Optimal transport for some symmetric, multidimensional integer partitions (Q6140238)

A partition of an integer number \(n \in \mathbb{N}\) is an ordered finite collection of integer numbers \((n_1, ..., n_k)\), with \(n_1 \geq n_2 \geq ... \geq n_k \geq 1\) and \(\sum_{i=1}^k = n\). In the paper [Discrete Appl. Math. 190--191, 75--85 (2015; Zbl 1326.49076)], \textit{S. Hohloch} has found a relation between the problem of finding integer partitions of a given number and the Monge-Kantorovich problem for the Euclidean distance; for a given partition of \(n\), one can construct a so-called Young diagram and associate to it a probability measure on the Young diagram. Then, some symmetry properties of the partition correspond to whether the identity map is optimal in the Monge problem. The present paper provides a generalisation of the notion of integer partitions to the multidimensional case and provides a similar result concerning the relation with the optimal transport problem.