Congruence properties of $S$-partition functions

Abstract

We study the function $p\left(S;n\right)$ that counts the number of partitions of $n$ with elements in $S$, where $S$ is a set of integers. Generalizing previous work of Kronholm, we find that given a positive integer $m$, the coefficients of the generating function of $p\left(S;n\right)$ are periodic modulo $m$, and we use this periodicity to obtain families of $S$-partition congruences. In particular, we obtain families of congruences between partition functions $p\left(S_1;n\right)$ and $p\left(S_2;n\right)$.