Two partition functions with congruences modulo 3, 5, 7, and 13

Abstract

We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use of generalized Lambert series identities and $q$-series techniques.