A congruence family modulo powers of 5 for generalized cubic partitions via the localization method

Abstract

Recently Amdeberhan, Sellers, and Singh introduced a new infinite family of partition functions called generalized cubic partitions. Given a positive integer $d$, they let $a_d(n)$ be the counting function for partitions of $n$ in which the odd parts are unrestricted and the even parts are $d$-colored. These partitions are natural generalizations of Chan's notion of cubic partitions, as they coincide when $d=2.$ Many Ramanujan-like congruences exist in the literature for cubic partitions, and in their work Amdeberhan, Sellers, and Singh proved a collection of congruences satisfied by $a_d(n)$ for various $d \geq 1$, including an infinite family with prime moduli. Our goal in this paper is to prove a family of congruences modulo powers of 5 for $a_3(n)$. More specifically, our main theorem asserts \[a_3\left(5^{2α}n +γ_α \right) \equiv 0 \pmod{5^α},\] where \[γ_α = 20 + \frac{19 \cdot 25 (25^{α-1}-1)}{24}.\] In order to prove these congruences, we use an approach centered around modular functions, as in the seminal work of Watson and Atkin on proving Ramanujan's congruences for the partition function $p(n)$. However, due to the complexity of the modular curve $X_0(10)$ associated to our modular functions, the classical method cannot be directly applied. Rather, we utilize the very recently developed localization method of Banerjee and Smoot, which is designed to treat congruence families over more complicated modular curves, such as $X_0(10).$