A Proof of a Partition Conjecture of Bateman and Erdos

Abstract

Bateman and Erdos found necessary and sufficient conditions on a set A for the kth differences of the partitions of n with parts in A, p(k)A(n), to eventually be positive; moreover, they showed that when these conditions occur p(k+1)A(n)/p(k)A(n) tends to zero as n tends to infinity. Bateman and Erdos conjectured that the ratio p(k+1)A(n)/p(k)A(n)=O(n -1/2). We prove this conjecture. © 2001 Academic Press.