Flag statistics from the Ehrhart series of multi-hypersimplices

Abstract

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.

18 pages

References in corpus (3)

• On the excedance number of colored permutation groups
• Roots of the Ehrhart polynomial of hypersimplices
• The face vector of a half-open hypersimplex