A uniform proof approach for congruences modulo 3 for partitions with $k$-colored odd parts

Abstract

In recent work, Hirschhorn and the second author defined $a_k(n)$ to be the number of partitions of $n$ wherein the even parts come in only one color, while the odd parts may be ``colored'' with one of $k$ colors for fixed $k\geq 1$. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying $a_k(n)$ has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.