Overcolored Partition Restricted by Parity of the Parts

TL;DR

This paper extends the concept of multicolored partitions to overpartitions, exploring the enumeration of such partitions with parity-based coloring constraints, building on recent definitions in partition theory.

Contribution

It introduces the extension of the function $a_{r,s}(n)$ to overpartitions, providing new combinatorial insights into parity-restricted multicolored overpartition enumeration.

Findings

Defined the overpartition analogue of $a_{r,s}(n)$

Derived formulas for counting overpartitions with parity-based coloring

Extended recent partition enumeration results to overpartitions

Abstract

Very recently, Thejitha, Sellers, and Fathima defined the function $a_{r,s}(n)$, which enumerates the number of multicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. In this paper, we extend the concept to overpartitions.