Some congruences for $(\ell, k)$ and $(\ell, k, r)$-regular partitions

Abstract

Let $b_{\ell, k}(n), b_{\ell, k, r}(n)$ count the number of $(\ell, k)$, $(\ell, k, r)$-regular partitions respectively. In this paper we shall derive infinite families of congruences for $b_{\ell, k}(n)$ modulo $2$ when $ (\ell, k) = (3,8), (4, 7)$, for $b_{\ell, k}(n)$ modulo $8$, modulo $9$ and modulo $12$ when $(\ell, k) = (4, 9)$ and $b_{\ell, k, r}(n)$ modulo $2$ when $(\ell, k, r) = (3, 5, 8)$.