On pairs of partitions with steadily decreasing parts
type: Item
instance of: scholarly article
title: On pairs of partitions with steadily decreasing parts
zbMATH Open document ID: 1020.11063
author: Christine Bessenrodt
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: If \(n\) and \(m\) are non-negative integers, let \(\alpha= (\alpha_1, \alpha_2, \dots,\alpha_r)\) be a partition of \(n\), while \(\beta= (\beta_1, \beta_2, \dots,\beta_s)\) is a partition of \(m\). If the pair of partitions \((\alpha,\beta)\) satisfies the conditions: \(\min(\alpha_i, \beta_i) \geq \max(\alpha_{i+1}, \beta_{i+1})\) for all \(i\), then \((\alpha,\beta)\) is said to have steadily decreasing parts. Such pairs of partitions have been previously been studied by Carlitz and by Andrews. The author represents such pairs of partitions graphically via a modified Young diagram. Her main result concerns \(R(n,m)\), the set of all pairs \((\alpha,\beta)\) of partitions of \(n,m\) with steadily decreasing parts such that in addition (1) the \(\alpha_i\) are distinct; (2) the \(\beta_j\) are distinct; (3) \(\alpha_i\neq\beta_i\) for all \(i\). For non-negative integers \(k,n,m\), let \(R(k)=\cup \{R(n,m): n+m=k\}\). Let \(D,O\) represent, respectively, the set of all partitions of non-negative integers into distinct parts, odd parts. Let \(Q(k)\) denote the set of all pairs \(\lambda,\mu\) of partitions such that \(\lambda\in D\cap O\), \(\mu\in O\), and \(|\lambda |+|\mu|=k\). Then \(|R(k)|=|Q(k)|\). The reader is cautioned that since the possibilities \(n,m=0\) are not excluded, some of the partitions under consideration are empty.
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 1888407
zbMATH Keywords: Young diagram
zbMATH Keywords: pairs of partitions
zbMATH Keywords: steadily decreasing parts
zbMATH Keywords: \(r\)-tuples of partitions
zbMATH Keywords: maximum
zbMATH Keywords: generating functions
reviewed by: Neville Robbins
MaRDI profile type: Publication
full work available at URL: jcta.2002.3267
OpenAlex ID: W2014264167
cites work: Q4146776
cites work: An Extension of Carlitz's Bipartition Identity
cites work: Some generating functions
cites work: Restricted bipartite partitions
DOI: 10.1006/JCTA.2002.3267
description: scientific article; zbMATH DE number 1888407