On Andrews' partitions with parts separated by parity

TL;DR

This paper generalizes a theorem on partitions with parts separated by parity, providing a bijective proof and exploring related identities to deepen understanding of parity-based partition functions.

Contribution

It introduces a new generalization of Andrews' theorem on parity-separated partitions and offers a bijective proof along with new identities for related partition functions.

Findings

Generalized Andrews' theorem on parity-separated partitions

Provided a bijective proof for the new theorem

Discovered new identities related to partition functions

Abstract

In this paper, we present a generalization of one of the theorems in [G. E. Andrews, Partitions with parts separated by parity, \textit{Annals of Combinatorics} \textbf{23}(2019), 241 - 248], and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting identities.