Dyson's Rank, overpartitions, and weak Maass forms

TL;DR

This paper constructs a new family of weak Maass forms from overpartitions, linking partition statistics to modular objects and providing combinatorial and modularity results related to Ramanujan-type congruences.

Contribution

It introduces a novel infinite family of Maass forms derived from overpartitions, expanding the class of explicit examples and applications.

Findings

New Maass forms from overpartitions

Decomposition of Ramanujan-type congruences

Modularity of rank differences

Abstract

In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decompositions of Ramanujan-type congruences for overpartitions as well as the modularity of rank differences in certain arithmetic progressions.