Some Open Problems in Combinatorial Physics

Some Op en Proble m s in Com binatorial Ph ysics G H E Duch amp a , H Cheballah a and the CIP team. a LIPN - UMR 7030 CNRS - Universit ´ e P aris 13 F-9343 0 Villetaneuse, F rance E-mail: ghed@l ipn-univ.paris13.fr , hayat. cheballah@lipn-univ.paris13.fr 02-11- 2018 0 8:47 Con ten t s 1 Problem A: Multiplicities in diag . 2 1.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem B: Comb inatorics of Riordan-Sheff er one-param eter groups. 4 2.1 Problem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Problem C: A corpus for com binatorial v ector fields. 4 3.1 Problem C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Problem D Probabilistic study of appro ximate substitutions 5 4.1 Problem D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Problem A: Multiplicities in diag. 1.1. Setting Let H ( F , G ) b e the Hadamard exponen tial pro duct as defined b elo w by F ( z ) = X n ≥ 0 a n z n n ! , G ( z ) = X n ≥ 0 b n z n n ! , H ( F , G ) := X n ≥ 0 a n b n z n n ! . (1) In the case of free exp onen tials, that is if w e write the functions as F ( z ) = exp ∞ X n =1 L n z n n ! ! , G ( z ) = exp ∞ X n =1 V n z n n ! ! , (2) and using the expansion with Bell p olynomials in the sets of v ariables L = { L n } , V = { V m } (see [6, 10] for details), w e obtain H ( F , G ) = X n ≥ 0 z n n ! X P 1 ,P 2 ∈ U P n L T y p e ( P 1 ) V T y p e ( P 2 ) (3) where U P n is the set of unordered partitio ns of [1 · · · n ]. ßAn unordered partit ion P of a set X is a subse t of P ⊂ P ( X ) − {∅}‡ (that is a n unordered collection of blo c ks, i. e. non-empty subsets of X ) suc h that • the union S Y ∈ P Y = X ( P is a co v ering) • P consists of disjoin t subsets, i. e. Y 1 , Y 2 ∈ P and Y 1 ∩ Y 2 6 = ∅ = ⇒ Y 1 = Y 2 . The ty p e of P ∈ U P n (denoted ab o v e b y T y pe ( P ) ) is the multi-index ( α i ) i ∈ N + suc h that α k is the n um b er o f k -blo c ks, that is the n um b er o f mem b ers of P with cardinalit y k . ßLet P 1 , P 2 b e t w o unordered partitions of the same set. T o eac h lab elling of the blo c ks P r = { B ( r ) i } 1 ≤ i ≤ n r ; r = 1 , 2 (4) one can asso ciate the intersec tion matr ix M =  card( B (1) i ∩ B (2) j )  1 ≤ i ≤ n 1 ; 1 ≤ j ≤ n 2 . (5) As ( P 1 , P 2 ) are, in essence, unla b elled, the arrow so constructed ( P 1 , P 2 ) 7→ cl ass ( M ) = d (6) aims at classes of pac k ed matrices [7] under p erm utations of rows and columns. These classes hav e b een shown [2 , 3 ] to b e in one to one corresp ondence with F eynman- Bender diagrams [1] whic h are bicoloured gra phs with p (= card( P 1 )) black sp ot s, q (= card( P 2 )) white sp ots, no isola ted v ertex and in teger m ultiplicities. W e denote the set of suc h diagrams by diag [8, 9]. Then, the corresp ondence go es as sho w ed b elo w. ‡ The set of subsets of X is denoted b y P ( X ) (th is notation [4] is that of the former German school). ❥ ❥ ❥ ❥ ③ ③ ③ { 1 } { 2 , 3 , 4 } { 5 , 6 , 7 , 8 , 9 } { 10 , 11 } { 2 , 3 , 5 } { 1 , 4 , 6 , 7 , 8 } { 9 , 10 , 11 } Fig 1 . — Di agr am fr om P 1 , P 2 (set p artitions of [1 · · · 11] ). P 1 = {{ 2 , 3 , 5 } , { 1 , 4 , 6 , 7 , 8 } , { 9 , 10 , 11 }} and P 2 = {{ 1 } , { 2 , 3 , 4 } , { 5 , 6 , 7 , 8 , 9 } , { 10 , 1 1 }} (r esp e ctively black sp ots for P 1 and white sp ots for P 2 ). The incidenc e matrix c orr esp ond ing to the dia gr am (as dr awn) o r th ese p artitions is   0 2 1 0 1 1 3 0 0 0 1 2   . Bu t, due to the fact that the defining p artitions ar e unor d er e d, one c an p ermute the sp ots (black and white, b etwe en themselves) and, so, the lines and c olumns of this matrix c an b e p ermute d. The diagr am c ould b e r e pr ese nte d by the matrix   0 0 1 2 0 2 1 0 1 0 3 1   as wel l. Noting mul t ( d ) the cardinality of eac h fibre of (6) , formula (3) reads H ( F , G ) = X n ≥ 0 z n n ! X d ∈ diag | d | = n mul t ( d ) L α ( d ) V β ( d ) (7) where α ( d ) (resp. β ( d )) is the “white spo ts t ype” (resp. t he “blac k sp ots t yp e”) i.e. the m ulti-index ( α i ) i ∈ N + (resp. ( β i ) i ∈ N + ) suc h that α i (resp. β i ) is the num b er of white sp ots (resp. blac k sp ot s) of degree i ( i lines connected to the sp ot) and mul t ( d ) is the n um b er of pairs o f unordered partitio ns of [1 · · · | d | ] (here | d | = | α ( d ) | = | β ( d ) | is the n um b er of lines of d ) with asso ciated diagram d . 1.2. Pr o blem A Giv e a form ula (a s smart as p ossible) for mul t ( d ) as a function o f d (in the language of [7], a s a f unction of the class of a pa c k ed matrix under the p ermutation of rows a nd columns). ß Hin t . — F or practical computations, one of the tw o partitions may b e k ept fixed, sa y P 1 and the result of the en umeration multiplied b y n ! | stab ( P 1 ) | . 2. Problem B: Comb inatorics of Riordan-Sheff er one-param eter groups. W e start with the (v ector) space C N × N of complex bi-infinite ma trices. Let RF ( N , C ) = ( C ( N ) ) N the space of ro w-finite matrices (i. e. matrices for w hic h ev ery row is finitely supp o rted). T o ev ery matrix T ∈ RF ( N , C ), one can asso ciate the sequence transformation ( a k ) k ∈ N 7→ ( b n ) n ∈ N (8) giv en b y b n = X k ∈ N T [ n, k ] a k (9) this sum is finitely supp orted as T ∈ RF ( N , C ). One can prov e that the set RF ( N , C ) is exactly t he a lgebra o f contin uous endomorphisms of C N endo w ed with t he t op ology of p oint wise con v ergence. This transformation can b e transp orted o n EGFs b y f = X k ∈ N a k z k k ! 7→ ˆ f = X n ∈ N b n z n n ! (10) and, in case ˆ f is giv en b y ˆ f ( z ) = Φ g ,φ [ f ]( z ) = g ( z ) f ( φ ( z )) . (11) with g ( z ) = 1 + highe r terms and φ ( z ) = z + higher terms . (12) w e sa y t hat the matrix is a matrix of substitutions with prefunction. In classical combinatorics (for OGF and EGF), the matr ices M g ,φ ( n, k ) a re kno wn as R ior dan matric es (see [11, 12] for example). One can prov e, using a Zariski-lik e argumen t, the follow ing prop osition [10, 5]. Prop osition 2.1 [10] L et M b e the matrix o f a substitution with pr efunc tion; then so is M t for al l t ∈ C . 2.1. Pr o blem B a) Prov ide a combinatorial pro of of the preceding prop osition f or t ∈ Q (without using the ”pr o-algebraic” structure o f the group o f substitutions with prefunctions, directly or indirectly). b) Giv e a com binatorial in terpretation of M 1 / 2 for some Sheffer matrices. 3. Problem C: A corpus for com binatorial v ector fields. With the preceding notations one can show that, if M is a matrix of substitution with prefunction, the limit lim q → + ∞ q ( M 1 q − I ) (13) exists (call it L ) and t he asso ciated transformation of sequences (see a b ov e) is the sum of a v ector field and a scalar field. One can see tha t M ∈ Q N × N = ⇒ L ∈ Q N × N . (14) in addition, if M is a matrix of substitution (i. e. the prefunction is ≡ 1) then the scalar field is zero and so the asso ciated differen tial op erator is a pure ve ctor field (with co efficien t s in Q if M is in Q N × N ). On the ot her hand, if C is a class of lab elled g raphs for whic h the exp onen tial f orm ula applies, the matrix M suc h that M [ n, k ] = Numb er of gr ap hs la b el le d by [1 ..n ] an d with k c onne cte d c omp onents (15) is a matrix o f substitution [10]. F or example with the g raphs o f equiv alence relations on finite sets, the substitution is z 7→ e z − 1; f or g raphs of idemp otent endofunctions, the substitution is z 7→ z e z . 3.1. Pr o blem C a) What is the com binatorial in t erpretation of the co efficien ts of the v ector field for the t w o preceding examples ? b) Can w e give a n y insight of the fo rm of this v ector field for general classes of graphs ? ß Hin t . — M z = e z log ( M ) where l og ( M ) is the matrix o f a differential o p erator of the form q ( z ) d dz + v ( z ). 4. Problem D Probabilistic study of appro ximate substitutions Our motiv ation, in this sec tion, consists in appro ximating the matrices of infinite substi- tutions by finite matrices of (approximate) substitutions. W e are then in terested in the probabilistic study of these matrices. T o this end, w e randomly generate unip ot en t (uni- triangular) matrices and observ e the num b er of o ccurrences of matrices of substitutions. W e start by giving some examples of our exp erimen t whic h are summarized in the table b elo w: Size Num ber of dra wings Range of v ariables Probabilit y [3 × 3] 300 [1 · · · 10] 1 [1 · · · 100] 1 [1 · · · 10000 ] 1 [4 × 4] 275 [1 · · · 10] 0 . 0473 [1 · · · 100] 0 . 0001 [1 · · · 10000 ] 0 + [10 × 10] 1500 [1 · · · 10] 0 . 0327 [1 · · · 100] 0 + [1 · · · 10000 ] 0 + According to the res ults obtained, w e observ e that the (approx imate) substitution matrices are not v ery frequen t. Ho w ev er, in meeting certain conditions suc h as size, the n um b er of drawings and the rang e of the v ariables, w e can obtain p ositiv e probabilities that these matrices app ear. Let us note t hat the smaller the size of the matrix the more probable one obtains a matrix of substitution in a reasonable n um ber of drawings. W e also notice that, if w e v a ry the range of v ariables, and this in an increasing wa y a nd b y k eeping unchanged the num b er of dra wings and size, the pro babilit y tends to zero. W e also notice t hat the unipot en t matrices of size 3 are all matrices of approximate substitutions. This is easy to see b ecause the exponential g enerating series of the 3 r d column will alwa ys hav e the form c k = x 2 2! . Th us, w e can say that t he test actually starts f rom the matrices of size higher o r equal to 4. Result 4.1 L et r r epr esent the c ar dinality of the r a nge of varia bles and n × n b e the size of the matrix. A c c or ding to the r esults obtaine d; we c an say that the pr ob abil ity p n of app e ar anc e of the matric es of substitutions dep ends on r and n a nd we have the fol lowing upp er b ound: p n ≤ r 2 n − 3 r n ( n − 1) 2 (16) which shows that p n − → 0 as n − → ∞ (17) 4.1. Pr o blem D One can conjecture that the effect of the range selection v anishes when n tends to infinit y . More precisely: p n ∼ r 2 n − 3 r n ( n − 1) 2 (18) References [1] C. M. Bender, D. C. Brod y, and B. K. Meister , Qua ntum field theory of par titions, J . Math. Phys. V ol 4 0 (199 9) [2] P . Blasiak, A. Horzela, K. A . Penson, G. H. E. Duchamp, A.I. Solomon , Boson normal or deri ng via substitu tions and Sheffer-T yp e Polynomials , Phys. Lett. A 338 (2005) 108 [3] P . Blasiak, K. A. Penson, A.I. Solomon , A. Horzela, G. H. E. Duchamp , Some useful formula for b osonic op er ators , Jour. Math. Phys. 46 0521 1 0 (20 05). [4] Bo urbaki N. , The ory of sets , Spring er [5] H . Cheballah, G. H. E. 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