Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts

TL;DR

This paper investigates the arithmetic properties of multicolored partitions with parity restrictions on parts, using generating functions and q-series techniques to uncover new mathematical insights.

Contribution

It introduces a detailed study of colored partitions with parity restrictions, applying elementary and classical q-series methods to analyze their arithmetic properties.

Findings

Derived new identities for multicolored partitions

Established congruence relations for partition counts

Connected partition properties to classical q-series results

Abstract

Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties satisfied by $a_{r,s}(n)$, using elementary generating function manipulations and classical $q$-series techniques.