A proof of a partition conjecture of Bateman and Erdos

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(A)((k))(n), t o eventually be positive; moreover, they showed that when these conditions occur p(A)((k+1)) (n)/p(A)((k))(n) tends to zero as n tends to infinity. Ba teman and Erdos conjectured that the ratio p(A)((k+1))(n)/p(A)((k))(n) = O( n(-1/1)). We prove this conjecture. (C) 2001 Academic Press.