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.
Highlights
A modern combinatorial definition of the Eulerian numbers An,k is given by counting descents in permutations: An,k := #{σ ∈ Sn : des(σ) = k − 1}. (1)the electronic journal of combinatorics 23(1) (2016), #P1.55Foata suggested in [6] the problem that we describe below
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 gave a generalization to colored permutations
We give another generalization to colored permutations, using the flag statistics introduced by Foata and Han
Summary
The set {v ∈ [0, 1]n : k vi k + 1} is a convex integral polytope known as the hypersimplex, and in this context we can consider the h∗-polynomial, which is a generalization of the volume. This led to a recent conjecture by Stanley about the h∗-polynomial of the hypersimplex (more precisely, a partially open version of the hypersimplex), which was proved by Nan Li [10] in two different ways. Our result is stated in terms of the flag descents and flag excedences in colored permutations, and relies on some related work by Foata and Han [5].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
