On the growth of restricted integer partition functions

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 $n$?

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