The Partition Function of Andrews and Stanley and Al-Salam-Chihara Polynomials

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1 Formal Power Series and Algebraic Combinatorics Séries Formelles et Combinatoire Algébriue San Diego California 006 The Partition Function of Andrews and Stanley and Al-Salam-Chihara Polynomials Masao Ishikawa and Jiang Zeng Abstract. For any partition λ let ω(λ denote the four parameter weight P ω(λ = a i 1 λi 1/ P b i 1 λi 1/ P c i 1 λi/ P d i 1 λi/ and let l(λ be the length of λ. We show that the generating function P ω(λz l(λ where the sum runs over all ordinary (resp. strict partitions with parts each N can be expressed by the Al-Salam-Chihara polynomials. As a corollary we prove G.E. Andrews result by specializing some parameters and C. Boulet s results when N +. In the last section we study the weighted sum P ω(λz l(λ P λ (x where P λ (x is Schur s P-function and the sum runs over all strict partitions. Résumé. Pour toute partition λ on définit ω(λ comme la fonction poids de uatre paramètres P ω(λ = a i 1 λi 1/ P b i 1 λi 1/ P c i 1 λi/ P d i 1 λi/ et désigne l(λ la longueur de λ. On démontre ue la fonction génératrice P ω(λz l(λ où la somme porte sur toutes les partitions ordinaires (resp. strictes avec chaue part N peut s exprimer par les polynômes d Al-Salam-Chihara. Comme corollaire on en déduit un résultat de G.E. Andrews en spécialisant certain parametres et ceux de C. Boulet uand N +. Dans la dernière section on étudie la somme pondérée P ω(λz l(λ P λ (x où P λ (x est la P-fonction de Schur et la somme porte sur toutes les partitions strictes. 1. Introduction Let λ be an integer partition and λ its conjugate. Let O(λ denote the number of odd parts of λ and λ the sum of its parts. R. Stanley ([13] has shown that if t(n denotes the number of partitions λ of n for which O(λ O(λ (mod 4 then t(n = 1 (p(n + f(n where p(n is the total number of partitions of n and n=0 f(n n = i 1 (1 + i 1 (1 4i (1 + 4i. In [1] G.E. Andrews has computed the generating function of ordinary partitions λ with parts each less than or eual to N with respect to the weight z O(λ y O(λ λ. We should note that in [1] A. Sills has given a combinatorial proof of this result and in [14] A. Yee has generalized this result to the generating function of ordinary partitions of parts N and length M. As a generalization of this weight we consider the following four parameter weight. Let a b c and d be commuting indeterminates. Define the following weight functions ω(λ on the set of all partitions (1.1 ω(λ = a P i 1 λi 1/ b P i 1 λi 1/ c P i 1 λi/ d P i 1 λi/ Key words and phrases. Partitions symmetric functions Al-Salam-Chihara polynomials basic hypergeometric functions Schur s Q-functions Pfaffians minor summation formula of Pfaffians.

2 Masao Ishikawa and Jiang Zeng where x (resp. x stands for the smallest (resp. largest integer greater (resp. less than or eual to x for a given real number x. For example if λ = ( then ω(λ is the product of the entries in the following diagram for λ. a b a b a c d c d a b a b c In [3] C. Boulet has obtained results on the generating functions for the weights ω(λ when λ runs over all ordinary partitions and when λ runs over all strict partitions In this paper we consider a refinement of these results i.e. the generating functions for the weights ω(λz l(λ where l(λ is the length of λ when λ runs over all ordinary partitions with parts each N and when λ runs over all strict partitions with parts each N and show that they are related to the basic hypergeometric series i.e. the Al-Salam-Chihara polynomials (see Theorem 3.4 and Theorem 4.3. In the last section we show the weighted sum ω(µz l(µ P µ (x of Schur s P-functions P µ (x (when z = this euals the weighted sum ω(µq µ (x of Schur s Q-functions Q µ (x can be expressed by a Pfaffian where µ runs over all strict partitions (with parts each N. A -shifted factorial is defined by. Preliminaries (a; 0 = 1 (a; n = (1 a(1 a (1 a n 1 n = We also define (a; = k=0 (1 ak. Since products of -shifted factorials occur very often to simplify them we shall use the compact notations (a 1... a m ; n = (a 1 ; n (a m ; n (a 1... a m ; = (a 1 ; (a m ;. We define an r+1 φ r basic hypergeometric series by ( a1 a...a r+1 r+1φ r ; z = b 1... b r n=0 (a 1 a... a r+1 ; n ( b 1...b r ; n z n. The Al-Salam-Chihara polynomial Q n (x = Q n (x; α β is by definition Q n (x; α β = (αβ; ( n n αu αu 1 α n 3φ ; αβ 0 = (αu; n u n ( n βu 1 α 1 n+1 u 1 ; α 1 u = (βu 1 ; n u n ( n αu β 1 n+1 u ; β 1 u 1 where x = u+u 1 (see [6] p.80. This is a specialization of the Askey-Wilson polynomials (see [] and satisfies the three-term recurrence relation (.1 xq n (x = Q n+1 (x + (α + β n Q n (x + (1 n (1 αβ n 1 Q n 1 (x with Q 1 (x = 0 Q 0 (x = 1. We also consider a more general recurrence relation: (. x Q n (x = Q n+1 (x + (α + βt n Qn (x + (1 t n (1 tαβ n 1 Q n 1 (x

3 THE PARTITION FUNCTION AND AL-SALAM-CHIHARA POLYNOMIALS which we call the associated Al-Salam-Chihara recurrence relation. Put ( t Q (1 n (x = u n 1 n βu 1 (.3 (tαu; n φ 1 t 1 α 1 n+1 u 1 ; α 1 u Q ( n (x = u n (t; ( n(tαβ; n t n+1 α 1 u (.4 (tβu; n tβ n+1 ; αu u where x = u+u 1. In [5] Ismail and Rahman have presented two linearly independent solutions of the associated Askey-Wilson recurrence euation (see also [4]. By specializing the parameters we conclude (1 ( that Q n (x and Q n (x are two linearly independent solutions of the associated Al-Salam-Chihara euation (. (see [4 p.03]. Here we use this fact and omit the proof. The series (.3 and (.4 are convergent if we assume u < 1 and < α < 1 (see [4 p.04]. Let (.5 W n = (1 ( (1 ( Q n (x Q n 1 (x Q n 1 (x Q n (x denote the Casorati determinant of the euation (.. Then we obtain (.6 W 1 = u 1 (tαu βu; (αu tβu;. In the following sections we need to find a polynomial solution of the recurrence euation (. which satisfies a given initial condition say Q 0 (x = Q 0 and Q 1 (x = Q (1 ( 1. Since Q n (x and Q n (x are linearly independent solutions of (. this Q n (x can be written as a linear combination of these functions say Q n (x = C 1 Q(1 n (x + C Q( n (x. If we substitute the initial condition Q 0 (x = Q 0 and Q 1 (x = Q 1 into this euation and solve the linear euation then we conclude that (.7 Q n (x = u(αu tβu; (tαu βu; [{ Q1 Q( 0 (x Q } Q( 0 1 (x Q(1 n (x { } ] + Q 0 Q (1 1 (x Q 1 Q (1 0 (x Q ( n (x and (.8 lim n un Qn (x = u(tβu αu; { Q1 (u ; Q( 0 (x Q } Q( 0 1 (x. 3. Strict Partitions A partition µ is strict if all its parts are distinct. One represents the associated shifted diagram of µ as a diagram in which the ith row from the top has been shifted to the right by i places so that the first column becomes a diagonal. A strict partition can be written uniuely in the form µ = (µ 1... µ n where n is an non-negative integer and µ 1 > µ > > µ n 0. The length l(µ is by definition the number of nonzero parts of µ. We define the weight function ω(µ exactly the same as in (1.1. For example if µ = (8 5 3 then l(µ = 3 ω(µ = a 6 b 5 c 3 d and its shifted diagram is as follows. Let (3.1 Ψ N = Ψ N (a b c d; z = ω(µz l(µ

4 Masao Ishikawa and Jiang Zeng where the sum is over all strict partitions µ such that each part of µ is less than or eual to N. For example we have Ψ 0 = 1 Ψ 1 = 1 + az Ψ = 1 + a(1 + bz + abcz Ψ 3 = 1 + a(1 + b + abz + abc(1 + a + adz + a 3 bcdz 3. In fact the only strict partition such that l(µ = 0 is the strict partitions µ such that l(µ = 1 and µ 1 3 are the following three: a a b a b a the strict partitions µ such that l(µ = and µ 1 3 are the following three: a b c a b a c a b a c d and the strict partition µ such that l(µ = 3 and µ 1 3 is the following one: a b a c d a. The sum of the weights of these strict partitions is eual to Ψ 3. In this section we always assume a b c d < 1. One of the main results of this section is that the even index terms and the odd index terms of Ψ N respectively satisfy the associated Al-Salam-Chihara recurrence relation: Theorem 3.1. Set = abcd. Let Ψ N = Ψ N (a b c d; z be as in (3.1 and put X N = Ψ N and Y N = Ψ N+1. Then X N and Y N satisfy (3. (3.3 X N+1 = { 1 + ab + a(1 + bcz N} X N ab(1 z N (1 acz N 1 X N 1 Y N+1 = { 1 + ab + abc(1 + adz N} Y N ab(1 z N (1 acz N Y N 1 where X 0 = 1 Y 0 = 1 + az X 1 = 1 + a(1 + bz + abcz and Y 1 = 1 + a(1 + b + abz + abc(1 + a + adz + a 3 bcdz 3. Especially if we put X N = (ab N X N and Y N = (ab N Y N then X N and Y N satisfy { } (ab 1 + (ab 1 X N = X N+1 a 1 b 1 (1 + bcz N X N (3.4 (3.5 { } + (1 z N (1 acz N 1 X N 1 (ab 1 + (ab 1 Y N = Y N+1 a 1 1 b c(1 + adz N Y N + (1 z N (1 a bc dz N 1 Y N 1 where X 0 = 1 Y 0 = 1 + az X 1 = (ab 1 + a 1 b 1 (1 + bz + (ab 1 cz and Y 1 = (ab a b (1 + b + abz + a b c(1 + a + adz + a 5 1 b cdz 3. One concludes that when a b c d < 1 the solutions of (3. and (3.3 are expressed by the linear combinations of (.3 and (.4 as follows. Theorem 3.. Assume a b c d < 1 and set = abcd. Let Ψ N = Ψ N (a b c d; z be as in (3.1.

5 THE PARTITION FUNCTION AND AL-SALAM-CHIHARA POLYNOMIALS (3.6 (3.7 (i Put X N = Ψ N. Then we have where X N = ( az abc; ( a abcz ; ( {(s X0 X 1 s X1 X 0( abcz N z b 1 ; N φ 1 (abc 1 N+1 z ; c 1 +(r1 X X 0 r0 X X 1(ab N (z acz ( ; N N+1 z c 1 } ( az ; N a N+1 z ; abc ( z r0 X = b 1 (abc 1 z ; c 1 ( z s X 0 = c 1 az ; abc r X 1 = (1 + abcz φ 1 ( z 1 b 1 (abc 1 z ; c 1 s X 1 = ab(1 z (1 acz 1 + az (ii Put Y N = Ψ N+1. Then we have where Y N = ( a bcdz abc; ( a bcd abcz ; ( z c 1 az ; abc { (s Y 0 Y 1 s Y 1 Y 0 ( abcz ; N φ 1 ( N z acd + (r Y 1 Y 0 r Y 0 Y 1 (ab N (z a bc dz ; N ( a bcdz ; N. (abc 1 N+1 z ; c 1 ( z r0 Y acd = φ 1 ( abc 1 z ; c 1 ( r1 Y = (1 + abcz 1 z ac φ 1 (abc 1 z ; c 1 ( z s Y c 1 0 = φ 1 a bcdz ; abc s Y 1 = ab(1 z (1 a bc dz 1 + a bcdz ( N+1 z c 1 a bcd N+1 ; abc z ( z c 1 a bcdz ; abc. If we take the limit N in (3.6 and (3.7 then by using (.8 we obtain the following generalization of Boulet s result (see Corollary 3.6. Corollary 3.3. Assume a b c d < 1 and set = abcd. Let s X i sy i X i Y i (i = 0 1 be as in the above theorem. Then we have ω(µz l(µ = ( abc az ; (s X 0 (ab; X 1 s X 1 X 0 (3.8 where the sum runs over all strict partitions. µ = ( abc a bcdz ; (ab; (s Y 0 Y 1 s Y 1 Y 0 Especially by substituting z = 1 into (3.6 and (3.7 we conclude that the solutions of the recurrence relations (3.4 and (3.5 with the above initial condition are exactly the Al-Salam-Chihara polynomials: } Theorem 3.4. Put u = ab x = u+u 1 and = abcd. Let Ψ N (a b c d; z be as in (3.1.

6 Masao Ishikawa and Jiang Zeng (3.9 (3.10 (i The polynomial Ψ N (a b c d; 1 is given by Ψ N (a b c d; 1 = (ab N QN (x; a 1 b 1 c a 1 b 1 (ii The polynomial Ψ N+1 (a b c d; 1 is given by ( N c = ( a; N φ 1 a 1 ; b N+1. Ψ N+1 (a b c d; 1 = (1 + a(ab N QN (x; a b c a b cd ( N c = ( a; N+1 φ 1 a 1 ; b. N If we substitute a = zy b = z 1 y c = zy 1 and d = z 1 y 1 into Theorem 3.4 then we immediately obtain the following corollary which is a strict version of Andrews result. (3.11 and (3.1 where Corollary 3.5. µ strict partitions µ 1 N µ strict partitions µ 1 N+1 z O(µ y O(µ µ = z O(µ y O(µ µ = N j=0 N j=0 [ ] N ( zy; 4 j ( zy 1 ; 4 N j (y N j j 4 [ ] N ( zy; 4 j+1 ( zy 1 ; 4 N j (y N j j 4 [ ] { N (1 N (1 N 1 (1 N j+1 = (1 j (1 j 1 (1 for 0 j N j 0 if j < 0 and j > N. If we put N in Corollary 3.4 then we immediately obtain the following corollary (cf. Corollary of [3]. We can also prove this corollary by setting z 1 in (3.8. Corollary 3.6. (Boulet Let = abcd then ω(µ = ( a; ( abc; (3.13 (ab; where the sum runs over all strict partitions µ. µ 4. Ordinary Partitions First we present a generalization of Andrews result in [1]. Let us consider (4.1 Φ N = Φ N (a b c d; z = λ λ 1 N ω(λz l(λ where the sum runs over all partitions λ such that each part of λ is less than or eual to N. For example the first few terms can be computed directly as follows: Φ 0 = 1 Φ 1 = 1 + az 1 acz 1 + a(1 + bz + abcz Φ = (1 acz (1 z Φ 3 = 1 + a(1 + b + abz + abc(1 + a + adz + a 3 bcdz 3 (1 z ac(1 z (1 z ac where = abcd as before. If one compares these with the first few terms of Ψ n one can easily guess the following theorem holds:

7 THE PARTITION FUNCTION AND AL-SALAM-CHIHARA POLYNOMIALS Theorem 4.1. Let N be a non-negative integer and let Φ N = Φ N (a b c d; z be as in (4.1. Then we have (4. Φ N (a b c d; z = Ψ N (a b c d; z (z ; N/ (z ac; N/ where Ψ N = Ψ N (a b c d; z is the generating function defined in (3.1. Note that Ψ N is explicitly given in terms of basic hypergeometric functions in Theorem 3.. First of all as an immediate corollary of Theorem 4.1 and Corollary 3.3 we obtain the following generalization of Boulet s result. Corollary 4.. Assume a b c d < 1 and set = abcd. Let s X i sy i X i Y i (i = 0 1 be as in Theorem 3.. Then we have ω(λz µ = ( abc az ; (4.3 (ab acz z (s X 0 X 1 s X 1 X 0 ; where the sum runs over all partitions λ. (4.4 (4.5 λ Theorem 4.1 and Theorem 3.4 also give the following corollary: Corollary 4.3. Put x = (ab 1 +(ab 1 and = abcd. Let Φ N = Φ N (a b c d; z be as in (4.1. (i The generating function Φ N (a b c d; 1 is given by Φ N (a b c d; 1 = (ab N Q N (x; a 1 b 1 c ab 1 1 (; N (ac; N ( ( a; N N c = (; N (ac; N a 1 ; b. N+1 (ii The generating function Φ N (a b c d; 1 is given by Φ N+1 (a b c d; 1 = (1 + a(ab N Q N (x; a b 1 c 1 a b 3 1 cd (; N (ac; N+1 ( ( a; N+1 N c = (; N (ac; N+1 a 1 ; b. N As before we immediately deduce the following corollary from Corollary 4.3. Let S N (n r s denote the number of partitions π of n where each part of π is N O(π = r O(π = s. Then we have the result of Andrews [1 Theorem 1]. (4.6 and (4.7 (4.8 Corollary 4.4. (Andrews nrs 0 nrs 0 S N (n r s n z r y s = S N+1 (n r s n z r y s = Corollary 4.5. (Boulet Let = abcd then λ partitions [ ] N N j=0 j ( zy; 4 j ( zy 1 ; 4 N j (y N j 4 ( 4 ; 4 N (z 4 ; 4 N [ ] N N j=0 j ( zy; 4 j+1 ( zy 1 ; 4 N j (y N j 4 ( 4 ; 4 N (z 4 ; 4. N+1 ω(λ = Here the sum runs over all partitions λ (cf. [3 Theorem 1]. First we show the following recurrence euations hold. ( a; ( abc; (; (ab; (ac;.

8 Masao Ishikawa and Jiang Zeng Proposition 4.6. Let Φ N = Φ N (a b c d; z be as before and = abcd. Then the following recurrences hold for any positive integer N. (4.9 (4.10 (1 z N Φ N = (1 + bφ N 1 b Φ N (1 z ac N Φ N+1 = (1 + aφ N aφ N A weighted sum of Schur s P-functions We use the notation X = X n = (x 1... x n for the finite set of variables x 1... x n. In [8] one of the authors used a Paffian expression of ω(λs λ (X to prove Stanley s open problem where the sum runs λ over all partitions λ and s λ (X stands for the Schur function with respect to a partition λ. The aim of this section is to give some determinantial formulas for the weighted sum ω(µz l(µ P µ (x where P µ (x is Schur s P-function. Let A n denote the skew-symmetric matrix (xi x j x i + x j 1 ij n and for each strict partition µ = (µ 1... µ l of length l n let Γ µ denote the n l matrix ( x µi j. Let ( An Γ A µ (x 1... x n = µ J l J t l Γ µ O l which is a skew-symmetric matrix of (n+l rows and columns. Define Pf µ (x 1...x n to be Pf A µ (x 1...x n if n + l is even and to be Pf A µ (x 1... x n 0 if n + l is odd. By Ex.13 p.67 [11] Schur s P-function P µ (x 1...x n is defined to be Pf µ (x 1...x n Pf (x 1... x n where it is well-known that Pf (x 1...x n = x i x j 1 i<j n x i+x j. Meanwhile by (8.7 p.53 [11] Schur s Q-function Q µ (x 1...x n is defined to be l(λ P µ (x 1... x n. In this section we consider a weighted sum of Schur s P-functions and Q-functions i.e. ξ N (a b c d; X n = µ µ 1 N η N (a b c d; X n = µ µ 1 N ω(µp µ (x 1... x n ω(µq µ (x 1...x n where the sums run over all strict partitions µ such that each part of µ is less than or eual to N. More generally we can unify these problems to finding the following sum: (5.1 ζ N (a b c d; z; X n = µ µ 1 N ω(µz l(µ P µ (x 1...x n where the sum runs over all strict partitions µ such that each part of µ is less than or eual to N. One of the main results of this section is that ζ N (a b c d; z; X n can be expressed by a Pfaffian. Further let us put (5. ζ(a b c d; z; X n = lim N ζ N(a b c d; z; X n = µ ω(µz l(µ P µ (X n where the sum runs over all strict partitions. We also write ξ(a b c d; X n = ζ(a b c d; 1; X n = µ ω(µp µ (X n where the sum runs over all strict partitions. Then we have the following theorem: Theorem 5.1. Let n be a positive integer. Then { Pf (γij ζ(a b c d; z; X n = 1 i<j n / Pf (X n if n is even (5.3 Pf (γ ij 0 i<j n / Pf (X n if n is odd

9 THE PARTITION FUNCTION AND AL-SALAM-CHIHARA POLYNOMIALS where (5.4 with (5.5 (5.6 if 1 i j n and (5.7 u ij = v ij = if 1 j n. Especially when z = 1 we have (5.8 where (5.9 γ ij = x i x j x i + x j + u ij z + v ij z xi + bx a det( i 1 abx i x j + bx j 1 abx j (1 abx i (1 abx j xi + ax abcx i x j det( i 1 a(b + dx i abdx3 i x j + ax j 1 a(b + dx j abdx3 j (1 abx i (1 abx j (1 abcdx i x j ξ(a b c d; X n = (5.10 ṽ ij = γ ij = γ 0j = 1 + ax j(1 + bx j 1 abx z j { Pf ( γij 1 i<j n / Pf (X n if n is even { 1+axj 1 abx j Pf ( γ ij 0 i<j n / Pf (X n x i x j x i+x j + ṽ ij if i = 0 if 1 i < j n with if n is odd xi + bx a det( i 1 b(a + cx i abcx3 i x j + bx j 1 b(a + cx j abcx3 j (1 abx i (1 abx j (1 abcdx i x j. We can generalize this result in the following theorem (Theorem 5. using the generalized Vandermonde determinant used in [9]. Let n be an non-negative integer and let X = (x 1... x n Y = (y 1... y n A = (a 1... a n and B = (b 1... b n be n-tuples of variables. Let V n (X Y A denote the n n matrix whose (i jth entry is a i x n j i y j 1 i for 1 i n 1 j n and let U n (X Y ; A B denote the n n matrix ( V n (X Y A V n (X Y B. For instance if n = then U (X Y ; A B is a 1 x 1 a 1 y 1 b 1 x 1 b 1 y 1 a x a y b x b y a 3 x 3 a 3 y 3 b 3 x 3 b 3 y 3. a 4 x 4 a 4 y 4 b 4 x 4 b 4 y 4 Hereafter we use the following notation for n-tuples X = (x 1 x n and Y = (y 1 y n of variables: X + Y = (x 1 + y 1... x n + y n X Y = (x 1 y 1... x n y n and for integers k and l X k = (x k 1... x k n X k Y l = (x k 1y1 l... x k nyn. l Let 1 denote the n-tuple ( For any subset I = {i 1... i r } ( [n] r let XI denote the r-tuple (x i1...x ir. (5.11 Theorem 5.. Let = abcd. If n is an even integer then we have ξ(a b c d; X n = n/ r=0 I ( [n] r ( 1 I (r+1 a r (r i I (1 abx i ij I i<j x i + x j (x i x j (1 x i x j detu r (X I 1 + X 4 I X I + bx I1 b(a + cx I abcx 3 I.

10 Masao Ishikawa and Jiang Zeng If n is an odd integer then we have n ξ(a b c d; X n = (5.1 (5.13 ij I i<j m=1 (n 1/ 1 + ax m 1 abx m r=0 I ( [n]\{m} r ( 1 I (r+1 a r (r x m + x i i I (1 abx i x m x i i I x i + x j (x i x j (1 x i x j detur (X I 1 + X4 I X I + bx I 1 b(a + cx I abcx3 I. Theorem 5.3. Let = abcd. If n is an even integer then ζ(a b c d; z; X n is eual to n/ z r r=0 I ( [n] r ( 1 I (r+1 (abc r (r i I x i i I (1 abx i ij I i<j detv r (X I 1 + X4 I X I + ax I 1 a(b + dx I abdx3 I n/ + z r 1 r=0 where I = I \ {k l}. I ( [n] r k<l kl I x i + x j (x i x j (1 x i x j ( 1 I (r 1 a r b r 1 c r 1 (r 1 {1 + b(xk + x l + abx k x l } i I x i i I (1 abx i ij I (x i + x j detv r 1 (XI 1 + X4 I X I + ax I 1 a(b + dx I i<j abdx3 I ij I (x i x j (1 x i x j i<j References [1] G. E. Andrews On a partition function of Richard Stanley Electron. J. Combin. 11( (004 #R1. [] G. Gasper and M. Rahman Basic Hypergeometric Series Cambridge University Press Cambridge Second edition 004. [3] C. Boulet A four parameter partition identity arxiv:math.co/ to appear in Ramanujan J. [4] D.P. Gupta and D.R. Masson Solutions to the associated -Askey-Wilson polynomial recurrence relation arxiv:math.ca/ [5] M.E.H. Ismail and M. Rahman The associated Askey-Wilson polynomials Trans. Amer. Math. Soc. 38 ( [6] R. Koekoek and R.F.Swarttouw The Askey-scheme of hypergeometric orthogonal polynomials and its -analogue Delft University of Technology Report no (1998. [7] C. Krattenthaler Advanced determinant calculus Seminaire Lotharingien Combin. 4 ( The Andrews Festschrift (1999 Article B4 67. [8] M. Ishikawa Minor summation formula and a proof of Stanley s open problem arxiv:math.co/ [9] M. Ishikawa S. Okada H. Tagawa and J. Zeng Generalizations of Cauchy s determinant and Schur s Pfaffian arxiv:math.co/ to appear in Adv. in Appl. Math. [10] M. Ishikawa and M. Wakayama Minor summation formula of Pfaffians Linear and Multilinear Algebra 39 ( [11] I.G. Macdonald Symmetric functions and Hall polynomials nd Edition Oxford University Press (1995. [1] A. Sills A combinatorial proof of a partition identity of Andrews and Stanley Journal of Mathematics and Mathematical Sciences 004/ [13] R. P. Stanley Some remarks on sign-balance and maj-balanced posets Adv. in Appl. Math 34( [14] A. J. Yee On partition functions of Andrews and Stanley J. Combin. Theory Ser.A 13 ( Faculty of Education Tottori University Koyama Tottori Japan address: ishikawa@fed.tottori-u.ac.jp Institut Camille Jordan Université Claude Bernard Lyon I 43 boulevard du 11 novembre Villeurbanne Cedex France address: zeng@math.univ-lyon1.fr