Flag statistics from the Ehrhart h* -polynomial of multi-hypersimplices

Guo-Niu Han (1, 2, 3) , Matthieu Josuat-Vergès (4, 5, 3)

1 IRMA - Institut de Recherche Mathématique Avancée
2 UNISTRA - Université de Strasbourg
3 CNRS - Centre National de la Recherche Scientifique
4 LIGM - Laboratoire d'Informatique Gaspard-Monge
5 UPEM - Université Paris-Est Marne-la-Vallée

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.

Domaines: Combinatoire [math.CO]

Guo-Niu Han, Matthieu Josuat-Vergès. Flag statistics from the Ehrhart h* -polynomial of multi-hypersimplices. The Electronic Journal of Combinatorics, 2016, 23 (1), Research Paper #P1.55. ⟨10.37236/5538⟩. ⟨hal-01396520⟩