Some New Congruences for $\ell$-Regular Multipartitions

Abstract

For a positive integer $n$, let $B_{\ell_1,\dots,\ell_r}(n)$ denote the number of $(\ell_1,\ell_2,\cdots,\ell_r)$-regular multipartitions of $n$. If $\ell_1=\ell_2=\cdots=\ell_r=\ell$, then we denote $B_{\ell_1,\dots,\ell_r}(n)$ as $B_\ell^{(r)}(n)$. In this paper, we prove several infinite families of congruences satisfied by $B_\ell^{(r)}(n)$ for different values of $\ell$ and $r$.