An inequality between finite analogues of rank and crank moments

TL;DR

This paper proves a conjectured inequality between finite analogues of rank and crank moments, extending previous work on partition statistics and their finite versions.

Contribution

It provides a proof for the conjectured inequality between finite rank and crank moments, advancing the understanding of partition statistics.

Findings

Proved the conjecture relating finite rank and crank moments.

Extended classical inequalities to finite analogues.

Contributed to the theory of vector partitions and partition identities.

Abstract

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews' famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan's conjecture. In the present paper, we give a proof of this conjecture.