A variation on a theme of Sylvester -- a smoother road to Göllnitz's (Big) theorem
type: Item
instance of: scholarly article
title: A variation on a theme of Sylvester -- a smoother road to Göllnitz's (Big) theorem
zbMATH Open document ID: 0946.11024
DOI: 10.1016/S0012-365X(98)00193-9
author: Krishnaswami Alladi
published in: Discrete Mathematics
publication date: 2000-10-26T00:00:00Z
review text: According to the celebrated theorem of \textit{H. Göllnitz} [J. Reine Angew. Math. 225, 154-190 (1967; Zbl 0166.00803)], the number of partitions of a positive integer \(n\) into distinct parts \(\equiv 2\), 4 or \(5\pmod 6\) is equal to the number of partitions of \(n\) in the form \(h_1+h_2+ \cdots +h_m\) such that \(h_m\neq 1\) or 3, and \(h_i-h_{i+1}\geq 6\) (for \(1\leq i<m)\) with strict inequality if \(h_i\equiv 0\), 1 or \(3\pmod 6\). \textit{K. Alladi}, \textit{G. E. Andrews} and \textit{B. Gordon} [J. Reine Angew. Math. 460, 165-188 (1995; Zbl 0810.11061)] obtained substantial generalizations and refinements of Göllnitz's theorem. In the paper under review the author gives a combinatorial proof of the following Theorem. The number of partitions of a positive integer \(n\) into distinct odd parts is equal to the number of partitions of \(n\) into parts \(\neq 2\) and differing by \(\geq 6\), where the inequality is strict whenever a part is even. The real interest in this Theorem comes from the fact that it has a three-parameter refinement which is much deeper. This refinement is stated as Theorem 2 in Section 4 and it is a reformulation of a three-parameter refinement of Göllnitz's theorem.
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 1321764
zbMATH Keywords: partitions with difference conditions and congruential restrictions
zbMATH Keywords: Göllnitz's theorem
reviewed by: Mihály Szalay
MaRDI profile type: Publication
cites work: A combinatorial correspondence related to Göllnitz’ (big) partition theorem and applications
cites work: Q4379428
cites work: Q4324101
cites work: Partitionen mit Differenzenbedingungen.
full work available at URL: s0012-365x(98)00193-9
OpenAlex ID: W2012447363
description: scientific article; zbMATH DE number 1321764