Generalizations of Dyson's Rank on Overpartitions

TL;DR

This paper introduces a new statistic called the $ar{k}$-rank for overpartitions, generalizing existing ranks, and explores its generating function and connections to mock theta functions, expanding the combinatorial understanding of overpartition ranks.

Contribution

It defines the $ar{k}$-rank for overpartitions, generalizes known partition ranks, and links the generating functions to tenth order mock theta functions.

Findings

The $ar{k}$-rank coincides with the Garvan $k$-rank when no overlined parts are present.

The $ar{k}$-rank reduces to the D-rank for overpartitions when $k=2$.

The generating function for the $ar{k}$-rank is explicitly given and related to mock theta functions.

Abstract

We introduce a statistic on overpartitions called the $\overline{k}$-rank. When there are no overlined parts, this coincides with the $k$-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when $k=2$. The generating function for the $\overline{k}$-rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth order mock theta functions $X(q)$ and $\chi(q)$.