Partitions of bipartite numbers
type: Item
instance of: scholarly article
title: Partitions of bipartite numbers
zbMATH Open document ID: 0869.11079
DOI: 10.1007/BF01202238
author: Jun Kyo Kim
author: Sang Geun Hahn
published in: Graphs and Combinatorics
publication date: 1997-08-19T00:00:00Z
review text: A bipartite number is defined to be a vector \((m,n)\), where \(m\) and \(n\) are non-negative integers. A partition of a bipartite number \((m,n)\) is a solution of the vector equation \(\sum^r_{k=1} (m_k,n_k) =(m,n)\), where each \((m_k, n_k)\) is a bipartite number other than \((0,0)\). Two partitions of \((m,n)\) which differ only in the order of the terms are considered to be indentical. Let \(p_j(m,n)\) be the number of partitions of \((m,n)\) into at most \(J\) parts. \textit{B. M. Landman}, \textit{E. A. Brown} and \textit{F. J. Portier} have conjectured [Graphs Comb. 8, 65-73 (1992; Zbl 0762.05005)] that for all \(j\) and \(n\), \(p_j(x,2n-x)\) is a maximum when \(x=n\). This conjecture is now proved, that is, it is shown that for all positive integers \(m,n,j\) with \(m\leq n\), \[ p_j(n,m)=p_j (m,n) \geq p_j (m-1,n+1). \]
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 989831
zbMATH Keywords: bipartite number
zbMATH Keywords: partitions
reviewed by: Betty Garrison
MaRDI profile type: Publication
cites work: Q4146776
cites work: Partitions of bi-partite numbers into at most \(j\) parts
description: scientific article; zbMATH DE number 989831