Ramanujan-type Congruences for $\ell$-Regular Partitions Modulo $3,5,11$ and $13$

Abstract

Let $b_{\ell }(n)$ be the number of $\ell$-regular partitions of $n$. Recently, Hou et al established several infinite families of congruences for $b_{\ell }(n)$ modulo $m$, where $(\ell ,m)=(3,3),(6,3),(5,5),(10,5)$ and $(7,7)$. In this paper, by the vanishing property given by Hou et al, we show an infinite family of congruence for $b_{11}(n)$ modulo $11$. Moreover, for $\ell =3,13$ and $25$, we obtain three infinite families of congruences for $b_{\ell }(n)$ modulo $3,5$ and $13$ by the theory of Hecke eigenforms.

arXiv comment13 pages