---
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.'
address:
- |
    Institut de Recherche Mathématique Avancée\
    Université de Strasbourg and CNRS\
    7 rue René Descartes\
    67084 Strasbourg\
    France
- |
    Institut Gaspard Monge\
    Université Paris-Est Marne-la-Vallée and CNRS\
    5 Boulevard Descartes, 77454 Marne-la-Vallée CEDEX\
    France
author:
- 'Guo-Niu Han'
- 'Matthieu Josuat-Vergès'
title: 'Flag statistics from the Ehrhart series of multi-hypersimplices'
---

Introduction
============

A modern combinatorial definition of the Eulerian numbers $A_{n,k}$ is given by counting descents in permutations: $$\label{defank}
  A_{n,k} := \# \{  \sigma \in \mathfrak{S}_n \, : \, {\rm des}(\sigma)=k-1  \}.$$ Foata suggested in [@foata] the problem that we describe below. It is known that the 