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 $(\ell ,k,r)=(3,5,8)$.