Flag Statistics from the Ehrhart $h^∗$-Polynomial of Multi-Hypersimplices

Guo-Niu Han, Matthieu Josuat-Vergès


It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley’s conjecture, and some combinatorial identities relating pairs of Eulerian statistics on colored permutations.


Ehrhart polynomial, Descents, Eulerian numbers, Colored permutations, Hypercube

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