Parity considerations in the number of parts

TL;DR

This paper explores parity properties of parts in overpartitions and partitions, extending previous work by analyzing the parity of parts relative to the largest overlined or non-overlined parts and the smallest odd part.

Contribution

It introduces new parity considerations for parts in overpartitions and partitions, expanding the understanding of their combinatorial properties.

Findings

Established parity relations for parts less than or equal to the largest overlined or non-overlined parts.

Analyzed the parity of even parts greater than the smallest odd part in partitions.

Abstract

Recently, Chen, He, Hu and Xie considered the parity of the number of non-overlined (resp. overlined) parts of size greater than or equal to the size of the smallest overlined (resp. non-overlined) part in an overpartition. In this article, we investigate the parity of the number of non-overlined (resp. overlined) parts of size less than or equal to the size of the largest overlined (resp. non-overlined) part in an overpartition. We also study the parity of the number of even parts greater than the smallest odd part in a partition.