Some congruences for $(\ell, k)$ and $(\ell, k, r)$-regular partitions

TL;DR

This paper establishes new infinite families of congruences for counts of specific regular partitions modulo small integers, expanding understanding of their arithmetic properties.

Contribution

It derives novel congruences for regular partition functions for various parameter sets, including infinite families modulo 2, 8, 9, and 12.

Findings

Congruences for $b_{3,8}(n)$ and $b_{4,7}(n)$ modulo 2.

Congruences for $b_{4,9}(n)$ modulo 8, 9, 12.

Congruences for $b_{3,5,8}(n)$ modulo 2.

Abstract

Let $b_{\ell, k}(n), b_{\ell, k, r}(n)$ count the number of $(\ell, k)$, $(\ell, k, r)$-regular partitions respectively. In this paper we shall derive infinite families of congruences for $b_{\ell, k}(n)$ modulo $2$ when $ (\ell, k) = (3,8), (4, 7)$, for $b_{\ell, k}(n)$ modulo $8$, modulo $9$ and modulo $12$ when $(\ell, k) = (4, 9)$ and $b_{\ell, k, r}(n)$ modulo $2$ when $(\ell, k, r) = (3, 5, 8)$.