A Combinatorial Proof of an Identity Involving Eulerian Numbers

Abstract

We give a combinatorial proof of an identity that involves Eulerian numbers and was obtained algebraically by Brenti and Welker (2009). To do so, we use alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the dual graph of the triangulation in the case of the dilated standard simplex and the hypersimplex, conjecture its structure for dilated hypersimplices. Also, we give a new proof of the classical result that the Eulerian numbers coincide with the normalized volumes of the hypersimplices.