  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 $n$?
