Rank and crank moments for overpartitions

Abstract

We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n).

14 pages

References in corpus (1)

• Partition statistics and quasiweak Maass forms

Cited by in corpus (19)

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• Variations of Andrews-Beck type congruences
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• Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E
• Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols
• The first positive rank and crank moments for overpartitions
• Generating Functions and Congruences for Some Partition Functions Related to Mock Theta Functions
• Higher order spt functions for overpartitions, overpartitions with smallest part even, and partitions without repeated odd parts
• The Unimodality of the Crank on Overpartitions
• Efficient computation of the overpartition function and applications
• Euler-like recurrences for smallest parts functions
• Universal mock theta functions and two-variable Hecke-Rogers identities
• $\ell$-adic properties of smallest parts functions
• On the number of even and odd strings along the over partitions of n
• Another proof of two modulo 3 congruences and another SPT crank for the number of smallest parts in overpartitions with even smallest part
• Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts
• A positivity conjecture related first positive rank and crank moments for overpartitions
• Monotonicity properties for ranks of overpartitions
• Rank-Crank type PDEs and generalized Lambert series identities