Asymptotic inequalities for positive crank and rank moments

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. Our main result states that the two families of moment functions are asymptotically equal, but the crank moments are also asymptotically larger than the rank moments. Andrews, Chan, and Kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function (and its analogs for higher moments), and we further determine the behavior of the ospt-function modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by relating its parity to Andrews’ spt-function.