Integer Partitions With Restricted Distinct Parts

Abstract

For any positive integers $s$ and $t$, let $Q_t^s(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for $Q_t^s(n)$ for some particular values of $s$ and $t$ by employing $q$-series and theta function identities.