Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts

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.