Asymptotic inequalities for positive crank and rank moments

TL;DR

This paper establishes the asymptotic behavior of modified crank and rank moments for integer partitions, showing they are asymptotically equal but crank moments are always larger, and also analyzes the ospt-function's asymptotics and parity.

Contribution

It proves the asymptotic equivalence of crank and rank moments and determines the asymptotic and parity properties of the ospt-function, extending prior results.

Findings

Crank moments are asymptotically larger than rank moments.

The asymptotic behavior of the ospt-function is characterized.

Parity of the ospt-function is related to the spt-function.

Abstract

Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases, and our main result states that while the two families of moment functions are asymptotically equal, the crank moments are always asymptotically larger than the rank moments. Andrews, Chan, and Kim primarily focused on one case, and proved the stronger result that the first crank moment is strictly larger than the first rank moment for all partitions by showing that the difference is equal to a combinatorial statistic on partitions that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function, and we further determine its behavior modulo 2 by relating its parity to Andrews spt-function.