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

TL;DR

This paper establishes new infinite families of Ramanujan-type congruences for $ ext{l}$-regular partition functions modulo 3, 5, 11, and 13, expanding the understanding of their arithmetic properties using modular form theory.

Contribution

It introduces novel infinite congruence families for $ ext{l}$-regular partitions modulo several primes, utilizing vanishing properties and Hecke eigenform theory.

Findings

Infinite congruences for $b_{11}(n)$ modulo 11.

Congruences for $b_{3}(n)$, $b_{13}(n)$, and $b_{25}(n)$ modulo 3, 5, and 13.

Extension of Ramanujan-type congruences to new partition functions.

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.