# Optimal transport and integer partitions

## TL;DR

This paper explores the connection between optimal transport theory and integer partitions, providing new characterizations of certain partition classes and suggesting potential for understanding higher-dimensional partitions.

## Contribution

It introduces a novel link between optimal transport and integer partitions, offering new characterizations and insights into partition classes.

## Key findings

- Characterization of symmetric partitions using optimal transport
- Reformulation of Euler's identity in transport terms
- Potential applications to higher-dimensional partitions

## Abstract

We link the theory of optimal transportation to the theory of integer partitions. Let $\mathscr P(n)$ denote the set of integer partitions of $n \in \mathbb N$ and write partitions $\pi \in \mathscr P(n)$ as $(n_1, \dots, n_{k(\pi)})$. Using terminology from optimal transport, we characterize certain classes of partitions like symmetric partitions and those in Euler's identity $|\{ \pi \in \mathscr P(n) |$ all $ n_i $ distinct $ \} | = | \{ \pi \in \mathscr P(n) | $ all $ n_i $ odd $ \}|$. Then we sketch how optimal transport might help to understand higher dimensional partitions.