Partitions of bipartite numbers

Abstract

Let p(j)(m, n) be the number of partitions of (m, n) into at most j parts. We prove Landman et al.'s conjecture: for all j and n, p(j)(x, 2n - x) is a maximum when x = n. More generally we prove that for all positive integers m, n and j, p(j)(n,m) = p(j)(m, n) greater than or equal to p(j)(m - 1, n + 1) if m less than or equal to n.