A combinatorial proof of an identity involving Eulerian numbers

TL;DR

This work provides a concrete combinatorial proof of a Brenti–Welker identity connecting C(r,d,i) counts and Eulerian numbers by interpreting both sides as counts of alcoves in alcoved triangulations of dilated hypersimplices. The authors construct two explicit bijections: (i) label alcoves of the dilated standard simplex using words and permutations, and (ii) label alcoves via pairs consisting of a composition and a permutation; these bijections extend to general hypersimplices through corresponding pairings of compositions with permutations. As a corollary, Eulerian numbers appear as normalized volumes of hypersimplices, while the dual graphs of the alcoved triangulations are described combinatorially and related to explicit permutation graphs; a natural graph-composition construction is proposed for assembling dual graphs of dilated hypersimplices. The results yield a transparent, constructive explanation of the identities, connect Ehrhart theory with Eulerian combinatorics, and suggest a structural conjecture for the dual graphs of general hypersimplices with potential implications for understanding Veronese-like transforms combinatorially.

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.

Figures (5)

• Figure 1: On the left, an alcoved polytope with its explicit $(H,z)$-representation; the dotted lines represent the elements of the affine Coxeter arrangement of type $A_2$. On the right, the image of the polytope under the map $\psi_2$ from \ref{['rem:equivalent_defs']} after translation by the vector $(1,1,1)$.
• Figure 2: Illustration of $G_{3,3}$, which is isomorphic to the dual graph of the alcoved triangulation of $3\Delta_{1,3}$. The words in different color correspond to alcoves that intersect the hyperplane $x_1 = 0$ in codimension $1$. Similarly, the words in boxes correspond to alcoves intersecting $x_2=0$ in codimension $1$.
• Figure 3: The dual graph of the alcoved triangulation of $\Delta_{2,4}$ in terms of sorted sets on the left; the solid and dotted edges correspond, respectively, to conditions $1.$ and $2.$ from \ref{['prop:dual_graph_hypersimplex']}. On the right, the same graph but in terms of permutations in $\mathfrak{A}(4,2)$; the colors of the edges represent the different hyperplanes that separate the corresponding alcoves.
• Figure 4: The construction of $G_{\mathcal{A},2,3}\langle G_{2,3} \rangle$ when choosing $W_j^\circ$ as connecting sets according to the specified labels. This graph is isomorphic to the dual graph of the alcoved triangulation of $2\Delta_{2,3}$.
• Figure :

Theorems & Definitions (61)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Remark 1.4
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6