On the growth of restricted integer partition functions

TL;DR

This paper investigates the growth rate of restricted integer partition functions, showing that under certain conditions, their growth cannot be purely polynomial, and raises questions about the nature of their asymptotic behavior.

Contribution

It establishes that the partition function with all multiplicities allowed cannot grow polynomially and explores conditions under which polynomial growth is possible.

Findings

If M includes all nonnegative integers, p(n,S,M) cannot have polynomial growth.

No sharper growth rate bounds are possible beyond the established results.

Raises open questions about polynomial growth when p(n,S,M) is positive for large n.

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 $n$?