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A compound determinant identity for minors of rectangular matrices is established. Given an (s+n-1)xsn matrix A with s blocks of n columns, we consider minors of A by picking up in each block the first consecutive columns specified by weak compositions at most s parts, and prove that the compound determinant of such nxn minors of A is equal to the product of maximal minors of A corresponding to compositions of s+n-1 with s parts. As an application, we obtain Vandermonde type product evaluations of determinants of classical group characters, including Schur functions.

For any partition @l let @w(@l) denote the four parameter weight @w(@l)=a^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^"^-^"^1^/^2^@?b^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^"^-^"^1^/^2^@?c^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^/^2^@?d^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^/^2^@?, and let @?(@l) be the length of @l. We show that the generating function @[email protected](@l)z^@?^(^@l^), 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 derive Andrews' result by specializing some parameters and Boulet's results by letting N->+~. In the last section we prove a Pfaffian formula for the weighted sum @[email protected](@l)z^@?^(^@l^)P"@l(x) where P"@l(x) is Schur's P-function and the sum runs over all strict partitions.

In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.

An ordered partition with $k$ blocks of $[n]:=\{1,2,\ldots, n\}$ is a sequence of $k$ disjoint and nonempty subsets, called blocks, whose union is $[n]$. Clearly the number of such ordered partitions is $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind. A statistic on ordered partitions of $[n]$ with $k$ blocks is called an Euler-Mahonian statistic if its generating polynomial is $[k]_q!S_q(n,k)$, which is a natural $q$-analogue of $k!S(n,k)$. Motivated by Steingrimsson's conjectures dating back to 1997, we consider two different methods to produce Euler-Mahonian statistics on ordered set partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of Francon-Viennot's bijection to derive many Euler-Mahonian statistics by expanding the generating function of $[k]_q!S_q(n,k)$ as an explicit continued fraction; (b) we encode ordered partitions by walks in some digraphs and then derive new Euler-Mahonian statistics by computing their generating functions using the transfer-matrix method. In particular, we prove several conjectures of Steingrimsson.

An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula, which implies, in particular, twelve more multiple q-identities of Rogers-Ramanujan type than those previously found by Stembridge and the last two authors.

We call an element of a Coxeter group fully covering (or a fully covering element) if its length is equal to the number of the elements it covers in the Bruhat ordering. It is easy to see that the notion of fully covering is a generalization of the notion of a 321-avoiding permutation and that a fully covering element is a fully commutative element. Also, we call a Coxeter group bi-full if its fully commutative elements coincide with its fully covering elements. We show that the bi-full Coxeter groups are the ones of type A n , D n , E n with no restriction on n. In other words, Coxeter groups of type E9, E10,.... are also bi-full. According to a result of Fan, a Coxeter group is a simply-laced FC-finite Coxeter group if and only if it is a bi-full Coxeter group.

We are developing the disassembling method considering the elementary composition ratio. However, the elementary composition ratio of any products and parts is not indicated, so that these information are not used in the disassembling system. Therefore, we proposed the estimation method of element composition of part by the X-ray fluorescence analysis equipment. (1) The elementary composition ratio of material is estimated based on the image measured by the X-ray fluorescence analysis equipment and image processing. (2) The elementary composition of basic component part (an electronic condenser, an integrated circuit and so on) is stored to the parts database in order to estimate the disassembling product. (3) All basic component part in the product are specified by image processing technique, and its elementary composition are estimated roughly based on its size and elementary composition by using the parts database.

The purpose of this paper is, to establish, by extensive use of the minor summation formula of pfaffians exploited in (Ishikawa, Okada, and Wakayama, J. Algebra183, 193–216) certain new generating functions involving Schur polynomials which have a product representation. This generating function gives an extension of the Littlewood formula. During the course of the proof we develop some techniques for computing sub-Pfaffians of a given skew-symmetric matrix. After the proof we present an open problem which generalizes our formula.

This paper presents some new product identities for certain summations of Schur functions. These identities are generalizations of some famous identities known to Littlewood and appearing in Macdonald’s book. We refer to these identities as the ‘‘Littlewood-type formulas.’’ In addition, analogues for summations of characters of the other classical groups are also given. The Littlewood-type formulas in this paper are separated into two classes, the rational Schur function series and the generalized Schur function series. An application of a rational Schur function series to the infinite product representation of the elliptic theta functions is also given. We prove these Littlewood-type formulas using the Cauchy]Binet formula. The Cauchy]Binet formula is a basic but powerful tool applicable in the present context, which can be derived from our Pfaffian formula, as we explain. Q 1998 Academic Press

The first and the third authors obtained a minor-summation formula of Pfaffian, which expresses a weighted sum of minors of any rectangular matrix in terms of a single Pfaffian. In this paper, as an application of this minor-summation formula, we give a new proof of Littlewood's formulas with additional parameters and generalize them for the classical groups.

Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n×n$ determinant $\det ((a+j-i)Γ (b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\mathrm{Pf}((j-i)Γ (b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a q-analogue of the Mehta–Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.

In this paper we establish a formula involving Pfaffians for a certain weighted sum of minors of an arbitrary given matrix. First we find a formula where the sum ranges for all columns, and secondly we obtain a formula where the sum ranges for both all rows and columns as the application of the first one. The first formula is stated in the framework of quantum matrix algebra (Matq(m,n)). These sums are weighted by "sub-Pfaffians" of any given (q)skew symmetric matrices. In the last section we provide an application of the minor summation formula to the generating functions of shifted tableaux.

We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula., 12 pages, to appear in Proceedings of the American Mathmatical Society

A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original approach does not work. As Zeilberger's approach is based on the Laplace expansion (cofactor expansion) of the determinant, we derive our approach from the cofactor expansion of the Pfaffian. To demonstrate the power of our method, we prove, using computer algebra algorithms, some conjectures proposed in the paper "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants" by Ishikawa, Tagawa, and Zeng. A minor summation formula related to partitions and Motzkin paths follows as a corollary., 7 pages, final version for the ISSAC proceedings; Proceedings of ISSAC 2012

Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions., 33 pages

We prove $$ {\rm Pf}\! \left( { x_i - x_j \over (x_i + x_j)^2 } \right)_{1 \le i, j \le 2n} = \prod_{1 \le i < j \le 2n}{ x_i - x_j \over x_i + x_j } {\rm Hf}\! \left( { 1 \over x_i + x_j } \right)_{1 \le i, j \le 2n} $$ (and its variants) by using complex analysis. This identity can be regarded as a Pfaffian–Hafnian analogue of Borchardt's identity and as a generalization of Schur's identity.

The initial purpose of this paper is to provide a combinatorial proof of the minor summation formula of Pfaffians based on the lattice path method. There we related Pl\"ucker relations with the minor summation formula of Pfaffians to simplify its proof by lattice paths. The second aim is to find a various applications of the minor summation formula. First we studied a variant of the Sundquist formula established in J. Alg. Combin. {\bf 5} (1996). Next we gave a simple proof of Kawanaka's formula concerning a $q$-series identity involving Schur functions in Osaka J. Math. {\bf 36} (1999). We also establish a certain identity similar to the Kawanaka formula, and give a combinatorial proof of the determinant version of his formula in Osaka J. Math. {\bf 38} (2001)., Comment: 39 pages, LaTeX; typos corrected

In this paper we treat the minor summation formulas of pfaffians presented in [IW1] and derive several basic formulas concerning pfaffians from it. We also present a pfaffian version of the Plucker relation and give a new pfaffian identity as its application.

We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition., Comment: 26 pages