Partitions into odd, unequal parts
type: Item
instance of: scholarly article
title: Partitions into odd, unequal parts
zbMATH Open document ID: 0584.05005
DOI: 10.1016/0022-4049(85)90003-9
author: Gert Almkvist
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \(q(j,n)\) denote the number of partitions of \(j\geq 2\) into odd unequal parts none exceeding \(2n-1.\) Then the author and R. P. Stanley conjecture that \(q(j,n)\) increases with j for \(2\leq j\leq n^ 2/2\). Here is the reviewer's proof of the conjecture. Let (1) \(j=u_ 1,u_ 2,u_ 3,\) \(u_ 4,...,u_ k,\) where the u's are distinct odd numbers arranged in ascending order of magnitude, be a partition of j into distinct odd parts. If 1 appears as a part on the right of (1), then (2) \(j+1=1\), \(u_ 1,u_ 2,u_ 3,\) \(u_ 4,...,u_ k\) provides a partition of \((j+1)\) which is not a partition of j. If 1 does not appear as a part on the right of (1) then \(j+2=u_ 1,u_ 2,u_ 3,\) \(u_ 4,...,(u_ k+2)\) provides a partition of \(j+2\) into distinct odd parts which is different from the partition of \((j+1)\) in (2) and also from the partition of j in (1). This proves the conjecture of the author and Stanley.
reviewed by: Hansraj Gupta
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 3935044
zbMATH Keywords: partitions
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90003-9
OpenAlex ID: W2081126768
cites work: On the unimodality of some partition polynomials
cites work: Some aspects of groups acting on finite posets
Wikidata QID: Q127673951
description: scientific article; zbMATH DE number 3935044