Counting partitions inside a rectangle

TL;DR

This paper derives precise asymptotic formulas for the number of integer partitions fitting inside a rectangle, refining previous results and providing new insights into the behavior of these partition counts.

Contribution

The authors develop a novel large deviation approach to obtain sharp asymptotics for rectangle-constrained partitions across a broad parameter regime.

Findings

Sharp asymptotics for partition counts in the regime rf6m rf6m

First asymptotic estimates on consecutive differences of these numbers

Refinement of Sylvester's unimodality theorem

Abstract

We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout the regime $\ell = \Theta (m)$ and $n = \Theta (m^2)$. Previously, sharp asymptotics were derived by Tak\'acs only in the regime where $|n - \ell m /2| = O(\sqrt{\ell m (\ell + m)})$ using a local central limit theorem. Our approach is to solve a related large deviation problem: we describe the tilted measure that produces configurations whose bounding rectangle has the given aspect ratio and is filled to the given proportion. Our results are sufficiently sharp to yield the first asymptotic estimates on the consecutive differences of these numbers when $n$ is increased by one and $m, \ell$ remain the same, hence significantly refining Sylvester's unimodality theorem.