# Mock modularity of the $M_d$-rank of overpartitions

## TL;DR

This paper explores the modular properties of a new family of partition ranks called the $M_d$-rank of overpartitions, revealing their connection to harmonic Maass forms and providing transformation formulas and identities.

## Contribution

It introduces the $M_d$-rank as a unifying framework for overpartition ranks and characterizes its modular behavior through harmonic Maass forms.

## Key findings

- The $M_d$-rank is the holomorphic part of a harmonic Maass form.
- Exact transformation formulas for the harmonic Maass form are provided.
- Identities relating the $M_d$-rank to known partition statistics are established.

## Abstract

We investigate the modular properties of a new partition rank, the $M_d$-rank of overpartitions. In fact this is an infinite family of ranks, indexed by the positive integer $d$, that gives both the Dyson rank of overpartitions and the overpartition $M_2$-rank as special cases. The $M_d$-rank of overpartitions is the holomorphic part of a certain harmonic Maass form of weight $\frac{1}{2}$. We give the exact transformation of this harmonic Maass form along with a few identities for the $M_d$-rank.