Overcolored Partition $k$-tuples Restricted by Parity of the Parts

In this paper, we study the combinatorial object ${\bar{b}}_{r,s}^k(n)$ which counts the overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. We extend results of Chacon and Sellers for several families of $r,s$ and $k$. We also establish divisibility properties for ${\bar{b}}_{r,s}^k(n)$ modulo prime $p$ and conclude with new congruences modulo powers of $2$. The techniques involved to obtain our results rely on theta function identities and modular forms.