A combinatorial proof of an identity involving Eulerian numbers

TL;DR

This paper provides a combinatorial proof of an Eulerian number identity using alcoved triangulations of dilated hypersimplices, linking geometric structures with algebraic identities and proposing conjectures on their dual graphs.

Contribution

It introduces a novel combinatorial proof of an algebraic identity involving Eulerian numbers and explores the geometric structure of hypersimplices.

Findings

Eulerian numbers equal normalized volumes of hypersimplices

Dual graph of triangulation described for the standard simplex

Conjecture on dual graph structure for general hypersimplices

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 study alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the dual graph of the triangulation in the case of the standard simplex, conjecture its structure for general hypersimplices, and prove combinatorially that the Eulerian numbers coincide with the normalized volumes of the hypersimplices.