On the growth of restricted integer partition functions

Abstract

We study the rate of growth of p(n,S,M), the number of partitions of n whose parts all belong to S and whose multiplicities all belong to M, where S (resp. M) are given infinite sets of positive (resp. nonnegative) integers. We show that if M is all nonnegative integers then p(n,S,M) cannot be of only polynomial growth and that no sharper statement can be made. We ask: if p(n,S,M) > 0 for all large enough n, can p(n,S,M) be of polynomial growth in n? © Springer Science+Business Media, LLC 2012.