Inequalities between ranks and cranks

Bringmann, Kathrin; Mahlburg, Karl

doi:10.1090/S0002-9939-09-09806-2

Inequalities between ranks and cranks
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by Kathrin Bringmann and Karl Mahlburg

Proc. Amer. Math. Soc. 137 (2009), 2567-2574

DOI: https://doi.org/10.1090/S0002-9939-09-09806-2

Published electronically: February 20, 2009

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Abstract:

Higher moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves the first several cases of (and strengthens) a conjecture due to Garvan, which states that the moments of the crank function are always larger than the moments of the rank function. Furthermore, asymptotic estimates for these differences are also proven.

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