On the parity of the number of $(a,b,m)$-copartitions of $n$

TL;DR

This paper investigates the parity properties of a generalized partition function called $(a,b,m)$-copartitions, identifying cases with predominantly even values and proving infinite oscillation between even and odd values.

Contribution

It provides new results on the parity distribution of $(a,b,m)$-copartitions, including cases with density 1 of even values and infinite parity oscillation.

Findings

Certain parameter choices yield copartitions with even values of density 1.

The sequence $ ext{cp}_{a,m-a,m}(n)$ takes both even and odd values infinitely often.

The generating function has a nice infinite product representation.

Abstract

We continue the study of the $(a,b,m)$-copartition function $\mathrm{cp}_{a,b,m}(n)$, which arose as a combinatorial generalization of Andrews' partitions with even parts below odd parts. The generating function of $\mathrm{cp}_{a,b,m}(n)$ has a nice representation as an infinite product. In this paper, we focus on the parity of $\mathrm{cp}_{a,b,m}(n)$. As with the ordinary partition function, it is difficult to show positive density of either even or odd values of $\mathrm{cp}_{a,b,m}(n)$ for arbitrary $a, b$, and $m$. However, we find specific cases of $a,b,m$ such that $\mathrm{cp}_{a,b,m}(n)$ is even with density 1. Additionally, we show that the sequence $\{\mathrm{cp}_{a,m-a,m}(n)\}_{n=0}^\infty$ takes both even and odd values infinitely often.