Optimal transport and integer partitions

Abstract

We link the theory of optimal transportation to the theory of integer partitions. Let $\mathcal{P}(n)$ denote the set of integer partitions of $n\in \mathbb{N}$ and write partitions $\pi \in \mathcal{P}(n)$ 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 $|\{\pi \in \mathcal{P}(n)|$ all $n_i$ distinct $\}|=|\{\pi \in \mathcal{P}(n)|$ all $n_i$ odd $\}|$. Then we sketch how optimal transport might help to understand higher dimensional partitions.

21 pages, 7 figures; In accordance with the journal's copyright, I am making a preprint version of my published paper available on the ArXiv

References in corpus (1)

• Optimal transport maps in Monge-Kantorovich problem