Elementary Proofs of Infinite Families of Congruences for Merca's Cubic Partitions

TL;DR

This paper provides elementary, classical proofs for two known congruences related to Merca's cubic partitions and establishes two infinite families of similar Ramanujan-like congruences modulo 3.

Contribution

It introduces elementary proofs for existing congruences and discovers new infinite families of Ramanujan-like congruences for Merca's cubic partition function.

Findings

Elementary proofs of two key congruences

Two infinite families of Ramanujan-like congruences modulo 3

Extension of known partition congruences

Abstract

Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\ A(27n+26) & \equiv 0 \pmod{3}. \end{align*} Here $A(n)$ is closely related to the function which counts the number of {\it cubic partitions}, partitions wherein the even parts are allowed to appear in two different colors. Indeed, $A(n)$ is defined as the difference between the number of cubic partitions of $n$ into an even numbers of parts and the number of cubic partitions of $n$ into an odd numbers of parts. In this brief note, we provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of non--nested Ramanujan--like congruences modulo 3 satisfied by $A(n)$ wherein Merca's original two congruences serve as the initial members of each family.