On the arithmetic properties of partitions into parts simultaneously $4$-regular and $9$-distinct

Abstract

In 2017, Keith presented a comprehensive survey on integer partitions into parts that are simultaneously regular, distinct, and/or flat. Recently, the authors initiated a study of partitions into parts that are simultaneously regular and distinct, examining them from both arithmetic and combinatorial perspectives. In particular, several Ramanujan-like congruences were obtained for $\myRD^{(\ell, t)}(n)$, the number of partitions of $n$ into parts that are simultaneously $\ell$-regular and $t$-distinct (parts appearing fewer than $t$ times), for various pairs $(\ell, t)$. In this paper, we focus on the case $(\ell, t)=(4,9)$ and conduct a thorough investigation of the arithmetic properties of $\myRD^{(4, 9)}(n)$. We establish several infinite families of congruences modulo $4$, $6$, and $12$, along with a collection of Ramanujan-like congruences modulo $24$.