Restricted Overpartitions and concave compositions: their modularity and asymptotics

TL;DR

This paper investigates restricted overpartitions and concave compositions, revealing their complex modular structures and deriving their asymptotic behaviors, along with related rank statistics.

Contribution

It uncovers the mixed modular nature of generating functions in restricted partition problems and provides asymptotic main terms.

Findings

Generating functions involve modular forms, mock theta, and false theta functions.

Asymptotic main terms for the studied functions are derived.

Related rank statistics are analyzed.

Abstract

In this paper we study restricted overpartitions and concave compositions. In several cases the resulting generating functions involve simultaneously modular forms, mock theta functions, mock Maass theta functions, and false theta functions, illustrating the appearance of mixed modular structures in restricted partition problems. Moreover, we obtain their asymptotic main terms. We also study related rank statistics.