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

Abstract

In recent work, authors defined ${\bar{b}}_{r,s}^k(n)$ to be the number of overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. This function generalizes the overcubic partition $k$-tuples studied by Chacon and Sellers. The present work gives a uniform framework for studying congruences for ${\bar{b}}_{r,s}^k(n)$ modulo odd primes. For example, we prove that, for $n\ge 1$, \begin{align*} \bar{b}_{4,2}^4(5n)\equiv 0\pmod{5}. \end{align*}