Arithmetic properties of $(\ell ,m)$-regular colored partitions

Abstract

Let $b_{\ell ,m}^k(n)$ denotes the number of $k-$colored partitions of $n$ into parts that are not multiples of $\ell$ or $m$. We establish several congruence relations for $b_{\ell ,m}(n)$. For instance, for any nonnegative integer $n$ $$b_{4,5}^2(8n+7)\equiv 0(\mathrm{mod}40).$$