On Congruences modulo powers of 3 for Simultaneously $l$-Regular and $t$-Distinct Partitions in $k$-Colors

Abstract

In 2017, Keith defined and studied a multi-restricted partition function $R{D}^{l,t}(n)$, the number of integer partitions of n into parts that are simultaneously $l$-regular and $t$-distinct if none of the parts is divisible by $l$ and parts appear fewer than $t$ times. Thereafter, in a series of papers Buragohain et al. and Nadji et al. examined combinatorial and arithmetic properties of $R{D}^{l,t}(n)$. In this article, we focus on simultaneously $l$-regular and $t$-distinct partitions in 3-colors, $R{D}_3^{l,t}(n)$ and establish infinite families of congruences modulo powers of 3 for specific values of $(l,t)$. For instance, for positive integers $n$ and $k$, we have \begin{align*} \sum_{n=0}^{\infty}RD_3^{3,3}\left(3^kn+\frac{3^k+1}{2}\right)q^n\equiv0 \pmod{3^{k+1}}. \end{align*} Our approach involves Hirschhorn's H operator technique.