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

TL;DR

This paper introduces a new crank generating function for (k+j)-colored partitions with j colors having distinct parts, extending previous work on partition crank functions and proposing a conjecture for a general case.

Contribution

It defines a new crank generating function for (k+j)-colored partitions with j distinct parts colors and proposes a conjecture for the general form.

Findings

Three infinite families of crank generating functions are provided.

The paper conjectures a general crank generating function for these partitions.

Extends previous crank concepts to more complex colored partition structures.

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.