A proof of the Bender-Knuth conjecture
type: Item
instance of: scholarly article
title: A proof of the Bender-Knuth conjecture
zbMATH Open document ID: 0533.05005
author: Basil Gordon
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let \(b_ r(n| I_ m)\) denote the number of r-rowed partitions of n whose parts lie in the set \(I_ m=\{1,2,...,m\}\) and decrease strictly along each row. \textit{E. A. Bender} and \textit{D. E. Knuth} conjectured [J. Comb. Theory, Ser. A 13, 40-54 (1972; Zbl 0246.05010)] that the generating function of \(b_ r(n| I_ m)\) is equal to \(\prod^{m}_{i=1}\prod^{i}_{j=1}(1-x^{r+i+j-1})/(1-x^{i+j- 1})\). This was derived from a related result by \textit{G. E. Andrews} [Pac. J. Math. 72, 283-291 (1977; Zbl 0376.10014)]. In this paper it is presented another, direct proof. It is based on the evaluation of certain determinants.
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 3845583
zbMATH Keywords: r-rowed partitions
zbMATH Keywords: generating function
Wikidata QID: Q123265966
reviewed by: Attila Pethő
MaRDI profile type: Publication
full work available at URL: pjm.1983.108.99
OpenAlex ID: W2053842820
DOI: 10.2140/PJM.1983.108.99
description: scientific article; zbMATH DE number 3845583