + 6 y 1 y 8 x 4 x 8 x 2 2 x 9 x 10 + 12 y 1 y 8 x 3 x 4 x 7 x 2 5 x 2 10 + 12 y 2 1 y 7 x 2 4 x 6 x 2 5 x 9 x 10 + 3 y 2 1 y 5 x 2 5 x 8 x 2 2 x 3 x 2 10 + y 2 2 y 5 x 2 6 x 8 x 2 1 x 3 x 2 7 + 2 y 2 y 5 y 7 x 2 6 x 1 x 3 x 8 + 3 y 2 1 y 5 y 2 7 x 2 6 x 3 x 8 x 2 9 + 3 y 8 x 3 x 2 7 x 2 9 x 5 x 8 x 3 10 + 9 y 2 7 x 2 x 2 4 x 2 6 x 2 7 x 3 5 x 3 8 + 3 y 2 7 x 3 4 x 3 6 x 2 7 x 3 x 3 5 x 3 8 + 3 y 2 8 x 2 x 2 3 x 2 7 x 3 5 x 8 x 2 10 + y 2 2 x 3 2 x 2 3 x 8 x 2 1 x 3 5 x 2 7 + y 2 3 x 3 2 x 2 3 x 8 x 2 1 x 3 5 x 2 6 65 + y 2 3 x 3 4 x 6 x 8 x 2 1 x 3 x 3 5 + 3 y 3 8 x 2 4 x 2 6 x 7 x 2 2 x 3 5 x 3 9 + 2 y 2 y 7 x 3 2 x 2 3 x 1 x 3 5 x 8 + 3 y 1 y 7 x 6 x 7 x 2 9 x 8 x 3 10 + 9 y 1 y 8 x 3 x 7 x 9 x 5 x 3 10 + 9 y 2 y 8 x 2 4 x 2 6 x 1 x 3 5 x 10 + 3 y 5 y 8 x 2 5 x 7 x 3 2 x 3 x 9 + 3 y 2 y 8 x 2 2 x 2 3 x 1 x 3 5 x 10 + 3 y 2 1 y 2 7 x 3 2 x 2 3 x 3 5 x 8 x 2 9 + 3 y 2 7 y 8 x 6 x 2 10 x 2 x 8 x 3 9 + 36 y 3 1 y 7 x 2 x 2 4 x 2 6 x 3 5 x 3 10 + 12 y 3 1 y 7 x 3 4 x 3 6 x 3 x 3 5 x 3 10 + y 5 y 3 8 x 2 6 x 7 x 3 2 x 3 x 3 9 + 6 x 2 4 x 6 x 3 7 x 4 9 x 2 5 x 2 8 x 4 10 + 36 y 2 1 x 2 4 x 6 x 7 x 2 9 x 2 5 x 4 10 + 24 y 3 1 x 2 4 x 6 x 8 x 9 x 2 5 x 4 10 + 6 y 4 1 x 2 4 x 6 x 2 8 x 2 5 x 7 x 4 10 + 12 y 3 1 y 8 x 2 2 x 2 3 x 8 x 3 5 x 4 10 + 36 y 3 1 y 8 x 2 4 x 2 6 x 8 x 3 5 x 4 10 + x 3 2 x 2 3 x 4 7 x 6 9 x 3 5 x 3 8 x 6 10 + y 5 x 2 6 x 4 7 x 6 9 x 3 x 3 8 x 6 10 + 20 y 3 1 x 3 2 x 2 3 x 7 x 3 9 x 3 5 x 6 10 + 15 y 4 1 x 3 2 x 2 3 x 8 x 2 9 x 3 5 x 6 10 + y 6 1 x 3 2 x 2 3 x 3 8 x 3 5 x 2 7 x 6 10 + 20 y 3 1 y 5 x 2 6 x 7 x 3 9 x 3 x 6 10 + 15 y 4 1 y 5 x 2 6 x 8 x 2 9 x 3 x 6 10 + y 6 1 y 5 x 2 6 x 3 8 x 3 x 2 7 x 6 10 + 3 y 1 y 8 x 3 x 8 x 2 x 6 x 9 x 10 + 3 y 2 y 8 x 3 x 8 x 1 x 5 x 7 x 9 + 3 y 3 y 8 x 3 x 8 x 1 x 5 x 6 x 9 + 3 y 3 y 8 x 4 x 8 x 1 x 2 x 5 x 9 + 2 y 3 y 5 y 7 x 6 x 7 x 1 x 3 x 8 + 3 y 1 y 3 y 5 y 7 x 6 x 1 x 3 x 9 + 3 y 7 x 2 4 x 7 x 10 x 2 x 5 x 8 x 9 + 3 y 2 x 2 x 3 x 4 x 9 x 1 x 2 5 x 10 + 6 y 8 x 3 x 4 x 2 7 x 9 x 2 5 x 8 x 2 10 + 3 y 5 x 2 5 x 2 7 x 2 9 x 2 2 x 3 x 8 x 2 10 + 12 y 7 x 2 4 x 6 x 2 7 x 9 x 2 5 x 2 8 x 10 + 3 y 2 8 x 4 x 6 x 7 x 2 2 x 5 x 9 x 10 + 6 y 2 7 x 2 4 x 6 x 7 x 2 10 x 2 5 x 2 8 x 2 9 + 24 y 1 y 8 x 2 4 x 6 x 7 x 2 x 2 5 x 2 10 + 24 y 1 y 7 x 2 4 x 6 x 7 x 2 5 x 8 x 10 + 6 y 1 y 5 x 2 5 x 7 x 9 x 2 2 x 3 x 2 10 + 3 y 1 y 2 8 x 3 x 8 x 2 x 5 x 2 9 x 10 + 6 y 2 1 y 8 x 3 x 4 x 8 x 2 5 x 9 x 2 10 + 6 y 2 1 y 7 x 2 x 3 x 4 x 2 5 x 9 x 10 + 3 y 1 y 2 7 x 4 x 6 x 10 x 5 x 8 x 2 9 + 3 y 5 y 2 8 x 2 6 x 2 7 x 2 2 x 3 x 8 x 2 10 + 3 y 1 y 2 7 x 2 x 3 x 10 x 5 x 8 x 2 9 + 6 y 1 y 7 y 8 x 4 x 6 x 2 x 5 x 2 9 + 3 y 2 y 5 y 8 x 2 6 x 1 x 2 x 3 x 10 + 2 y 2 y 3 y 5 x 6 x 8 x 2 1 x 3 x 7 + 6 y 1 y 5 y 2 7 x 2 6 x 7 x 3 x 2 8 x 9 + 3 y 7 x 3 2 x 2 3 x 3 7 x 3 9 x 3 5 x 3 8 x 3 10 + 3 y 8 x 3 4 x 6 x 7 x 3 2 x 3 x 5 x 9 + 3 y 2 2 x 2 x 2 4 x 2 6 x 8 x 2 1 x 3 5 x 2 7 + 9 y 2 7 x 2 2 x 3 x 4 x 6 x 2 7 x 3 5 x 3 8 + 3 y 2 3 x 2 2 x 3 x 4 x 8 x 2 1 x 3 5 x 6 + 3 y 2 8 x 3 4 x 2 6 x 7 x 3 2 x 3 x 2 5 x 2 9 + 9 y 2 8 x 3 x 4 x 6 x 2 7 x 3 5 x 8 x 2 10 + 9 y 2 8 x 2 4 x 2 6 x 2 7 x 2 x 3 5 x 8 x 2 10 + 3 y 2 1 x 3 4 x 6 x 8 x 2 2 x 3 x 5 x 2 10 + y 2 2 x 3 4 x 3 6 x 8 x 2 1 x 3 x 3 5 x 2 7 + y 3 7 x 3 2 x 2 3 x 7 x 3 10 x 3 5 x 3 8 x 3 9 + 3 y 3 8 x 3 x 4 x 6 x 7 x 2 x 3 5 x 3 9 + y 3 8 x 3 4 x 3 6 x 7 x 3 2 x 3 x 3 5 x 3 9 + 6 y 7 y 8 x 2 2 x 2 3 x 2 7 x 3 5 x 2 8 x 10 + 3 y 5 y 7 x 2 6 x 3 7 x 3 9 x 3 x 3 8 x 3 10 + 6 y 2 y 7 x 2 x 2 4 x 2 6 x 1 x 3 5 x 8 + 2 y 2 y 7 x 3 4 x 3 6 x 1 x 3 x 3 5 x 8 + 9 y 3 y 8 x 2 4 x 6 x 7 x 1 x 3 5 x 10 + 18 y 7 y 8 x 2 4 x 2 6 x 2 7 x 3 5 x 2 8 x 10 + 6 y 1 y 2 x 3 2 x 2 3 x 2 9 x 1 x 3 5 x 3 10 + 9 y 2 1 y 8 x 4 x 6 x 8 x 2 x 5 x 3 10 + 6 y 1 y 2 7 x 3 2 x 2 3 x 7 x 3 5 x 2 8 x 9 + 9 y 2 1 y 2 7 x 2 x 2 4 x 2 6 x 3 5 x 8 x 2 9 + 3 y 2 1 y 2 7 x 3 4 x 3 6 x 3 x 3 5 x 8 x 2 9 + 3 y 5 y 2 8 x 5 x 6 x 7 x 3 2 x 3 x 2 9 + 6 y 1 y 2 8 x 2 x 2 3 x 7 x 3 5 x 9 x 2 10 + 3 y 2 1 y 2 8 x 2 x 2 3 x 8 x 3 5 x 2 9 x 2 10 + 3 y 3 1 y 8 x 3 x 2 8 x 5 x 7 x 9 x 3 10 + 36 y 3 1 y 7 x 2 2 x 3 x 4 x 6 x 3 5 x 3 10 + y 5 y 3 7 x 2 6 x 7 x 3 10 x 3 x 3 8 x 3 9 + 6 y 1 y 2 y 5 x 2 6 x 2 9 x 1 x 3 x 3 10 + 6 y 2 1 y 7 y 8 x 2 2 x 2 3 x 3 5 x 2 9 x 10 + 18 y 2 1 y 7 y 8 x 2 4 x 2 6 x 3 5 x 2 9 x 10 + 3 x 2 x 3 x 4 x 3 7 x 4 9 x 2 5 x 2 8 x 4 10 + 3 y 8 x 2 2 x 2 3 x 3 7 x 3 9 x 3 5 x 2 8 x 4 10 + 24 y 1 x 2 4 x 6 x 2 7 x 3 9 x 2 5 x 8 x 4 10 + 9 y 8 x 2 4 x 2 6 x 3 7 x 3 9 x 3 5 x 2 8 x 4 10 + 18 y 2 1 x 2 x 3 x 4 x 7 x 2 9 x 2 5 x 4 10 + 12 y 3 1 x 2 x 3 x 4 x 8 x 9 x 2 5 x 4 10 + 3 y 4 1 x 2 x 3 x 4 x 2 8 x 2 5 x 7 x 4 10 + 18 y 2 1 y 8 x 2 2 x 2 3 x 7 x 9 x 3 5 x 4 10 + 54 y 2 1 y 8 x 2 4 x 2 6 x 7 x 9 x 3 5 x 4 10 + 12 y 3 1 y 5 y 8 x 2 6 x 8 x 2 x 3 x 4 10 + 3 x 2 x 2 4 x 2 6 x 4 7 x 6 9 x 3 5 x 3 8 x 6 10 + x 3 4 x 3 6 x 4 7 x 6 9 x 3 x 3 5 x 3 8 x 6 10 + 6 y 1 x 3 2 x 2 3 x 3 7 x 5 9 x 3 5 x 2 8 x 6 10 + 15 y 2 1 x 3 2 x 2 3 x 2 7 x 4 9 x 3 5 x 8 x 6 10 + 60 y 3 1 x 2 x 2 4 x 2 6 x 7 x 3 9 x 3 5 x 6 10 + 20 y 3 1 x 3 4 x 3 6 x 7 x 3 9 x 3 x 3 5 x 6 10 + 45 y 4 1 x 2 x 2 4 x 2 6 x 8 x 2 9 x 3 5 x 6 10 + 15 y 4 1 x 3 4 x 3 6 x 8 x 2 9 x 3 x 3 5 x 6 10 + 6 y 5 1 x 3 2 x 2 3 x 2 8 x 9 x 3 5 x 7 x 6 10 + 3 y 6 1 x 2 x 2 4 x 2 6 x 3 8 x 3 5 x 2 7 x 6 10 + y 6 1 x 3 4 x 3 6 x 3 8 x 3 x 3 5 x 2 7 x 6 10 + 6 y 1 y 5 x 2 6 x 3 7 x 5 9 x 3 x 2 8 x 6 10 + 15 y 2 1 y 5 x 2 6 x 2 7 x 4 9 x 3 x 8 x 6 10 + 6 y 5 1 y 5 x 2 6 x 2 8 x 9 x 3 x 7 x 6 10 + 3 y 7 y 8 x 4 x 6 x 7 x 2 x 5 x 8 x 9 + 3 y 2 y 7 x 2 x 3 x 10 x 1 x 5 x 7 x 9 + 3 y 3 y 7 x 2 x 3 x 10 x 1 x 5 x 6 x 9 + 3 y 2 y 7 x 4 x 6 x 10 x 1 x 5 x 7 x 9 + 3 y 3 y 5 x 5 x 7 x 9 x 1 x 2 x 3 x 10 + 3 y 2 y 5 x 5 x 6 x 9 x 1 x 2 x 3 x 10 + 3 y 1 y 3 y 5 x 5 x 8 x 1 x 2 x 3 x 10 + 3 y 3 y 5 y 8 x 6 x 7 x 1 x 2 x 3 x 10 + 12 y 8 x 2 4 x 6 x 2 7 x 9 x 2 x 2 5 x 8 x 2 10 + 6 y 7 x 2 x 3 x 4 x 2 7 x 9 x 2 5 x 2 8 x 10 + 3 y 2 7 x 2 x 3 x 4 x 7 x 2 10 x 2 5 x 2 8 x 2 9 + 6 y 1 y 2 x 2 4 x 6 x 8 x 1 x 2 5 x 7 x 10 + 6 y 7 y 8 x 3 x 4 x 7 x 10 x 2 5 x 8 x 2 9 + 12 y 1 y 7 x 2 x 3 x 4 x 7 x 2 5 x 8 x 10 + 3 y 1 y 7 x 4 x 6 x 7 x 9 x 5 x 8 x 2 10 + 3 y 5 y 7 x 2 5 x 7 x 10 x 2 2 x 3 x 8 x 9 + 3 y 1 y 7 x 2 x 3 x 7 x 9 x 5 x 8 x 2 10 + 12 y 2 1 y 8 x 2 4 x 6 x 8 x 2 x 2 5 x 9 x 2 10 + 3 y 1 y 2 8 x 4 x 6 x 8 x 2 2 x 5 x 2 9 x 10 + 12 y 1 y 5 y 8 x 5 x 6 x 7 x 2 2 x 3 x 2 10 + 6 y 5 y 7 y 8 x 2 6 x 2 7 x 2 x 3 x 2 8 x 10 + 6 y 1 y 5 y 2 8 x 2 6 x 7 x 2 2 x 3 x 9 x 2 10 + 3 y 2 1 y 5 y 2 8 x 2 6 x 8 x 2 2 x 3 x 2 9 x 2 10 + 6 y 2 1 y 5 y 7 x 5 x 6 x 2 x 3 x 9 x 10 + 3 y 1 y 2 y 5 y 7 x 2 6 x 1 x 3 x 7 x 9 + 6 y 2 1 y 5 y 7 y 8 x 2 6 x 2 x 3 x 2 9 x 10 + 3 x 3 4 x 6 x 2 7 x 2 9 x 2 2 x 3 x 5 x 8 x 2 10 + 2 y 2 x 3 2 x 2 3 x 7 x 3 9 x 1 x 3 5 x 8 x 3 10 + 3 y 8 x 4 x 6 x 2 7 x 2 9 x 2 x 5 x 8 x 3 10 + 3 y 7 x 3 4 x 3 6 x 3 7 x 3 9 x 3 x 3 5 x 3 8 x 3 10 + 3 y 2 x 3 4 x 2 6 x 9 x 1 x 2 x 3 x 2 5 x 10 + 6 y 1 x 3 4 x 6 x 7 x 9 x 2 2 x 3 x 5 x 2 10 + 9 y 7 x 2 x 2 4 x 2 6 x 3 7 x 3 9 x 3 5 x 3 8 x 3 10 + 3 y 2 2 x 2 2 x 3 x 4 x 6 x 8 x 2 1 x 3 5 x 2 7 + 3 y 2 8 x 3 4 x 3 6 x 2 7 x 2 2 x 3 x 3 5 x 8 x 2 10 + 3 y 3 7 x 2 x 2 4 x 2 6 x 7 x 3 10 x 3 5 x 3 8 x 3 9 + y 3 7 x 3 4 x 3 6 x 7 x 3 10 x 3 x 3 5 x 3 8 x 3 9 + 9 y 1 y 8 x 4 x 6 x 7 x 9 x 2 x 5 x 3 10 + 6 y 2 y 7 x 2 2 x 3 x 4 x 6 x 1 x 3 5 x 8 + 18 y 1 y 2 x 2 x 2 4 x 2 6 x 2 9 x 1 x 3 5 x 3 10 + 6 y 3 y 7 x 2 2 x 3 x 4 x 7 x 1 x 3 5 x 8 + 3 y 2 y 8 x 3 4 x 3 6 x 1 x 2 x 3 x 3 5 x 10 + 12 y 1 y 8 x 3 4 x 2 6 x 7 x 2 2 x 3 x 2 5 x 2 10 + 2 y 3 y 7 x 3 2 x 2 3 x 7 x 1 x 3 5 x 6 x 8 + 2 y 2 y 3 x 3 2 x 2 3 x 8 x 2 1 x 3 5 x 6 x 7 + 6 y 3 y 7 x 2 x 2 4 x 6 x 7 x 1 x 3 5 x 8 + 12 y 1 y 7 x 3 2 x 2 3 x 2 7 x 2 9 x 3 5 x 2 8 x 3 10 + 6 y 1 y 2 x 3 4 x 3 6 x 2 9 x 1 x 3 x 3 5 x 3 10 + 2 y 3 y 7 x 3 4 x 2 6 x 7 x 1 x 3 x 3 5 x 8 + 2 y 2 y 3 x 3 4 x 2 6 x 8 x 2 1 x 3 x 3 5 x 7 + 9 y 2 y 8 x 2 x 3 x 4 x 6 x 1 x 3 5 x 10 + 9 y 3 y 8 x 2 x 3 x 4 x 7 x 1 x 3 5 x 10 + 6 y 2 y 3 x 2 2 x 3 x 4 x 8 x 2 1 x 3 5 x 7 + 6 y 2 y 3 x 2 x 2 4 x 6 x 8 x 2 1 x 3 5 x 7 + 2 y 2 y 5 x 2 6 x 7 x 3 9 x 1 x 3 x 8 x 3 10 + 2 y 3 y 5 x 6 x 2 7 x 3 9 x 1 x 3 x 8 x 3 10 + 3 y 3 y 8 x 2 2 x 2 3 x 7 x 1 x 3 5 x 6 x 10 + 9 y 2 7 y 8 x 2 4 x 2 6 x 7 x 2 10 x 3 5 x 2 8 x 3 9 + 18 y 1 y 2 8 x 2 4 x 2 6 x 7 x 2 x 3 5 x 9 x 2 10 + 9 y 2 1 y 2 8 x 2 4 x 2 6 x 8 x 2 x 3 5 x 2 9 x 2 10 + 3 y 7 y 2 8 x 2 x 2 3 x 7 x 10 x 3 5 x 8 x 3 9 + 18 y 2 1 y 7 x 3 2 x 2 3 x 7 x 9 x 3 5 x 8 x 3 10 + 18 y 1 y 2 7 x 2 x 2 4 x 2 6 x 7 x 3 5 x 2 8 x 9 + 18 y 1 y 2 8 x 3 x 4 x 6 x 7 x 3 5 x 9 x 2 10 + 9 y 2 1 y 2 8 x 3 x 4 x 6 x 8 x 3 5 x 2 9 x 2 10 + 9 y 2 1 y 2 7 x 2 2 x 3 x 4 x 6 x 3 5 x 8 x 2 9 66 + 6 y 1 y 2 7 x 3 4 x 3 6 x 7 x 3 x 3 5 x 2 8 x 9 + 6 y 2 1 y 7 x 3 4 x 2 6 x 2 x 3 x 2 5 x 9 x 10 + 3 y 2 7 y 8 x 2 2 x 2 3 x 7 x 2 10 x 3 5 x 2 8 x 3 9 + 2 y 3 1 y 2 x 3 2 x 2 3 x 2 8 x 1 x 3 5 x 2 7 x 3 10 + 3 y 3 1 y 7 x 4 x 6 x 8 x 5 x 7 x 9 x 2 10 + 3 y 3 1 y 7 x 2 x 3 x 8 x 5 x 7 x 9 x 2 10 + 3 y 1 y 3 y 7 x 3 2 x 2 3 x 1 x 3 5 x 6 x 9 + 9 y 1 y 3 y 7 x 2 2 x 3 x 4 x 1 x 3 5 x 9 + 9 y 1 y 3 y 7 x 2 x 2 4 x 6 x 1 x 3 5 x 9 + 3 y 1 y 3 y 7 x 3 4 x 2 6 x 1 x 3 x 3 5 x 9 + 3 y 1 y 2 y 7 x 3 2 x 2 3 x 1 x 3 5 x 7 x 9 + 6 y 1 y 3 y 5 x 6 x 7 x 2 9 x 1 x 3 x 3 10 + 12 y 1 y 5 y 7 x 2 6 x 2 7 x 2 9 x 3 x 2 8 x 3 10 + 6 y 2 1 y 3 y 5 x 6 x 8 x 9 x 1 x 3 x 3 10 + 18 y 2 1 y 5 y 7 x 2 6 x 7 x 9 x 3 x 8 x 3 10 + 2 y 3 1 y 3 y 5 x 6 x 2 8 x 1 x 3 x 7 x 3 10 + 2 y 3 1 y 2 y 5 x 2 6 x 2 8 x 1 x 3 x 2 7 x 3 10 + 3 x 3 4 x 2 6 x 3 7 x 4 9 x 2 x 3 x 2 5 x 2 8 x 4 10 + 3 y 5 x 5 x 6 x 3 7 x 4 9 x 2 x 3 x 2 8 x 4 10 + 12 y 1 x 2 x 3 x 4 x 2 7 x 3 9 x 2 5 x 8 x 4 10 + 18 y 2 1 x 3 4 x 2 6 x 7 x 2 9 x 2 x 3 x 2 5 x 4 10 + 12 y 3 1 x 3 4 x 2 6 x 8 x 9 x 2 x 3 x 2 5 x 4 10 + 3 y 4 1 x 3 4 x 2 6 x 2 8 x 2 x 3 x 2 5 x 7 x 4 10 + 12 y 1 y 8 x 2 2 x 2 3 x 2 7 x 2 9 x 3 5 x 8 x 4 10 + 36 y 1 y 8 x 2 4 x 2 6 x 2 7 x 2 9 x 3 5 x 8 x 4 10 + 3 y 5 y 8 x 2 6 x 3 7 x 3 9 x 2 x 3 x 2 8 x 4 10 + 18 y 2 1 y 5 x 5 x 6 x 7 x 2 9 x 2 x 3 x 4 10 + 36 y 3 1 y 8 x 2 x 3 x 4 x 6 x 8 x 3 5 x 4 10 + 12 y 3 1 y 8 x 3 4 x 3 6 x 8 x 2 x 3 x 3 5 x 4 10 + 12 y 3 1 y 5 x 5 x 6 x 8 x 9 x 2 x 3 x 4 10 + 3 y 4 1 y 7 x 3 2 x 2 3 x 8 x 3 5 x 7 x 9 x 3 10 + 3 y 4 1 y 8 x 2 2 x 2 3 x 2 8 x 3 5 x 7 x 9 x 4 10 + 3 y 4 1 y 5 x 5 x 6 x 2 8 x 2 x 3 x 7 x 4 10 + 9 y 4 1 y 8 x 2 4 x 2 6 x 2 8 x 3 5 x 7 x 9 x 4 10 + 18 y 2 1 y 5 y 8 x 2 6 x 7 x 9 x 2 x 3 x 4 10 + 3 y 4 1 y 5 y 7 x 2 6 x 8 x 3 x 7 x 9 x 3 10 + 3 x 2 2 x 3 x 4 x 6 x 4 7 x 6 9 x 3 5 x 3 8 x 6 10 + 18 y 1 x 2 x 2 4 x 2 6 x 3 7 x 5 9 x 3 5 x 2 8 x 6 10 + 6 y 1 x 3 4 x 3 6 x 3 7 x 5 9 x 3 x 3 5 x 2 8 x 6 10 + 45 y 2 1 x 2 x 2 4 x 2 6 x 2 7 x 4 9 x 3 5 x 8 x 6 10 + 15 y 2 1 x 3 4 x 3 6 x 2 7 x 4 9 x 3 x 3 5 x 8 x 6 10 + 60 y 3 1 x 2 2 x 3 x 4 x 6 x 7 x 3 9 x 3 5 x 6 10 + 45 y 4 1 x 2 2 x 3 x 4 x 6 x 8 x 2 9 x 3 5 x 6 10 + 18 y 5 1 x 2 x 2 4 x 2 6 x 2 8 x 9 x 3 5 x 7 x 6 10 + 6 y 5 1 x 3 4 x 3 6 x 2 8 x 9 x 3 x 3 5 x 7 x 6 10 + 3 y 6 1 x 2 2 x 3 x 4 x 6 x 3 8 x 3 5 x 2 7 x 6 10 + 3 y 2 y 8 x 4 x 6 x 8 x 1 x 2 x 5 x 7 x 9 + 12 y 1 y 5 y 7 x 5 x 6 x 7 x 2 x 3 x 8 x 10 + 3 y 3 x 2 x 3 x 4 x 7 x 9 x 1 x 2 5 x 6 x 10 + 6 y 5 y 7 x 5 x 6 x 2 7 x 9 x 2 x 3 x 2 8 x 10 + 12 y 7 y 8 x 2 4 x 6 x 7 x 10 x 2 x 2 5 x 8 x 2 9 + 3 y 1 y 2 x 2 x 3 x 4 x 8 x 1 x 2 5 x 7 x 10 + 3 y 1 y 3 x 2 x 3 x 4 x 8 x 1 x 2 5 x 6 x 10 + 6 y 5 y 8 x 5 x 6 x 2 7 x 9 x 2 2 x 3 x 8 x 2 10 + 3 y 5 y 2 7 x 5 x 6 x 7 x 2 10 x 2 x 3 x 2 8 x 2 9 + 6 y 2 1 y 5 y 8 x 5 x 6 x 8 x 2 2 x 3 x 9 x 2 10 + 2 y 2 x 3 4 x 3 6 x 7 x 3 9 x 1 x 3 x 3 5 x 8 x 3 10 + 2 y 3 x 3 4 x 2 6 x 2 7 x 3 9 x 1 x 3 x 3 5 x 8 x 3 10 + 2 y 3 x 3 2 x 2 3 x 2 7 x 3 9 x 1 x 3 5 x 6 x 8 x 3 10 + 9 y 7 x 2 2 x 3 x 4 x 6 x 3 7 x 3 9 x 3 5 x 3 8 x 3 10 + 6 y 2 x 2 x 2 4 x 2 6 x 7 x 3 9 x 1 x 3 5 x 8 x 3 10 + 6 y 3 x 2 x 2 4 x 6 x 2 7 x 3 9 x 1 x 3 5 x 8 x 3 10 + 3 y 3 x 3 4 x 6 x 7 x 9 x 1 x 2 x 3 x 2 5 x 10 + 6 y 3 x 2 2 x 3 x 4 x 2 7 x 3 9 x 1 x 3 5 x 8 x 3 10 + 3 y 7 x 3 4 x 6 x 7 x 10 x 2 2 x 3 x 5 x 8 x 9 + 6 y 8 x 3 4 x 2 6 x 2 7 x 9 x 2 2 x 3 x 2 5 x 8 x 2 10 + 6 y 7 x 3 4 x 2 6 x 2 7 x 9 x 2 x 3 x 2 5 x 2 8 x 10 + 3 y 2 7 x 3 4 x 2 6 x 7 x 2 10 x 2 x 3 x 2 5 x 2 8 x 2 9 + 3 y 3 7 x 2 2 x 3 x 4 x 6 x 7 x 3 10 x 3 5 x 3 8 x 3 9 + 18 y 1 y 2 x 2 2 x 3 x 4 x 6 x 2 9 x 1 x 3 5 x 3 10 + 18 y 1 y 3 x 2 2 x 3 x 4 x 7 x 2 9 x 1 x 3 5 x 3 10 + 3 y 1 y 3 x 3 4 x 6 x 8 x 1 x 2 x 3 x 2 5 x 10 + 18 y 7 y 8 x 2 x 3 x 4 x 6 x 2 7 x 3 5 x 2 8 x 10 + 6 y 7 y 8 x 3 4 x 3 6 x 2 7 x 2 x 3 x 3 5 x 2 8 x 10 + 6 y 1 y 3 x 3 2 x 2 3 x 7 x 2 9 x 1 x 3 5 x 6 x 3 10 + 18 y 1 y 3 x 2 x 2 4 x 6 x 7 x 2 9 x 1 x 3 5 x 3 10 + 6 y 1 y 3 x 3 4 x 2 6 x 7 x 2 9 x 1 x 3 x 3 5 x 3 10 + 36 y 1 y 7 x 2 x 2 4 x 2 6 x 2 7 x 2 9 x 3 5 x 2 8 x 3 10 + 12 y 1 y 7 x 3 4 x 3 6 x 2 7 x 2 9 x 3 x 3 5 x 2 8 x 3 10 + 3 y 3 y 8 x 3 4 x 2 6 x 7 x 1 x 2 x 3 x 3 5 x 10 + 12 y 1 y 7 x 3 4 x 2 6 x 7 x 2 x 3 x 2 5 x 8 x 10 + 18 y 2 1 y 3 x 2 2 x 3 x 4 x 8 x 9 x 1 x 3 5 x 3 10 + 9 y 7 y 2 8 x 2 4 x 2 6 x 7 x 10 x 2 x 3 5 x 8 x 3 9 + 6 y 1 y 2 8 x 3 4 x 3 6 x 7 x 2 2 x 3 x 3 5 x 9 x 2 10 + 3 y 2 1 y 2 8 x 3 4 x 3 6 x 8 x 2 2 x 3 x 3 5 x 2 9 x 2 10 + 6 y 2 1 y 8 x 3 4 x 2 6 x 8 x 2 2 x 3 x 2 5 x 9 x 2 10 + 6 y 2 1 y 2 x 3 2 x 2 3 x 8 x 9 x 1 x 3 5 x 7 x 3 10 + 6 y 2 1 y 3 x 3 2 x 2 3 x 8 x 9 x 1 x 3 5 x 6 x 3 10 + 18 y 2 1 y 3 x 2 x 2 4 x 6 x 8 x 9 x 1 x 3 5 x 3 10 + 6 y 2 1 y 3 x 3 4 x 2 6 x 8 x 9 x 1 x 3 x 3 5 x 3 10 + 9 y 7 y 2 8 x 3 x 4 x 6 x 7 x 10 x 3 5 x 8 x 3 9 + 54 y 2 1 y 7 x 2 x 2 4 x 2 6 x 7 x 9 x 3 5 x 8 x 3 10 + 18 y 1 y 2 7 x 2 2 x 3 x 4 x 6 x 7 x 3 5 x 2 8 x 9 + 18 y 2 1 y 7 x 3 4 x 3 6 x 7 x 9 x 3 x 3 5 x 8 x 3 10 + 3 y 3 1 y 8 x 4 x 6 x 2 8 x 2 x 5 x 7 x 9 x 3 10 + 6 y 3 1 y 3 x 2 2 x 3 x 4 x 2 8 x 1 x 3 5 x 7 x 3 10 + 6 y 3 1 y 2 x 2 x 2 4 x 2 6 x 2 8 x 1 x 3 5 x 2 7 x 3 10 + 2 y 3 1 y 2 x 3 4 x 3 6 x 2 8 x 1 x 3 x 3 5 x 2 7 x 3 10 + 2 y 3 1 y 3 x 3 2 x 2 3 x 2 8 x 1 x 3 5 x 6 x 7 x 3 10 + 6 y 3 1 y 3 x 2 x 2 4 x 6 x 2 8 x 1 x 3 5 x 7 x 3 10 + 2 y 3 1 y 3 x 3 4 x 2 6 x 2 8 x 1 x 3 x 3 5 x 7 x 3 10 + 12 y 1 y 7 y 8 x 2 2 x 2 3 x 7 x 3 5 x 8 x 9 x 10 + 9 y 1 y 3 y 8 x 2 4 x 6 x 8 x 1 x 3 5 x 9 x 10 + 9 y 1 y 2 y 7 x 2 x 2 4 x 2 6 x 1 x 3 5 x 7 x 9 + 3 y 1 y 2 y 7 x 3 4 x 3 6 x 1 x 3 x 3 5 x 7 x 9 + 36 y 1 y 7 y 8 x 2 4 x 2 6 x 7 x 3 5 x 8 x 9 x 10 + 6 y 2 1 y 7 y 8 x 3 4 x 3 6 x 2 x 3 x 3 5 x 2 9 x 10 + 18 y 2 1 y 7 y 8 x 2 x 3 x 4 x 6 x 3 5 x 2 9 x 10 + 3 y 5 y 2 7 y 8 x 2 6 x 7 x 2 10 x 2 x 3 x 2 8 x 3 9 + 3 y 5 y 7 y 2 8 x 2 6 x 7 x 10 x 2 2 x 3 x 8 x 3 9 + 6 y 2 1 y 2 y 5 x 2 6 x 8 x 9 x 1 x 3 x 7 x 3 10 + 3 y 8 x 3 4 x 3 6 x 3 7 x 3 9 x 2 x 3 x 3 5 x 2 8 x 4 10 + 9 y 8 x 2 x 3 x 4 x 6 x 3 7 x 3 9 x 3 5 x 2 8 x 4 10 + 12 y 1 x 3 4 x 2 6 x 2 7 x 3 9 x 2 x 3 x 2 5 x 8 x 4 10 + 12 y 1 y 5 x 5 x 6 x 2 7 x 3 9 x 2 x 3 x 8 x 4 10 + 54 y 2 1 y 8 x 2 x 3 x 4 x 6 x 7 x 9 x 3 5 x 4 10 + 18 y 2 1 y 8 x 3 4 x 3 6 x 7 x 9 x 2 x 3 x 3 5 x 4 10 + 9 y 4 1 y 7 x 2 x 2 4 x 2 6 x 8 x 3 5 x 7 x 9 x 3 10 + 3 y 4 1 y 7 x 3 4 x 3 6 x 8 x 3 x 3 5 x 7 x 9 x 3 10 + 12 y 1 y 5 y 8 x 2 6 x 2 7 x 2 9 x 2 x 3 x 8 x 4 10 + 3 y 4 1 y 5 y 8 x 2 6 x 2 8 x 2 x 3 x 7 x 9 x 4 10 + 18 y 1 x 2 2 x 3 x 4 x 6 x 3 7 x 5 9 x 3 5 x 2 8 x 6 10 + 45 y 2 1 x 2 2 x 3 x 4 x 6 x 2 7 x 4 9 x 3 5 x 8 x 6 10 + 18 y 5 1 x 2 2 x 3 x 4 x 6 x 2 8 x 9 x 3 5 x 7 x 6 10 + 3 y 1 y 2 y 5 x 5 x 6 x 8 x 1 x 2 x 3 x 7 x 10 + 3 y 1 y 3 y 5 y 8 x 6 x 8 x 1 x 2 x 3 x 9 x 10 + 6 y 5 y 7 y 8 x 5 x 6 x 7 x 10 x 2 2 x 3 x 8 x 2 9 + 12 y 1 y 5 y 7 y 8 x 2 6 x 7 x 2 x 3 x 8 x 9 x 10 + 6 y 2 x 2 2 x 3 x 4 x 6 x 7 x 3 9 x 1 x 3 5 x 8 x 3 10 + 6 y 7 y 8 x 3 4 x 2 6 x 7 x 10 x 2 2 x 3 x 2 5 x 8 x 2 9 + 36 y 1 y 7 x 2 2 x 3 x 4 x 6 x 2 7 x 2 9 x 3 5 x 2 8 x 3 10 + 3 y 1 y 2 x 3 4 x 2 6 x 8 x 1 x 2 x 3 x 2 5 x 7 x 10 + 3 y 2 7 y 8 x 3 4 x 3 6 x 7 x 2 10 x 2 x 3 x 3 5 x 2 8 x 3 9 + 18 y 2 1 y 2 x 2 x 2 4 x 2 6 x 8 x 9 x 1 x 3 5 x 7 x 3 10 + 6 y 2 1 y 2 x 3 4 x 3 6 x 8 x 9 x 1 x 3 x 3 5 x 7 x 3 10 + 3 y 7 y 2 8 x 3 4 x 3 6 x 7 x 10 x 2 2 x 3 x 3 5 x 8 x 3 9 + 54 y 2 1 y 7 x 2 2 x 3 x 4 x 6 x 7 x 9 x 3 5 x 8 x 3 10 + 9 y 2 7 y 8 x 2 x 3 x 4 x 6 x 7 x 2 10 x 3 5 x 2 8 x 3 9 + 6 y 3 1 y 2 x 2 2 x 3 x 4 x 6 x 2 8 x 1 x 3 5 x 2 7 x 3 10 + 3 y 1 y 2 y 8 x 2 2 x 2 3 x 8 x 1 x 3 5 x 7 x 9 x 10 + 3 y 1 y 3 y 8 x 2 2 x 2 3 x 8 x 1 x 3 5 x 6 x 9 x 10 + 9 y 1 y 3 y 8 x 2 x 3 x 4 x 8 x 1 x 3 5 x 9 x 10 + 9 y 1 y 2 y 8 x 2 4 x 2 6 x 8 x 1 x 3 5 x 7 x 9 x 10 + 9 y 1 y 2 y 7 x 2 2 x 3 x 4 x 6 x 1 x 3 5 x 7 x 9 + 36 y 1 y 8 x 2 x 3 x 4 x 6 x 2 7 x 2 9 x 3 5 x 8 x 4 10 + 12 y 1 y 8 x 3 4 x 3 6 x 2 7 x 2 9 x 2 x 3 x 3 5 x 8 x 4 10 + 9 y 4 1 y 8 x 2 x 3 x 4 x 6 x 2 8 x 3 5 x 7 x 9 x 4 10 67 + 3 y 4 1 y 8 x 3 4 x 3 6 x 2 8 x 2 x 3 x 3 5 x 7 x 9 x 4 10 + 9 y 4 1 y 7 x 2 2 x 3 x 4 x 6 x 8 x 3 5 x 7 x 9 x 3 10 + 3 y 1 y 2 y 5 y 8 x 2 6 x 8 x 1 x 2 x 3 x 7 x 9 x 10 + 18 y 2 1 y 2 x 2 2 x 3 x 4 x 6 x 8 x 9 x 1 x 3 5 x 7 x 3 10 + 3 y 1 y 3 y 8 x 3 4 x 2 6 x 8 x 1 x 2 x 3 x 3 5 x 9 x 10 + 12 y 1 y 7 y 8 x 3 4 x 3 6 x 7 x 2 x 3 x 3 5 x 8 x 9 x 10 + 36 y 1 y 7 y 8 x 2 x 3 x 4 x 6 x 7 x 3 5 x 8 x 9 x 10 + 9 y 1 y 2 y 8 x 2 x 3 x 4 x 6 x 8 x 1 x 3 5 x 7 x 9 x 10 + 3 y 1 y 2 y 8 x 3 4 x 3 6 x 8 x 1 x 2 x 3 x 3 5 x 7 x 9 x 10 . F or letting all v ariables x i = x and y j = y , w e can calculate the desired expansion form ula for diagonal γ ′ = ( x (1) 3 , x (5) 2 ): x (1) 3 = 4 + 11 x + 12 x 2 + 8 x 3 + 98 y + 440 y 2 + 390 y 3 + 132 xy + 249 xy 2 + 73 x 2 y + 1 + 32 y + 300 y 2 + 572 y 3 + 285 y 4 x + 3 y + 48 y 2 + 225 y 3 + 264 y 4 + 102 y 5 x 2 + 3 y 2 + 20 y 3 + 51 y 4 + 49 y 5 + 17 y 6 x 3 + y 3 + y 4 + 3 y 5 + 3 y 6 + y 7 x 4 ; x (5) 2 = 4 y + 33 y 2 + 34 y 3 + 5 xy + 21 xy 2 + 4 x 2 y + y + 19 y 2 + 39 y 3 + 14 y 4 x + 2 y 2 + 12 y 3 + 8 y 4 + 2 y 5 x 2 + y 3 + y 4 x 3 . In general, for all vertices: x (1) 1 = 2 y ; x (5) 2 = 4 y + 33 y 2 + 34 y 3 + 5 xy + 21 xy 2 + 4 x 2 y + y + 19 y 2 + 39 y 3 + 14 y 4 x 2 y 2 + 12 y 3 + 8 y 4 + 2 y 5 x 2 + y 3 + y 4 x 3 ; x (1) 3 = 4 + 11 x + 12 x 2 + 8 x 3 + 98 y + 440 y 2 + 390 y 3 + 132 xy + 249 xy 2 + 73 x 2 y + 1 + 32 y + 300 y 2 + 572 y 3 + 285 y 4 x + 3 y + 48 y 2 + 225 y 3 + 264 y 4 + 102 y 5 x 2 + 3 y 2 + 20 y 3 + 51 y 4 + 49 y 5 + 17 y 6 x 3 + y 3 + y 4 + 3 y 5 + 3 y 6 + y 7 x 4 ; x (1) 4 = y + 2 y 2 + y 3 x ; x (3) 5 = 4 y 2 + 23 y 3 + 20 y 4 + 5 y 5 + 4 xy 2 + 4 xy 3 + xy 4 + 2 y 2 + 26 y 3 + 51 y 4 + 32 y 5 + 7 y 6 x + 5 y 3 + 20 y 4 + 21 y 5 + 8 y 6 + y 7 x 2 + 4 y 4 + 4 y 5 + y 6 x 3 ; x (1) 6 = 1 + 3 x + 3 x 2 + x 3 + 23 y + 67 y 2 + 45 y 3 + 25 xy + 30 xy 2 + 9 x 2 y + 3 y + 40 y 2 + 69 y 3 + 30 y 4 x + 3 y 2 + 21 y 3 + 25 y 4 + 9 y 5 x 2 + y 3 + 3 y 4 + 3 y 5 + y 6 x 3 ; x (12) 7 = 12 y 3 + 90 y 4 + 131 y 5 + 99 y 6 + 9 y 7 + 8 xy 3 + 12 xy 4 + 14 xy 5 + xy 6 + 11 y 3 + 466 y 5 + 557 y 6 + 275 y 7 + 30 y 8 x + 4 y 3 + 104 y 4 + 594 y 5 + 1201 y 6 + 1038 y 7 + 373 y 8 + 45 y 9 x 2 + 34 y 4 + 355 y 5 + 1085 y 6 + 1420 y 7 + 872 y 8 + 258 y 9 + 30 y 10 x 3 + 3 y 4 + 54 y 5 + 305 y 6 + 629 y 7 + 600 y 8 + 80 y 10 + 9 y 11 x 4 + 3 y 5 + 26 y 6 + 110 y 8 + 81 y 9 + 36 y 10 + 9 y 11 + y 12 x 5 + y 6 + 3 y 7 + 3 y 8 + y 9 x 6 ; x (6) 8 = 8 y 4 + 12 y 5 + 6 y 6 + y 7 + 12 y 4 + 52 y 5 + 64 y 6 + 30 y 7 + 5 y 8 x + y 3 + 7 y 4 + 70 y 5 + 143 y 6 + 121 y 7 + 47 y 8 + 7 y 9 x 2 + 6 y 4 + 27 y 5 + 72 y 6 + 86 y 7 + 48 y 8 + 12 y 9 + y 10 x 3 + 12 y 5 + 36 y 6 + 39 y 7 + 18 y 8 + 3 y 9 x 4 + 8 y 6 + 12 y 7 + 6 y 8 + y 9 x 5 ; x (4) 9 = 1 + 2 x + x 2 + 13 y + 13 y 2 + 7 xy + 2 y + 6 y 2 + 3 y 3 x + y 2 x 2 ; x (2) 10 = 3 y + 3 y 2 + xy + y 2 + y 3 x . Clearly , tw o given flips are differen t, so in case we do a general triangulation on a surface, we need to clarify whic h type of flips from each quadrilateral. 68 A Recursion form ula via go o d lattice The following section demonstrates further observ ation of the recurrence relation discussed in Section 3.4 , partially solving Prop osition 3.15 . Note that from Prop ositions 3.7 and 3.9 , we can rewrite the following v alues in case either a = 1 or b = 1: f (1 , b, i, j ) = X ( l 1 +1) ,l i ,u i , ( u k +1) ,k ∈ N : P k i =1 ( l i + u i )= b x i − P k s =1 l s ,j + P k t =1 u t k Y v =1 x i +1 − P v − 1 s =1 l s ,j + P v − 1 t =1 u t x i +1 − P v s =1 l s ,j + P v − 1 t =1 u t x i − P v s =1 l s ,j − 1+ P v − 1 t =1 u t x i − P v s =1 l s ,j − 1+ P v t =1 u t ! ; f ( a, 1 , i, j ) = X ( r 1 +1) ,r i ,d i , ( d k +1) ,k ∈ N : P k i =1 ( r i + d i )= a x i + P k s =1 r s ,j − P k t =1 d t k Y v =1 x i − 1+ P v − 1 s =1 r s ,j − P v − 1 t =1 d t x i − 1+ P v s =1 r s ,j − P v − 1 t =1 d t x i + P v s =1 r s ,j +1 − P v − 1 t =1 d t x i + P v s =1 r s ,j +1 − P v t =1 d t ! for all a, b ≥ 1. W e shall introduce the notion of go o d lattic e and all its related features. Definition A. 1. F or any ( a, b ) ∈ N 2 , denote L ( a, b ) := { ( p, q ) ∈ Z 2 | − a ≤ min {− p, q , − p + q } ≤ max {− p, q , − p + q } ≤ b } \ { ( a, b ) , ( − b, − a ) } ⊂ Z 2 called the go o d lattic e of typ e ( a, b ). Additionally for a, b ≥ 3 w e also define the degenerate cases: L (0 , 0) = L (0 , 0) = L (1 , 0) = L (2 , 0) = L (0 , 1) = L (0 , 2) := { (0 , 0) } , L (0 , b ) := { ( − p, q ) ∈ Z 2 | p, q ∈ N , p + q ≤ b − 1 } ∪ { (0 , 0) } and L ( a, 0) := { ( p, − q ) ∈ Z 2 | p, q ∈ N , p + q ≤ a − 1 } ∪ { (0 , 0) } . F urthermore, for any ( a, b ) ∈ N 2 , we shall call the conv ex hull of a go o d lattice of type ( a, b ) its b oundary and denote it by ∂ L ( a, b ). See Figure 44 for examples. (0 , 0) (0 , 0) (0 , 0) Figure 44: All p oin ts in L (3 , 5) (left), L (0 , 6) (middle), L (4 , 0) (right) and the corresp onding b oundary of L (3 , 5) (red), using the grid lab eling of Definition 4.1 . F or each ( a, b ) ∈ N 2 , we further define subsets ∂ □ L ( a, b ) with □ ∈ {← , → , ↓ , ↑} of the b oundary ∂ L ( a, b ), where ∂ ← L ( a, b ) (resp. → , ↓ , ↑ ) is defined to b e the set of all b oundary p oin ts that are strictly to the left (resp. right, b elow, ab ov e) the p oin t (0 , 0), see Figure 45 for examples. 69 (0 , 0) ∂ ← L (3 , 5) (0 , 0) ∂ ↓ L (3 , 5) (0 , 0) ∂ → L (3 , 5) (0 , 0) ∂ ↑ L (3 , 5) Figure 45: All 4 sets ∂ □ L (3 , 5) (red) for □ ∈ {← , → , ↓ , ↑} W e can directly compute the follo wing v alues for any a, b ≥ 1: |L ( a, b ) | = ab + ( a + b + 1)( a + b + 2) 2 − 2; | ∂ L ( a, b ) | = 3( a + b ) − 2; |L (0 , b ) | = 1 + ( b − 1)( b − 2) 2 ; |L ( a, 0) | = 1 + ( a − 1)( a − 2) 2 ; | ∂ ← L ( a, b ) | = a + 2 b − 2; | ∂ → L ( a, b ) | = 2 a + b − 2; | ∂ ↓ L ( a, b ) | = 2 a + b − 2; | ∂ ↑ L ( a, b ) | = a + 2 b − 2 . F or m ∈ N , we consider the standard lexicographical ordering on Z m : for any ( a 1 , a 2 , ..., a m ) , ( b 1 , b 2 , ..., b m ) ∈ Z m , we ha ve ( a 1 , a 2 , ..., a m ) > ( b 1 , b 2 , ..., b m ) if and only if there exists 1 ≤ k ≤ m suc h that a i = b i for i = 1 , 2 , ..., k − 1 and a k > b k . Now we can write each v alue g ( a, b, i, j ) (for a, b ∈ Z ≥ 0 and i, j ∈ Z ) of form: g ( a, b, i, j ) = 2 ab X N =1 Y ( k,l ) ∈L ( a,b ) x c N,k ,l a,b i + k,j + l where w e hav e a total of 2 ab exp onential ve ctors c N ,a,b = ( c N ,a,b − 1 a,b , c N ,a,b − 2 a,b , ..., c N , − b, − a +1 a,b ) that are arranged in the lexicographical order of Z ab + ( a + b +1)( a + b +2) 2 − 2 , with all terms c N ,k,l a,b arranged in the lexicographical order of Z 2 for pairs ( k , l ) in the goo d lattice of ( a, b ) for each N = 1 , 2 , ..., 2 ab . W e also define x N ,a,b i,j := Y ( k,l ) ∈L ( a,b ) x c N,k ,l a,b i + k,j + l 70 for all ( a, b, i, j ) ∈ Z 2 ≥ 0 × Z 2 and N = 1 , 2 , ..., 2 ab . W e hav e the following particular v alues: c 1 , 0 ,b = (1 , 1 , ..., 1); c 1 ,a, 0 = (1 , 1 , ..., 1); c 1 , 1 , 1 = (1 , 0 , 0 , 0 , 1); c 2 , 1 , 1 = (0 , 1 , 0 , 1 , 0); c 1 , 1 , 2 = (1 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 1 , 0); c 2 , 1 , 2 = (0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0); c 3 , 1 , 2 = (0 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 1); c 4 , 1 , 2 = (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 0); c 1 , 2 , 1 = (1 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0); c 2 , 2 , 1 = (0 , 1 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 1); c 3 , 2 , 1 = (0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0); c 4 , 2 , 1 = (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 0); F or all a, b ≥ 1, w e can compute: g ( a + 1 , b + 1 , i, j ) = 2 ( a +1)( b +1) X P =1 Y ( k,l ) ∈L ( a +1 ,b +1) x c P,k ,l a +1 ,b +1 i + k,j + l = 2 ( a +1)( b +1) X P =1 x c P,a,b a +1 ,b +1 i + a,j + b x c P, − b, − a a +1 ,b +1 i − b,j − a ·   Y ( k,l ) ∈L ( a,b ) x c P,k ,l a +1 ,b +1 i + k,j + l   ·   Y ( k,l ) ∈ ∂ L ( a +1 ,b +1) x c P,k ,l a +1 ,b +1 i + k,j + l   ; g ( a + 1 , b, i − 1 , j ) g ( a, b + 1 , i + 1 , j ) =   2 ( a +1) b X M =1 Y ( k,l ) ∈L ( a +1 ,b ) x c M,k ,l a +1 ,b i − 1+ k,j + l   ·   2 a ( b +1) X N =1 Y ( k,l ) ∈L ( a,b +1) x c N,k ,l a,b +1 i +1+ k,j + l   =   2 ( a +1) b X M =1 x c M, − b, − a a +1 ,b i − b,j − a ·   Y ( k,l ) ∈L ( a,b ) x c M,k ,l a +1 ,b i + k,j + l   ·   Y ( k,l ) ∈ ∂ ← L ( a +1 ,b +1) x c M,k ,l a +1 ,b i + k,j + l     ·   2 a ( b +1) X N =1 x c N,a,b a,b +1 i + a,j + b ·   Y ( k,l ) ∈L ( a,b ) x c N,k ,l a,b +1 i + k,j + l   ·   Y ( k,l ) ∈ ∂ → L ( a +1 ,b +1) x c N,k ,l a,b +1 i + k,j + l     ; g ( a, b + 1 , i, j − 1) g ( a + 1 , b, i, j + 1) =   2 a ( b +1) X N =1 Y ( k,l ) ∈L ( a,b +1) x c N,k ,l a,b +1 i + k,j − 1+ l   ·   2 ( a +1) b X M =1 Y ( k,l ) ∈L ( a +1 ,b ) x c M,k ,l a +1 ,b i + k,j +1+ l   =   2 a ( b +1) X N =1 x c N, − b, − a a,b +1 i − b,j − a ·   Y ( k,l ) ∈L ( a,b ) x c N,k ,l a,b +1 i + k,j + l   ·   Y ( k,l ) ∈ ∂ ↓ L ( a +1 ,b +1) x c N,k ,l a,b +1 i + k,j + l     ·   2 ( a +1) b X M =1 x c M,a,b a +1 ,b i + a,j + b ·   Y ( k,l ) ∈L ( a,b ) x c M,k ,l a +1 ,b i + k,j + l   ·   Y ( k,l ) ∈ ∂ ↑ L ( a +1 ,b +1) x c M,k ,l a +1 ,b i + k,j + l     . Then by the Prop osition 3.15 , w e hav e the following computations: LHS = g ( a + 1 , b + 1 , i, j ) g ( a, b, i, j ) =   2 ab X Q =1 Y ( k,l ) ∈L ( a,b ) x c Q,k,l a,b i + k,j + l   ·   2 ( a +1)( b +1) X P =1 x c P,a,b a +1 ,b +1 i + a,j + b x c P, − b, − a a +1 ,b +1 i − b,j − a ·   Y ( k,l ) ∈L ( a,b ) x c P,k ,l a +1 ,b +1 i + k,j + l   ·   Y ( k,l ) ∈ ∂ L ( a +1 ,b +1) x c P,k ,l a +1 ,b +1 i + k,j + l     = 2 ab X Q =1 2 ( a +1)( b +1) X P =1 x c P,a,b a +1 ,b +1 i + a,j + b x c P, − b, − a a +1 ,b +1 i − b,j − a ·   Y ( k,l ) ∈L ( a,b ) x c P,k ,l a +1 ,b +1 + c Q,k,l a,b i + k,j + l   ·   Y ( k,l ) ∈ ∂ L ( a +1 ,b +1) x c P,k ,l a +1 ,b +1 i + k,j + l   ; 71 RHS = x i,j − a x i,j + b · g ( a + 1 , b, i − 1 , j ) g ( a, b + 1 , i + 1 , j ) + x i + a,j x i − b,j · g ( a, b + 1 , i, j − 1) g ( a + 1 , b, i, j + 1) = x i,j − a x i,j + b · 2 ( a +1) b X M =1 x c M, − b, − a a +1 ,b i − b,j − a · Y ( k,l ) ∈L ( a,b ) x c M,k ,l a +1 ,b i + k,j + l ! · Y ( k,l ) ∈ ∂ ← L ( a +1 ,b +1) x c M,k ,l a +1 ,b i + k,j + l !! · 2 a ( b +1) X N =1 x c N,a,b a,b +1 i + a,j + b · Y ( k,l ) ∈L ( a,b ) x c N,k ,l a,b +1 i + k,j + l ! · Y ( k,l ) ∈ ∂ → L ( a +1 ,b +1) x c N,k ,l a,b +1 i + k,j + l !! + x i + a,j x i − b,j · 2 a ( b +1) X N =1 x c N, − b, − a a,b +1 i − b,j − a · Y ( k,l ) ∈L ( a,b ) x c N,k ,l a,b +1 i + k,j + l ! · Y ( k,l ) ∈ ∂ ↓ L ( a +1 ,b +1) x c N,k ,l a,b +1 i + k,j + l !! · 2 ( a +1) b X M =1 x c M,a,b a +1 ,b i + a,j + b · Y ( k,l ) ∈L ( a,b ) x c M,k ,l a +1 ,b i + k,j + l ! · Y ( k,l ) ∈ ∂ ↑ L ( a +1 ,b +1) x c M,k ,l a +1 ,b i + k,j + l !! = x i,j − a x i,j + b · 2 ( a +1) b X M =1 2 a ( b +1) X N =1 x c N,a,b a,b +1 i + a,j + b x c M, − b, − a a +1 ,b i − b,j − a · Y ( k,l ) ∈L ( a,b ) x c M,k ,l a +1 ,b + c N,k ,l a,b +1 i + k,j + l ! · Y ( k,l ) ∈ ∂ ← L ( a +1 ,b +1) x c M,k ,l a +1 ,b i + k,j + l ! · Y ( k,l ) ∈ ∂ → L ( a +1 ,b +1) x c N,k ,l a,b +1 i + k,j + l !! + x i + a,j x i − b,j · 2 ( a +1) b X M =1 2 a ( b +1) X N =1 x c M,a,b a +1 ,b i + a,j + b x c N, − b, − a a,b +1 i − b,j − a · Y ( k,l ) ∈L ( a,b ) x c M,k ,l a +1 ,b + c N,k ,l a,b +1 i + k,j + l ! · Y ( k,l ) ∈ ∂ ↓ L ( a +1 ,b +1) x c N,k ,l a,b +1 i + k,j + l ! · Y ( k,l ) ∈ ∂ ↑ L ( a +1 ,b +1) x c M,k ,l a +1 ,b i + k,j + l !! . B Maple co de for computation of recurrence relation The following Maple co de computes the recurrence relation discussed in Section 3.4 . restart; # Function f: discrete 4D Hirota-Miwa equation f := proc(a, b, i, j) local res; option remember; if a = 0 or b = 0 then res := x[i, j]; elif a = 1 and b = 1 then res := (x[i - 1, j]*x[i + 1, j] + x[i, j - 1]*x[i, j + 1])/x[i, j]; else res := expand(factor( (f(a, b - 1, i - 1, j)*f(a - 1, b, i + 1, j) + f(a - 1, b, i, j - 1)*f(a, b - 1, i, j + 1)) / f(a - 1, b - 1, i, j) )); end if; return res; end proc: # Function X_{i,j}^{a,b}: product over the specified range 72 X := proc(a, b, i, j) local p, q, product_result, min_val, max_val; option remember; # Handle the (0,0) case if a = 0 and b = 0 then return x[i, j]; end if; product_result := 1; # For (a,b) \neq (0,0), compute the product over integers p,q # with 1-a \leq min{p,q,p+q} \leq max{p,q,p+q} \leq b-1 for p from 1-a to b-1 do for q from 1-a to b-1 do min_val := min(p, q, p+q); max_val := max(p, q, p+q); if min_val >= 1-a and max_val <= b-1 then product_result := product_result * x[i - p, j + q]; end if; end do; end do; return product_result; end proc: # Function g(a,b,i,j) = X_{i,j}^{a,b} * f(a,b,i,j) g := proc(a, b, i, j) local x_product, f_value; option remember; x_product := X(a, b, i, j); f_value := f(a, b, i, j); return expand(x_product * f_value); end proc: # Function to verify the recurrence relation for g verify_g_recurrence := proc(a, b, i, j) local LHS, RHS, term1, term2; # Left-hand side of the recurrence LHS := g(a + 1, b + 1, i, j) * g(a, b, i, j); # Right-hand side: two terms term1 := x[i, j - a] * x[i, j + b] * g(a + 1, b, i - 1, j) * g(a, b + 1, i + 1, j); term2 := x[i + a, j] * x[i - b, j] * g(a, b + 1, i, j - 1) * g(a + 1, b, i, j + 1); RHS := expand(term1 + term2); # Check if LHS equals RHS if simplify(LHS - RHS) = 0 then return true; 73 else return false, LHS, RHS; end if; end proc: F rom the code ab o ve, using Prop osition 3.15 yields the v alidit y of the recurrence relation for g , i.e. we alw ays get “ true ” as output from the call verify_g_recurrence . 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