# Limit Shape of Subpartition Maximizing Partitions

## TL;DR

This paper proves a limit shape theorem for partitions that maximize subpartitions, providing explicit formulas and connecting to Vershik's limit shape for uniform partitions.

## Contribution

It introduces a limit shape theorem for maximizing subpartition partitions, combining large deviations, convex analysis, and Hardy-Ramanujan asymptotics.

## Key findings

- Explicit limit shape for maximizing subpartition partitions.
- Growth rate of the number of such subpartitions determined.
- Limit shape matches Vershik's for uniform partitions.

## Abstract

This is an expository note answering a question posed to us by Richard Stanley, in which we prove a limit shape theorem for partitions of $n$ which maximize the number of subpartitions. The limit shape and the growth rate of the number of subpartitions are explicit. The key ideas are to use large deviations estimates for random walks, together with convex analysis and the Hardy-Ramanujan asymptotics. Our limit shape coincides with Vershik's limit shape for uniform random partitions.