On a generalized crank for $k$-colored partitions

TL;DR

This paper introduces a generalized crank for $k$-colored partitions, deriving inequalities and positivity results for its moments, thereby extending the understanding of partition statistics and their properties.

Contribution

It defines a new $k$-crank for $k$-colored partitions and establishes inequalities and positivity results for its moments, advancing partition theory research.

Findings

Inequalities between $k$-crank counts for $m=2,3,4$

Positivity of symmetrized even $k$-crank moments for $k=2,3$

Remarks on future research directions

Abstract

A generalized crank ($k$-crank) for $k$-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$-crank. Namely, we first obtain some inequalities between the $k$-crank counts $M_{k}(r,m,n)$ for $m=2,3$ and $4$, then we prove the positivity of symmetrized even $k$-crank moments weighted by the parity for $k=2$ and $3$. We conclude with several remarks on furthering the study initiated here.