Partitions with prescribed hook differences
type: Item
instance of: scholarly article
title: Partitions with prescribed hook differences
zbMATH Open document ID: 0643.05006
DOI: 10.1016/S0195-6698(87)80041-0
author: William H. Burge
author: George E. Andrews
author: R. J. Baxter
author: David M. Bressoud
author: Peter J. Forrester
author: Xavier Viennot
published in: European Journal of Combinatorics
publication date: 1987-01-01T00:00:00Z
review text: The hook difference at the (i,j)-th node of Ferrers graph of a partition P is defined as the number of nodes in the i-th row of P minus the number of nodes in the j-th column of P. The (i,j)-th node lies on diagonal c if \(i-j=c\). Let \(p_{k,i}(N,M;\alpha,\beta;n)\) be the number of partitions of n into at most M parts each \(\leq\) N that the hook difference on diagonal 1-\(\beta\) are \(\geq -i+\beta +1\) and on diagonal \(\alpha\)-1 are \(\leq K-i-\alpha -1.\) Theorem 1. Let \(1\leq i\leq K/2,\alpha >0\), \(\beta >0\), \(\alpha +\beta <K\) with M,N,i,\(\alpha\),\(\beta\) integers. For \(-i+\beta \leq N-M\leq K-i- \alpha,\) \[ \sum_{n\geq 0}P_{K,i}(N,M;\alpha,\beta;n)q^ n=\sum^{\infty}_{\mu =-\infty}q^{\mu (K\mu +i)(\alpha +\beta)- K\beta \mu}\left[ \begin{matrix} N+M\\ N-K\mu \end{matrix} \right]_ q- \sum^{\infty}_{\mu =-\infty}q^{\mu (k\mu -i)(\alpha +\beta)-K\beta \mu +\beta i}\left[ \begin{matrix} N+M\\ N-K\mu +i\end{matrix} \right]_ q \] where \(\left[\begin{matrix} A\\ B\end{matrix} \right]_ q\) is the Gaussian q-binomial coefficient. Theorem 2. For \(1\leq i\leq K/2,\alpha >0\), \(\beta >0\), \(\alpha +\beta <K\) all integers \[ \sum^{\infty}_{n=0}P_{K,i}(\alpha,\beta;n)q^ n=\frac{1}{(q)_{\infty}}\sum^{\infty}_{\mu =-\infty}q^{\mu (K\mu -i)(\alpha +\beta)+K\beta \mu}(1-q^{\beta i+2i(\alpha +\beta)\mu)}= \] \[ \frac{1}{(q)_{\infty}}(\prod^{\infty}_{n=0}(1-q^{2K(\alpha +\beta)(n+1)})(1+q^{K\beta +(\alpha +\beta)(2Kn+K-i)})(1+q^{-K\beta +(\alpha +\beta)(2Kn+K+i)}- \] \[ q^{\beta i}\prod^{\infty}_{n=0}(1-q^{2K(\alpha +\beta)(n+1)})(1+q^{K\beta +(\alpha +\beta)(2Kn+K+i)})(1+q^{-K\beta +(\alpha +\beta)(2Kn+K-i)}) \] where \((q)_{\infty}=\prod^{\infty}_{n=1}(I-q^ n)\), \(P_{K,i}(\alpha,\beta;n)=\lim_{M,N\to \infty}P_{K,i}(N,M;\alpha,\beta;n).\) Theorem 3. For \(1\leq i\), \(\alpha\leq K/2\), 2i\(\neq K\), \[ \sum^{\infty}_{n=0}P_{K,i}(\alpha,\alpha;n)q^ n=\prod^{\infty}_{n=1,n\not\equiv 0,\pm \alpha_ i(mod \alpha K)}(1- q^ n)^{-1}; \] Theorem 4. Let K,\(\alpha\),…
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 4047728
zbMATH Keywords: partition
zbMATH Keywords: hook difference
zbMATH Keywords: q-binomial coefficient
MaRDI profile type: Publication
cites work: Q5635503
cites work: Sieves in the Theory of Partitions
cites work: Q4146776
cites work: The hard-hexagon model and Rogers—Ramanujan type identities
cites work: Q3344206
cites work: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
cites work: A NOTE ON RANKS AND CONJUGACY OF PARTITIONS
cites work: On the Simplification of Some Identities of the Rogers-Ramanujan Type
cites work: Q3324796
cites work: Extension of the partition sieve
cites work: A correspondence between partitions related to generalizations of the Rogers-Ramanujan identities
cites work: A three-way correspondence between partitions
cites work: Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities
cites work: Q5508372
description: scientific article; zbMATH DE number 4047728