Optimal transport and integer partitions

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.