Partitions of bipartite numbers
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 \(m\leq n\), \[ p_j(n,m)=p_j (m,n) \geq p_j (m-1,n+1). \]
scientific article; zbMATH DE number 989831