A proposed crank for $(k+j)$-colored partitions, with $j$ colors having distinct parts

Abstract

In 1988, George Andrews and Frank Garvan discovered a crank for $p(n)$. In 2020, Larry Rolen, Zack Tripp, and Ian Wagner generalized the crank for p(n) in order to accommodate Ramanujan-like congruences for $k$-colored partitions. In this paper, we utilize the techniques used by Rolen, Tripp, and Wagner for crank generating functions in order to define a crank generating function for $(k + j)$-colored partitions where $j$ colors have distinct parts. We provide three infinite families of crank generating functions and conjecture a general crank generating function for such partitions.