Determinants and Perfect Matchings

DETERMINA NTS AND PER FEC T MA TCHINGS AR VIND A YYER Abstract. W e give a com binatorial interpretation of the deter- minant of a matrix as a gener a ting function ov er Brauer diagrams in tw o different but related w ays. The sign of a p ermutation as- so ciated to its num ber o f inversions in the L e ibniz for m ula for the determinant is replaced by the num ber of crossings in the Brauer diagram. This in terpretation natura lly explains wh y the determi- nant of an even a n tisymmetric matrix is the square of a Pfaffian. 1. Introduction There are man y differen t form ulas for ev aluating the determinant of a matrix. Apart from the familiar Leibniz form ula, there is Laplace form ula, D o dgson’s condensation and Gaussian elimination. How ev er, there is no formula to the b est of our kno wle dge in whic h Ca yley’s celebrated form ula [Ca y47] relating Pfaffians to determinants is trans- paren t. In this work, w e give a new form ula whic h do es precisely this. The form ula uses the notion of Bra uer diagrams. These parametrize the ba sis elemen ts of the so-called Brauer algebra [Bra37], whic h is imp ortant in the represen tation theory of the orthogonal group. Brauer diagrams are p erfect matc hings on a certain kind of planar graph. W e shall pro ve in Theorem 1 (to b e stated formally in Sec- tion 3) that the determinan t o f an n × n matrix can b e expanded a s a sum ov er all Brauer diagrams o f a certain w eight function. Since p erfect matc hings are related to Pfaffians, w e obtain a natural com- binatorial in terpretation of Ca yley’s b eautiful result relating Pfaffians and determinan ts. There hav e b een some connections no ted in the literature b etw een Bra uer diagrams and com binatorial ob jects suc h as Y oung tableaux [Sun86 , T er01, HL06], a nd Dyc k paths [MM11] in t he past. The connection b et we en determinan ts and p erfect matc hings came up while studying the num b er of terms (including repetitions) in the determinan ts of Hermitian matrices, whic h turns out to b e (2 n − 1)!!. The num b er of distinct terms in the determinan t of symmetric and sk ew-symmetric matr ices, on the other hand, is classical. This has b een Date : O ctob er 23, 2018. 1 2 AR VIN D A YYER studied, among others, b y Ca yley and Sylv ester [Mui60]. In particular, Sylv ester sho wed that the n um b er o f distinct t erms in the determinan t of a sk ew-symmetric matrix of size 2 n is given b y (2 n − 1)!! v n , where v n satisfies (1.1) v n = (2 n − 1) v n − 1 − ( n − 1) v n − 2 , v 0 = v 1 = 1 . Aitk en [Ait44] has also studied recurrences for the num b er of terms in symmetric and sk ew-symmetric determinan ts. The num b er of terms in the symmetric determinant also app ears in a problem in the American Mathematical Monthly prop osed by Richard Stanley [SR72]. The spirit of this work is similar to those on combinatorial in ter- pretations of identities and formulas in linear algebra [Jac77, F oa80, Str83, Zei85], combinatorial form ulas for determinants [Zei97], and f o r Pfaffians [Hal6 6, Knu96, MSV04, E˘ ge90]. The plan of the pap er is as follows. Tw o non-standard r epresen- tations of a matrix are give n in Section 2. W e recall the definition of Brauer diagrams in Section 3. W e will a lso define the we igh t and the crossing num b er of a Brauer dia g ram, and state the main theorem there. W e will then digress to giv e a differen t combinatorial explana- tion for the n um b er of terms in t he determinant of these non-standar d matrices in Section 4. The main idea of the pro of is a bijection b etw een terms in b oth determinan t expansions and Brauer diagrams, whic h will b e giv en in Section 5. W e define the crossing n um b er for a Brauer dia- gram and prov e some prop erties ab out it in Section 6. The main r esult is then pro ved in Section 7. 2. Two Different M a tr ix Rep resent a tions A word a b out nota tion: throughout, w e will use ı as the complex n umber √ − 1 and i as an indexing v ariable. Let A b e a symmetric matrix and B b e a sk ew-symmetric matrix. An y mat r ix can b e decom- p osed in t wo w ays as a linear com bination of A and B , namely A + B and A + ı B . W e denote the former b y M F and the la tter by M B . The terminology will b e explained lat er. That is, (2.1) ( M F ) i,j =      a i,j + b i,j i < j, a j,i − b j,i i > j, a i,i i = j, ; ( M B ) i,j =      a i,j + ı b i,j i < j, a j,i − ı b j,i i > j, a i,i i = j, DETERMINANTS AND PERFECT M A TCHINGS 3 where a i,j and b i,j are complex indeterminates. F or example, a generic 3 × 3 matrix can b e written in these t w o w a ys, M (3) F =   a 1 , 1 a 1 , 2 + b 1 , 2 a 1 , 3 + b 1 , 3 a 1 , 2 − b 1 , 2 a 2 , 2 a 2 , 3 + b 2 , 3 a 1 , 3 − b 1 , 3 a 2 , 3 − b 2 , 3 a 3 , 3   , M (3) B =   a 1 , 1 a 1 , 2 + ı b 1 , 2 a 1 , 3 + ı b 1 , 3 a 1 , 2 − ı b 1 , 2 a 2 , 2 a 2 , 3 + ı b 2 , 3 a 1 , 3 − ı b 1 , 3 a 2 , 3 − ı b 2 , 3 a 3 , 3   . (2.2) Notice tha t a i,j is defined when i ≤ j and b i,j is defined when i < j . The determinan t of the matrices is clearly a p olynomial in these indeterminates. F or example, the determinan t o f the matrices in (2.2) is given b y det( M (3) F ) = a 1 , 1 a 2 , 2 a 3 , 3 − a 1 , 1 a 2 , 3 2 − a 2 , 2 a 1 , 3 2 − a 3 , 3 a 1 , 2 2 + a 1 , 1 b 2 , 3 2 + a 2 , 2 b 1 , 3 2 + a 3 , 3 b 1 , 2 2 + 2 a 1 , 2 a 2 , 3 a 1 , 3 − 2 a 1 , 2 b 2 , 3 b 1 , 3 + 2 a 1 , 3 b 1 , 2 b 2 , 3 − 2 a 2 , 3 b 1 , 2 b 1 , 3 , det( M (3) B ) = a 1 , 1 a 2 , 2 a 3 , 3 − a 1 , 1 a 2 , 3 2 − a 2 , 2 a 1 , 3 2 − a 3 , 3 a 1 , 2 2 − a 1 , 1 b 2 , 3 2 − a 2 , 2 b 1 , 3 2 − a 3 , 3 b 1 , 2 2 + 2 a 1 , 2 a 2 , 3 a 1 , 3 + 2 a 1 , 2 b 2 , 3 b 1 , 3 − 2 a 1 , 3 b 1 , 2 b 2 , 3 + 2 a 2 , 3 b 1 , 2 b 1 , 3 , (2.3) in these t w o decomp ositions. The n um b er of terms in eac h of the form ula s in (2.3) is seen to b e 15, whic h is equal to 5!!. 3. B ra uer Diagrams One of the most common represen tations o f p ermutations is the tw o- line represen tation or t wo-line diagram of a p erm utation. This is also an example of a p erfect ma t ching on a complete bipartite graph. 1 2 3 4 5 6 7 r r r r r r r 1 2 3 4 5 6 7 r r r r r r r ✑ ✑ ✑ ✑ ✑ ✟ ✟ ✟ ✟ ✟ ✟ ❅ ❅ ❅ ❆ ❆ ❆    ❍ ❍ ❍ ❍ ❍ ❍ ❅ ❅ ❅ Figure 1. A tw o- line diagram for the p erm utation 36 41725. One of the adv antages of a t w o-line diag ram is that the in v ersion n umber of a p ermu tation is simply the num b er of pairwise in tersections of the n lines. In Figure 1 ab ov e, there are 10 in tersections, whic h is the inv ersion num b er of the p erm utation 3 6 41725. 4 AR VIN D A YYER W e will consider the complete gr aph on 2 n ve rtices arranged in a t wo-line represen tation. R ecall that a p erfect matc hing of a graph is a set of pairwise non-adjacen t edges which matc hes all the v ertices of a graph. The visual represen tatio ns of suc h perf ect matc hings are called Brauer diagra ms and are defined formally b elo w. Definition 1. L et T and B b e the set of ve rtic es in the top and b ottom r ow r esp e ctively, with n p oints e ach, forming a two-line di a gr am. An unlabeled Brauer diagram of size n , µ , is a p erfe ct ma tchin g w her e an e dge joining two p oints in T is c al le d a c up ; an e dge joining two p oi n ts in B is c al l e d a cap a n d an e dge joining a p oint in T with a p oi n t in B is c al le d an arc . F or c onvenienc e, we c al l the form er horizontal e dges, and the latter, vertic al . The e d ges satisfy the fol lowing c on d itions. (1) Two c aps may interse ct in a t most one p oint. (2) Two cups may interse ct in at most on e p oint. (3) A c ap and a cup may not interse ct. (4) A n a r c me ets an ar c or a c ap or a cup in at mo s t one p oint. Let B n b e the set of unlab eled Brauer diagrams of size n . Figure 2 depicts a n unlab eled Brauer diag r a m of size sev en. W e no w define r r r r r r r r r r r r r r ✟ ✟ ✟ ✟ ✟ ✟ ❅ ❅ ❅ ❅ ❅ ❅ Figure 2. An unlab eled Brauer diagram of size 7 with sev en crossings. t wo t yp es of lab eled Brauer diag r a ms. Definition 2. L et µ ∈ B n and let T b e lab e le d with the inte gers 1 thr ough n fr om left to right. An F - Brauer diagram (f or forwar d) is a Br auer diagr am w her e the inte gers 1 thr ough n ar e lab ele d left to right and an B -Brauer diagram ( f o r b ackw a r d) is a Br auer diagr am wher e the inte gers 1 thr ough n ar e lab ele d right to le f t. The F -Brauer diagram has the same lab eling as the usual t w o-line diagram fo r a p erm uta tion. Let ( B F ) n (resp. ( B B ) n ) b e the set of F - Brauer diagrams (resp. B -Brauer diagrams) of size n . Figure 3 sho ws an example of each type. W e dra w all members of B 3 and lab el the matc hings in T able 1. DETERMINANTS AND PERFECT M A TCHINGS 5 1 2 3 4 5 6 7 r r r r r r r 1 2 3 4 5 6 7 r r r r r r r ✟ ✟ ✟ ✟ ✟ ✟ ❅ ❅ ❅ ❅ ❅ ❅ 7 6 5 4 3 2 1 r r r r r r r 1 2 3 4 5 6 7 r r r r r r r ✟ ✟ ✟ ✟ ✟ ✟ ❅ ❅ ❅ ❅ ❅ ❅ Figure 3. The same Brauer diagram in Figure 2 con- sidered as an elemen t of ( B F ) 7 on the left and ( B B ) 7 on the rig h t. r r r r r r r r r r r r ❅ ❅   r r r r r r ❍ ❍ ❍     r r r r r r ✟ ✟ ✟   r r r r r r   r r r r r r   ❅ ❅ r r r r r r ❅ ❅ ❅ ❅ ✟ ✟ ✟ r r r r r r ❍ ❍ ❍ ✟ ✟ ✟ r r r r r r   r r r r r r r r r r r r r r r r r r ❅ ❅ r r r r r r ❍ ❍ ❍ r r r r r r r r r r r r ❅ ❅ T able 1. All Br a uer diag r a ms b elonging to B 3 . Let µ ∈ ( B F ) n or ( B B ) n . F urther, let µ T (resp. µ B ) contain cups (resp. caps) and µ T B con tain ar cs. By con v en tion, edges will b e desig- nated a s ordered pairs. When the edges b elong to µ T or µ B , they will b e written in increasing order and when they b elong to µ T B , the v ertex in the top row will b e written first. The crossing n um b er χ ( µ ) of µ is the n umber of pairwise interse ctions among edges in µ . 0 1 2 0 1 1 2 3 1 2 0 1 0 0 1 T able 2. Crossing n umbers for all t he Brauer diagrams in B 3 according to T able 1. W e no w asso ciate a w eight to µ , consisting of edges µ T , µ B and µ T B . Let a i,j (resp. b i,j ) b e unknow ns defined for 1 ≤ i ≤ j ≤ n (resp. 1 ≤ i < j ≤ n ) and let ( c i, j ) = (min( i, j ) , max( i, j )). The w eight of 6 AR VIN D A YYER µ , w ( µ ), is giv en b y (3.1) w ( µ ) = Y ( i,j ) ∈ µ T b i,j Y ( i,j ) ∈ µ B b i,j Y ( i,j ) ∈ µ T B a c i,j . Note that this w eight depends on whethe r w e consider µ a s an elemen t of ( B F ) n or ( B B ) n . How ev er, the formal expression is the same in b oth cases. F or completeness, w e list the w eigh ts of all Brauer diagrams in B 3 according as whether they b elong in ( B F ) n and ( B B ) n resp ectiv ely . a 1 , 1 a 2 , 2 a 3 , 3 a 3 , 3 a 2 1 , 2 a 1 , 2 a 1 , 3 a 2 , 3 a 1 , 3 b 1 , 2 b 2 , 3 a 1 , 2 b 1 , 3 b 2 , 3 a 1 , 1 a 2 2 , 3 a 1 , 2 a 1 , 3 a 2 , 3 a 2 , 2 a 2 1 , 3 a 2 , 3 b 1 , 2 b 1 , 3 a 2 , 2 b 2 1 , 3 a 1 , 1 b 2 2 , 3 a 1 , 2 b 1 , 3 b 2 , 3 a 1 , 3 b 1 , 2 b 2 , 3 a 3 , 3 b 2 1 , 2 a 2 , 3 b 1 , 2 b 1 , 3 a 2 , 2 a 2 1 , 3 a 1 , 2 a 1 , 3 a 2 , 3 a 1 , 1 a 2 2 , 3 a 3 , 3 b 2 1 , 2 a 2 , 3 b 1 , 2 b 1 , 3 a 1 , 2 a 1 , 3 a 2 , 3 a 3 , 3 a 2 1 , 2 a 1 , 1 a 2 , 2 a 3 , 3 a 2 , 3 b 1 , 2 b 1 , 3 a 2 , 2 b 2 1 , 3 a 1 , 3 b 1 , 2 b 2 , 3 a 1 , 2 b 1 , 3 b 2 , 3 a 1 , 1 b 2 2 , 3 a 1 , 3 b 1 , 2 b 2 , 3 a 1 , 2 b 1 , 3 b 2 , 3 T able 3. W eigh t s of all the Brauer diagrams of size n = 3 according to T able 1. The first table describes the w eigh ts for ( B F ) n and the second, for ( B B ) n . W e are no w in a p osition to state the main theorem. Theorem 1. T he determinant of an n × n m a trix c an b e written as a sum of Br auer d iagr am s a s, det( M F ) = X µ ∈ ( B F ) n ( − 1) χ ( µ ) w ( µ ) , det( M B ) = ( − 1) ( n 2 ) X µ ∈ ( B B ) n ( − 1) χ ( µ ) w ( µ ) . (3.2) One can v erify that Theorem 1 is v alid f or n = 3 in b oth cases b y adding all t he w eights in T a ble 3 times the corresp onding crossing num- b ers in T able 2 for a ll t he Brauer diag r ams in T able 1, and comparing with (2 .3). 4. The number of terms in the det erminant exp ansion W e show b y a quic k argumen t that the num b er of monomials in the determinan t of an n × n matrix M F (and for the same reason, for M B ) is giv en by (2 n − 1)!!. This calculation is somewhat redundan t b ecause DETERMINANTS AND PERFECT M A TCHINGS 7 of Theorem 1. The reason for this short demonstration is that it sho ws wh y determinan ts should be related to p erfect matchings. T o start, let M b e either M F or M B . R ecall the Leibniz for mula for the determinant of M , (4.1) det( M ) = X π ∈ S n ( − 1) inv( π ) ( M ) 1 ,π (1) . . . ( M ) n,π ( n ) , where S n is the set of p erm utations in n letters and in v ( π ) is the n umber of in vers ion of the p erm utation. Usually , this w ould giv e us n ! terms, of course. In the new nota t io n, (2 .1), w e obta in ma ny mor e terms b ecause eac h factor ( M ) i,π ( i ) giv es t wo terms whenev er π ( i ) 6 = i . T o see ho w man y terms w e now hav e, it is b est to think of p er- m utations according to the n um b er and length of cycles they con ta in, π = C 1 . . . C k . If a cyc le C is of length 1, C = ( i ), then it corresp onds to a diagonal elemen t a i,i , whic h con tributes one term. If, on the o ther hand, C con tains j entries , then there are j off diagonal elemen t s, whic h giv e 2 j terms, coun ting m ultiplicities, exactly half of whic h con tain an o dd n umber of b i,j ’s. These terms will be cancelled b y the permuta- tion π ′ whic h has all other cycles the same, and C replaced b y C ′ , the rev erse of C . Therefore, if C con ta ins j en t ries, w e effectiv ely g et a con tribution of 2 j − 1 terms. The num b er of terms can b e written as a sum o ver p erm uta tions with k disjoin t cycles. When there are k cycles, w e get 2 n − k terms. Since t he n umber of permutations with k disjoint cycles is t he unsigned Stirling n umber o f the first kind, s ( n, k ), the total n umber of terms is g iven by (4.2) n X k =1 s ( n, k )2 n − k . Since t he generating function of the unsigned Stirling n um b ers of the first kind are giv en b y the Pochhamme r sym b ol or rising factoria l, (4.3) n X k =1 s ( n, k ) x k = ( x ) ( n ) ≡ x ( x + 1) · · · ( x + n − 1) , w e can calculate the more general sum, (4.4) n X k =1 s ( n, k ) x n − k = (1 + x )(1 + 2 x ) · · · ( 1 + ( n − 1) x ) . Substituting x = 2 in the a b o ve equation g ives (2 n − 1)!!, the desired answ er. 8 AR VIN D A YYER 5. B ijection between terms and Labeled Brauer diagrams W e now describe the bijection b et w een lab eled Brauer diagrams on the one hand and p ermu tations leading to a pro duct of a i,j ’s and b i,j ’s on the other. The algor it hm is indep enden t of whether w e consider B F or B B . L et µ b e a lab eled Brauer diagram. W e first state the algorithm constructing the latter from the former. Algorithm 1. We s tart with the thr e e sets of matchings µ T , µ B and µ T B . (1) F or e ach term ( i, j ) in µ T and µ B , write the term b i,j and for ( i, j ) i n µ T B , write the term a c i,j . (2) Start with π = ∅ . (3) Find the smal lest inte ger i 1 ∈ T n o t yet in π and find its p artner i 2 . That is , either ( i 1 , i 2 ) ∈ µ T B or \ ( i 1 , i 2 ) ∈ µ T . If i 2 = i 1 , then app end the cycle ( i 1 ) to π and r ep e at Step 3. O therw i s e move on to S tep 4. (4) If i k is in T (r esp. B ), lo ok for the p artner of the other i k in B (r esp . T ) and c a l l it i k +1 . Note that i k +1 c an b e in T or B in b oth c a ses. (5) R ep e a t Step 4 for k fr om 2 until m such that i m +1 = i 1 . App e n d the cycle ( i 1 , i 2 , . . . , i m ) to π . (6) R ep e a t Steps 3 - 5 until π is a p ermutation on n letters in cycle notation. Therefore, w e obtained the desired pro duct in Step 1 and the p er- m utation at the end of Step 6. Here is a simple conseque nce of the algorithm. Lemma 2. By the c onstruction of Algorithm 1, if the triplet ( µ T , µ B , µ T B ) le ads to π , then ( µ B , µ T , µ T B ) le ads to π − 1 . Pr o of. Eac h cycle ( i 1 , i 2 , . . . , i m ) constructed according to Algorithm 1 b y the triplet ( µ T , µ B , µ T B ) will b e constructed as ( i 1 , i m , . . . , i 2 ) by the triplet ( µ B , µ T , µ T B ). Since each cycle will b e rev ersed, this is the in ve rse of the original permutation.  W e no w describe the rev erse algorithm. Algorithm 2. We start with a pr o duct of a i,j ’s and b i,j ’s, an d a p er- mutation π = C 1 . . . C m written in cycle notation such that 1 ∈ C 1 , the smal lest inte ger in π \ C 1 b elo ngs to C 2 , and so on. (1) F or e ach b i,j , we obtain a term d ( i, j ) which b elo n gs either to µ T or µ B and for e ach a i,j , we obtain one o f ( i, j ) or ( j, i ) which b elo ngs to µ T B . DETERMINANTS AND PERFECT M A TCHINGS 9 (2) Start with µ T = µ B = µ T B = ∅ . S et k = 1 . (3) Find the first entry i 1 in C k and lo ok for either a i 1 ,i 2 or b i 1 ,i 2 . If the forme r, assign i 2 to B and app end ( i 1 , i 2 ) to µ T B and otherwise, assign i 2 to T and app end ( i 1 , i 2 ) to µ T . Set l = 2 . (4) Find either a i l ,i l +1 or b i l ,i l +1 . Assign i l +1 to one of T or B and ( i l , i l +1 ) to o ne of µ T , µ B or µ T B ac c or ding to the fol lo w ing table. i l T erm i l +1 ( i l , i l +1 ) Next i l +1 T a B µ T B T T b T µ T B B a T µ T B B B b B µ B T Incr ement l by o n e. (5) R ep e a t Step 4 until you r eturn to i 1 , whi c h wil l ne c essaril y b e- long to B , sinc e ther e ar e an even n umb er of b i,j ’s in the term. (6) Incr emen t k by 1. (7) R ep e a t Steps 3-6 until k = m , i.e., until a l l cycles ar e exhauste d. The follo wing result is no w an easy consequence. Lemma 3. A lgorithms 1 and 2 ar e inverses of e ach other. 6. The Cro ssing Number No w that w e ha ve established a bijection b et we en terms in the de- terminan t expansion a nd lab eled Brauer diagrams, w e need to sho w that the sign asso ciated to b oth of these are the same. W e start with a lab eled Brauer diagram µ , whic h leads to a p ermutation π = C 1 . . . C m and a pro duct of a ’s and b ’s according to Algorithm 1 . L et τ b e t he same pro duct obtained from the determinan t expansion of the matrix using p erm uta tion π including the sign . F rom the definition of the matrix (2.1), w e will first write a for m ula for the sign associated to τ . Let C j = ( n ( j ) 1 , . . . , n ( j ) l ( j ) ). Then, define the sequences β ( j ) (resp. γ ( j ) ) of length l ( j ) consisting of t erms ± 1 (resp. ± i ) according to the follo wing definition. (6.1) β ( j ) i = ( +1 n ( j ) i < n ( j ) i +1 , − 1 n ( j ) i > n ( j ) i +1 , ; γ ( j ) i = ( + i n ( j ) i < n ( j ) i +1 , − i n ( j ) i > n ( j ) i +1 , where n ( j ) l ( j )+1 ≡ n ( j ) 1 . Then the sign asso ciated to the term τ depends on whether µ b elongs to ( B F ) n or ( B B ) n . In the former case, w e ha ve 10 AR VIN D A YYER the fo rm ula (6.2) sgn( τ ) = ( − 1) inv( π ) m Y j = 1 l ( j ) Y i =1 b \ n ( j ) i ,n ( j ) i +1 ∈ τ β ( j ) i . and in the la t t er, (6.3) sgn ( τ ) = ( − 1) inv( π ) m Y j = 1 l ( j ) Y i =1 b \ n ( j ) i ,n ( j ) i +1 ∈ τ γ ( j ) i . Since the num b er of b ’s in the second pro duct is ev en for a ll j , the pro duct in (6.3) will necessarily b e real and equal to ± 1. First we lo ok at Brauer diagrams with no cups or caps. There are no b i,j ’s in the associated term in the determinan t expansion. Lemma 4. Supp os e µ is a lab ele d Br auer diagr a m such that µ T = µ B = ∅ and let π b e the asso ciate d p e rmutation. Then, if µ ∈ ( B F ) n , then (6.4) in v ( π ) = χ ( µ ) , and if µ ∈ ( B B ) n , then (6.5) in v ( π ) + χ ( µ ) =  n 2  . Pr o of. The former is obvious since µ is identical to the tw o - line dia gram for π . The latter requires just a little more w ork. F or a matc hing with only arcs, the edges are exactly giv en by ( i, π i ) for i ∈ [ n ]. No w con- sider t wo edges ( i, π i ) and ( j, π j ) where i < j , without lo ss of g enerality . Recall that i, j ∈ T and π i , π j ∈ B b y con v en t io n. Then ( i, π i ) in t er- sects ( j, π j ) if and o nly if π i < π j b ecause of the rig ht-to-left n umbering con ve ntion in B . Th us, (6.6) χ ( µ ) = |{ ( i, j ) | i < j, π i < π j }| . On the other ha nd, the definition of an in v ersion n umber is (6.7) in v ( π ) = |{ ( i, j ) | i < j, π i > π j }| . Since these tw o count disjoint cases, whic h span all p ossible pairs ( i, j ), they m ust sum up to the total n umber of p ossibilities ( i, j ) where i < j , whic h is exactly  n 2  .  No w w e will see what happ ens to the crossing n um b er o f a ma t ching when a cup a nd a cup are con verted to t wo arcs. DETERMINANTS AND PERFECT M A TCHINGS 11 Lemma 5. A l l other e dges r emaining the same, for any i, j, k , l , the fol lowing r esults hold. (a) ( − 1) χ k l i j ! = ( − 1) χ ✁ ✁ ✁ ❆ ❆ ❆ k l i j ! . (b) ( − 1) χ k l i j ! = − ( − 1) χ ✁ ✁ ✁ ❆ ❆ ❆ k l i j ! . (c) ( − 1) χ i k j l ! = − ( − 1) χ i k j l ! . (d) ( − 1) χ i l j k ! = − ( − 1) χ i l j k ! . Pr o of. W e will pro ve the result o nly for (a ) . The idea o f the pro of is iden tical f o r all other cases. W e consider a ll p ossible edges t ha t could in tersect with an y of the 4 edges ( i, j ) , ( k , l ) , ( i, l ) and ( j, k ) illustrated ab ov e. W e group them according to their p osition. (1) Let n ij (resp. n k l ) be the n um b er of edges suc h that exactly one of its endp o in ts lies b et wee n i a nd j (resp. k and l ), and the o ther endp oint do es not lie b etw een k and l (resp. i and j ). These edges in tersect ( i, j ) (resp. ( k , l )) and do not in tersect ( k , l ) (r esp. ( i, j )). They also in tersect exactly one among ( i, l ) and ( j, k ). (2) Let n ij k l b e the num b er of edges one of whose endp oin t s lies b et wee n i and j , and the other, b et wee n k a nd l . These in tersect b oth ( i, j ) and ( k , l ). (3) Let n LR b e the n umber of edges, one of whose endp oin ts is less than k if it b elongs to the top ro w and more t ha n j in the b ottom row, and the other is more than l in the top row or 12 AR VIN D A YYER less than i in the b ottom row. These are edges whic h do not in tersect either ( i, j ) or ( k , l ), but inters ect b oth ( i, l ) and ( j, k ). No w, the con tribution of the edges ( i, j ) and ( k , l ) to χ in t he left hand side of (6.8) is n i,j + n k l + 2 n ij k l , where as tha t t o the r ig h t hand side of (6.8) is n ij + n k l + 2 n LR . Since all o ther edges are the same, the difference b et we en the crossing num b er of the configuration on the left and tha t on the rig h t is 2 n ij k l − 2 n LR and hence, the parit y of b oth crossing num b ers is the same.  7. The Main R esul t W e no w pro ve t he theorem in a purely comb inatorial wa y . The pro of will dep end on whether the Brauer diagram b elongs to ( B F ) n or ( B B ) n , but the idea is ve ry similar in b o th cases. W e will pr ov e the former and p oin t out the essen tia l difference in the pro of of the latter at the v ery end. Pro of of Theorem 1: F rom Lemma 3, w e ha v e s ho wn that ev ery term in the expansion of the determinan t corresponds, in an in v ertible w ay , to a Brauer diagram. W e will now show the signs are also equal b y p erforming an induction on the n um b er of cups, or equiv a len tly caps, since b oth are the same. Consider a F -Brauer diagram µ ∈ ( B F ) n with at least o ne cup and cap eac h. Using the bijection of Lemma 3, construct the associated p erm utatio n π . By the construction in Algorithm 1, there ha v e to b e at least t w o b ’s in the same cycle C , sa y . W e pic k tw o of them such that ( i, j ) ∈ µ B is a cup and ( k , l ) ∈ µ T is a cap. W e hav e to show that ( − 1) χ ( µ ) = sgn( τ ) using (6.3). W e no w get a new Brauer diagram µ ′ ∈ ( B F ) n b y replacing the cup ( i, j ) and the cap ( k , l ) b y the arcs ( i, k ) and ( j, l ) using Lemma 5(a). This replaces the asso ciated we igh ts b i,j b k ,l with a c i,k a b j,l , a nd the sign remains the same, ( − 1) χ ( µ ) = ( − 1) χ ( µ ′ ) . Now w e use the same algo- rithm to construct the p ermutation π ′ asso ciated to the new term, and lo ok a t how the cycle C c hanges to C ′ . Let τ and τ ′ b e terms obtained in the determinan t expansion of M F including the sign. There a re fo ur w ays in whic h these 4 n umbers are arranged in C . W e list these and the w ay they transform in T able 4. In eac h case, the links { i, j } and { k , l } are brok en and the links { i, k } and { j, l } are formed. Recall that i < j and k < l according t o Lemma 5(a). W e no w need an result from undergraduate comb inatorics. When n is o dd (resp. ev en), a p erm utation π of size n is o dd if and only if the num b er of cyc les is even ( r esp. o dd) in its cycle decomp o sition. Therefore, the parity of the p erm utatio n π ′ is differen t from π in cases DETERMINANTS AND PERFECT M A TCHINGS 13 C ∈ π C ′ ∈ π ′ F actors in π Relativ e sign ( i, j, . . . , k , l , . . . ) ( i, k , . . . , j, l , . . . ) b i,j b k ,l +1 ( i, j, . . . , l, k , . . . ) ( i, k , . . . )( j, . . . , l ) b i,j ( − b k ,l ) − 1 ( j, i, . . . , k , l , . . . ) ( j, l , . . . )( i, . . . , k ) ( − b i,j ) b k ,l − 1 ( j, i, . . . , l, k , . . . ) ( j, l , . . . , i, k , . . . ) ( − b i,j )( − b k ,l ) +1 T able 4. Comparison b et w een the difference o f the n umber of cycles in C and C ′ , a nd the relativ e sign b e- t we en the factor in π and a c i,k a b j,l ∈ π ′ . (1) a nd (4) and the same as that of π in cases (2) and ( 3 ). Notice that the relative signs also fo llo w the same pattern. T o summarize, we ha ve sho wn that ( − 1) χ ( µ ) = sgn( τ ) holds if and only if ( − 1) χ ( µ ′ ) = sgn( τ ′ ) ho lds when µ, µ ′ ∈ ( B F ) n . But t his is pre- cisely the induction step since µ ′ and µ ′′ ha ve one less cup and one less cap that µ . F rom Lemma 4, w e hav e already sho wn that the terms whic h corresp ond to Bra uer diagra ms with only arcs ha v e the correct sign. This completes the pro of. W e follow the same strat egy when µ b elongs to ( B B ) n . The difference is t ha t l < k a nd tha t b i,j and b k ,l come with additional factors of ı . The in terested reader can c hec k that these t w o con t r ibute opposing signs leading to the same result.  F or ev en an tisymmetric matrices, this gives a natural combinatorial in terpretation of Ca yley’s theorem differen t f r om the ones give n b y Halton [Hal66 ] and E˘ gecio˘ glu [E˘ ge90]. Corollary 6 (Ca yley 1847, [Ca y47]) . F o r an antisymmetric matrix M of size n , (7.1) det M = ( (pf M ) 2 n even , 0 n o dd . Pr o of. F rom (2.1), w e see that all a i,j ’s are zero for an antisy mmetric matrix for b oth M F and M B . W e consider only the former represen ta- tion since the argumen t is iden tical f o r the latter. The only F -Brauer diagrams in ( B F ) n that con tribute are those with no arcs. If n is o dd, this is clearly not p ossible. Th us the determinan t is zero. If n is ev en, w e hav e the sum in Theorem 1 ov er all Brauer diagrams with only cups and cups. This sum now factors into tw o dis tinct sums for cups and for caps. But for eac h of these cases, w e kno w that the answ er is the same since they are indep enden t sums. Moreo v er, eac h o f these is the Pfaffian [Ste90].  14 AR VIN D A YYER It w o uld b e in teresting to find a n analogous express ion fo r t he p er- manen t of a matrix. This migh t en tail finding a different planar graph instead of a Brauer diagr am or a differen t analog of the crossing n um- b er or b oth. F or example, the p ermanen t of the matrix in (2.2) is given b y P erm ( M (3) F ) = a 2 1 , 3 a 2 , 2 + a 2 2 , 3 a 1 , 1 + a 2 1 , 2 a 3 , 3 − b 2 1 , 2 a 3 , 3 − b 2 1 , 3 a 2 , 2 − b 2 2 , 3 a 1 , 1 + 2 a 1 , 2 a 1 , 3 a 2 , 3 + a 1 , 1 a 2 , 2 a 3 , 3 − 2 a 2 , 3 b 1 , 2 b 1 , 3 − 2 a 1 , 2 b 1 , 3 b 2 , 3 + 2 a 1 , 3 b 1 , 2 b 2 , 3 . (7.2) Note that not a ll signs in t he p ermanen t expansion of are p ositiv e. A cknowledgements This w ork w as motiv at ed b y discussions with Craig T racy , whom w e thank for encouragemen t and supp ort. W e also thank Ira Gessel, D a vid M. Jac kson, Christian Kratten thaler, Greg Kup erb erg, Dan R omik, Alexander Soshnik ov and D oron Zeilb erger for constructiv e feedbac k. W e also thank a referee for a v ery careful reading of the manuscript whic h led to man y improv emen ts. Reference s [Ait44] A. C. Aitk en. On the num b er of distinct ter ms in the expansion of sym- metric a nd skew determinan ts. Edinbur gh Math. Notes , 194 4(34):1–5 , 1944. [Bra37 ] Richard Brauer . On a lgebras which are co nnected with the se misimple contin uous groups. Ann. of Math. (2) , 38 (4):857–8 72, 1937. [Cay47] A. Ca yley . Sur les determinants gauches. J. r eine angew. Math. , 38:93 –96, 1847. [E˘ ge90] ¨ Omer E˘ gecio˘ glu. Skew-symmetric matrices and the Pfaffian. Ars Combi n . , 29:107 –116, 1990. [F o a80] Do minique F o ata. A combinatorial pro of of Jaco bi’s iden tity . A nn. Dis- cr ete Math. , 6:125–13 5, 19 80. Co mbinatorial mathematics, optimal de- signs and their a pplications (Pro c. Symp os. Com bin. Math. and Optimal Design, Colorado State Univ., F ort Collins, Colo ., 19 78). [Hal66] J ohn H. Halton. A combinatorial pro of of Cayley’s theorem on Pfaffia ns. J. Combina torial The ory , 1:224–2 32, 1966. [HL06] T o m Halverson and Tim Lewando ws ki. RSK insertion for set partitions and dia g ram alg ebras. Ele ctr on. J. Combin. , 11 (2):Research Paper 24, 24 pp. (electronic), 2004 /06. [Jac77] D. M. J a ckson. The combinatorial in ter pretation of the Jacobi identit y from Lie alg e bras. J. Combina torial The ory Ser. A , 23(3):23 3 –256, 1977. [Knu96] Donald E. Knuth. O verlapping Pfa ffia ns. Ele ctr on. J. Combin. , 3(2):Re- search Paper 5, approx. 13 pp. (electro nic), 1 996. The F o ata F es tsch rift. [MM11] Rob ert J. Mar sh and Paul Ma rtin. Tiling bijections betw een paths and Brauer diagra ms . J. Algebr aic Combin. , 33(3):427–4 53, 20 11. DETERMINANTS AND PERFECT M A TCHINGS 15 [MSV04] Meena Maha jan, P . R. Subramanya, and V. Vinay . The combinatorial approach yields an NC algorithm for computing Pfa ffia ns. Discr ete Appl. Math. , 143(1- 3):1–16, 2004. [Mui60] Thomas Muir. The The ory of the Determinant in the Historic al Or der of Development. Dov er Publications, 19 6 0. [SR72] Richard Sta nle y and J. Riordan. E2297 . The Ameri c an Mathematic al Monthly , 79(5):pp. 519 – 520, 1972 . [Ste90] John R. Stembridge. Nonintersecting pa ths, pfaffians, a nd pla ne par ti- tions. A dvanc es in Mathematics , 8 3 (1):96 – 1 31, 1990 . [Str83] How a rd Stra ubing. A combinatorial pr o of of the Ca yley-Hamilton theo- rem. Discr ete Math. , 43(2-3):27 3–279 , 1983 . [Sun86] Sheila Sundaram. On the c ombinatorics of r epr esent ations of Sp(2n,C) . PhD thesis, Massach usetts Institute of T echnology , 1 986. [T er 01] Itaru T erada. Bra uer diagrams , updown tablea ux and nilp otent matrices. J. A lgebr aic Combin. , 14(3 ):2 29–26 7, 200 1. [Zei85] Doro n Zeilb erger. A com bina torial approa ch to matrix algebra. Discr ete Math. , 56(1):61 – 72, 1985. [Zei97] Doro n Zeilb erge r. Do dgson’s determinant-ev aluation r ule proved by tw o- timing men a nd women. Ele ctr on. J. Combin. , 4(2):Resear c h Paper 22, approx. 2 pp. (electro nic), 19 9 7. The Wilf F estschrift (Philadelphia, P A, 1996). Ar vind A yyer, Dep ar tment of Ma thema tics, University of Califor- nia, Da vis, CA 95616 E-mail addr ess : ayyer@ math.u cdavis.edu