Generalisations of the Andrews-Olsson partition identity and applications
type: Item
instance of: scholarly article
title: Generalisations of the Andrews-Olsson partition identity and applications
zbMATH Open document ID: 0839.05009
DOI: 10.1016/0012-365X(93)E0189-B
author: Christine Bessenrodt
published in: Discrete Mathematics
publication date: 1996-06-05T00:00:00Z
review text: The celebrated 1926 partition theorem of Schur is the statement that the number of partitions of \(n\) into distinct parts \(\equiv 1\) or \(2\pmod 3\) is equal to the number of partitions of \(n\) into parts differing by \(\geq 3\) and where consecutive multiples of 3 cannot occur as parts. In the 1960's, George Andrews obtained an extension of this result where the modulus 3 is replaced by an integer \(M\geq 3\) and the residue classes \(1, 2\pmod 3\) are replaced by \(r_i\pmod M\) where the \(r_i\) satisfy certain conditions. In 1991, \textit{G. E. Andrews} and \textit{J. B. Olsson} [J. Reine Angew. Math. 413, 198-212 (1991; Zbl 0704.20003)] obtained an important reformulation of this general result by using conjugation of Ferrers graphs. Also in 1991, the author [Eur. J. Comb. 12, No. 4, 271-276 (1991; Zbl 0743.05003)] gave a combinatorial proof of the Andrews-Olsson partition identity. In this paper the author obtains a further generalization by removing the conditions on the residues \(r_i\pmod M\) that Andrews and Olsson had. In doing so, the author obtains as a special case, a generalization of the Schur partition theorem due to the reviewer and \textit{B. Gordon} [Manuscr. Math. 79, No. 2, 113-126 (1993; Zbl 0799.11043)].
reviewed by: Krishnaswami Alladi
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 778520
zbMATH Keywords: Sylvester bijection
zbMATH Keywords: number of partitions
zbMATH Keywords: Andrews-Olsson partition identity
zbMATH Keywords: Schur partition theorem
MaRDI profile type: Publication
cites work: Generalizations of Schur's partition theorem
cites work: On the Geometry of Numbers in Elementary Number Theory
cites work: Q3483443
cites work: A combinatorial proof of a refinement of the Andrews-Olsson partition identity
cites work: A bijection for Lebesgue's partition identity in the spirit of Sylvester
cites work: A Combinatorial Proof of Schur's 1926 Partition Theorem
cites work: Q3943030
description: scientific article; zbMATH DE number 778520