# Asymptotic equidistribution and convexity for partition ranks

## TL;DR

This paper investigates the distribution of partition ranks, proving their asymptotic equidistribution and monotonicity, and confirms a conjecture on their convexity, advancing understanding of partition rank functions.

## Contribution

It establishes the asymptotic equidistribution and monotonicity of the Dyson rank function and proves a conjecture on its convexity, providing new insights into partition ranks.

## Key findings

- Dyson rank function is monotonic in n.
- Dyson rank function becomes equidistributed as n approaches infinity.
- Convexity conjecture of N(r,t;n) is proven.

## Abstract

We study the Dyson rank function $N(r,t;n)$, the number of partitions with rank congruent to $r$ modulo $t$. We first show that it is monotonic in $n$, and then show that it equidistributed as $n \rightarrow \infty$. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of $N(r,t;n)$.