Flag statistics from the Ehrhart series of multi-hypersimplices

TL;DR

This paper generalizes Stanley's conjecture on hypersimplices Ehrhart series to colored permutations using flag statistics, providing new proofs and combinatorial identities involving Eulerian statistics.

Contribution

It introduces a new generalization of Stanley's conjecture to colored permutations via flag statistics, offering alternative proofs and identities.

Findings

New proof of Stanley's conjecture on hypersimplices

Generalization to colored permutations using flag statistics

Combinatorial identities relating Eulerian statistics

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.