Congruences for the difference of even and odd number of parts of the cubic and some analogous partition functions

TL;DR

This paper simplifies the proofs of existing congruences related to cubic partitions and introduces analogous partition functions with multiple colors for specific integers, expanding the understanding of partition congruences.

Contribution

It provides simplified formulas for generating functions of cubic partitions and their congruences, and explores new colored partition functions for certain integers.

Findings

Simplified proofs of Merca's and da Silva and Sellers' congruences using Ramanujan's theta identities.

Derived formulas for generating functions of cubic partitions and their congruences.

Studied analogous colored partition functions for k in {3,5,7,23}.

Abstract

Partitions wherein the even parts appear in two different colours are known as cubic partitions. Recently, Merca introduced and studied the function $A(n)$, which is defined as the difference between the number of cubic partitions of $n$ into an even number of parts and the number of cubic partitions of $n$ into an odd number of parts. In particular, using Smoot's \textsf{RaduRK} Mathematica package, Merca proved the following congruences by finding the exact generating functions of the respective functions. For all $n\ge0$, \begin{align*}A(9n+5)\equiv 0\pmod 3,\\ A(27n+26)\equiv 0\pmod 3. \end{align*} By using generating function manipulations and dissections, da Silva and Sellers proved these congruences and two infinite families of congruences modulo 3 arising from these congruences. In this paper, by employing Ramanujan's theta function identities, we present simplified formulas of the generating functions from which proofs of the congruences of Merca as well as those of da Silva and Sellers follow quite naturally. We also study analogous partition functions wherein multiples of $k$ appear in two different colours, where $k\in\{3,5,7,23\}$.