# Ranks of overpartitions: Asymptotics and inequalities

## TL;DR

This paper derives asymptotic formulas for overpartition rank generating functions, proves their equidistribution modulo c, and confirms certain inequalities between ranks for specific n values, advancing understanding of overpartition statistics.

## Contribution

It provides the first asymptotic analysis of overpartition rank generating functions and proves conjectured inequalities, offering new insights into overpartition distribution and rank relations.

## Key findings

- Overpartition ranks are asymptotically equidistributed modulo c.
- Confirmed inequalities between overpartition ranks for n=6 and n=10.
- Derived asymptotic formulas for overpartition rank generating functions.

## Abstract

In this paper we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions. Using these results, we show that $ \overline{N}(a,c,n), $ the number of overpartitions of $ n $ with rank congruent to $ a $ modulo $ c, $ is equidistributed with respect to $ 0\le a< c, $ for any $ c\ge2, $ as $ n\to\infty $ and, in addition, we prove some inequalities between ranks of overpartitions conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018) for $ n=6 $ and $ n=10. $