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arXiv · 2201.04247

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

Hannah E. Burson·Dennis Eichhorn
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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.

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Hannah E. Burson, Dennis Eichhorn. 2022-01-12. On the parity of the number of $(a,b,m)$-copartitions of $n$. https://arxiv.org/abs/2201.04247

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