Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

TL;DR

This paper provides combinatorial interpretations of residual cranks for overpartitions and partitions without repeated odd parts, connecting these to mock theta functions and simplifying their definitions.

Contribution

It introduces new combinatorial interpretations of residual cranks, removing the need for adjusted weights, and explores their connections with mock theta functions.

Findings

New combinatorial interpretations of residual cranks for overpartitions.

Connections established between crank statistics and tenth order mock theta functions.

Simplified definitions of crank statistics without adjusted weights.

Abstract

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.