The Andrews–Stanley partition function and Al-Salam–Chihara polynomials

Abstract

For any partition λ let ω(λ) denote the four parameter weightand let (λ) be the length of λ. We show that the generating function ω(λ)z (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 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 ω(λ)z (λ) P λ (x) where P λ (x) is Schur's P-function and the sum runs over all strict partitions.

Weighted Rogers–Ramanujan partitions and Dyson crank

“…We note that [8], and [10] makes it possible to refine the results involving the partition statistic O by imposing bounds on the number of parts and the largest part of partitions. Moreover, there are many more fundamental statistics of partitions similar to O, and E. It would be of interest to see results and weights related to the rank of a partition and similar known classical partition statistics.…”

“…Writing generating functions with some natural partition statistics such as O and E can be studied directly from the the results of [8] , [9], and [10]. One decorates the Ferrers diagrams by writing on the dots on the rows.…”

A unifying combinatorial approach to refined little Göllnitz and Capparelli's companion identities

“…These generating functions come most naturally by considering rows of the labelled Ferrers' diagram, even though the block types are defined by columns. Indeed, fix 0 ≤ i ≤ N, the series (−c; Q) i (ab) i (Q; Q) i generates exactly i blocks of type II (distinct) and III, while the series In [19], Ishikawa and the second author considered the bounded version of Boulet's formula and obtained the following series expansion via application of results on associated Al-Salam-Chihara polynomials.…”

“…Actually they proved a finite version [9, Theorem 4.1] of the above result using recurrence and a special case of a finite version of (1.2) due to Ishikawa and the second author [19,Corollary 3.4]. One of our aims is to give a combinatorial proof of their finite version using a variant of Boulet's bijection (see Section 4.1).…”

Minor summation formula and a proof of Stanley’s open problem

“…At the end of this paper we state some corollaries which generalize Stanley's open problem to the big Schur functions, and to certain polynomials arising from the Macdonald polynomials. Furthermore, in the forthcoming paper [10], we study a finite version of Boulet's theorem and present certain relations with orthogonal polynomials and the basic hypergeometric series. In the paper we find more applications of the Pfaffain expression of Stanley's weight ω(λ) obtained in this paper, and also study a certain summation of Schur's Q-functions weighted by ω(λ).…”