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

TL;DR

This paper investigates the arithmetic properties of partitions into parts that are simultaneously 4-regular and 9-distinct, revealing multiple infinite families of congruences and Ramanujan-like congruences modulo various integers.

Contribution

It provides a detailed study of the specific case (4,9) and establishes new infinite families of congruences for the partition function.

Findings

Several infinite families of congruences modulo 4, 6, and 12.

A collection of Ramanujan-like congruences modulo 24.

Enhanced understanding of the arithmetic structure of (4,9)-regular and 9-distinct partitions.

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 $24$.