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 $r$.