Optimal transport and integer partitions

We link the theory of optimal transportation to the theory of integer partitions. Let $\mathcal{P}(n)$ denote the set of integer partitions of $n \in \mathbb N$andwritepartitions$\pi \in \mathscr P(n)$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 $\mathrm{\mid }\{\pi \in \mathcal{P}(n)\mathrm{\mid }$ 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.