Eulerian Polynomials for Digraphs

TL;DR

This paper introduces a new polynomial for digraphs that generalizes Eulerian and Mahonian polynomials, providing combinatorial interpretations at specific evaluations, especially for bipartite graphs.

Contribution

It defines a generating function for digraph labelings weighted by descents and interprets its value at -1 through generalized permutations for bipartite graphs.

Findings

Provides a combinatorial interpretation of |A_D(-1)| for bipartite graphs.

Generalizes classical polynomials to digraphs with new combinatorial insights.

Connects polynomial evaluations to generalized permutation structures.

Abstract

Given an $n$-vertex digraph $D$ and a labeling $\sigma:V(D)\to [n]$, we say that an arc $u\to v$ of $D$ is a descent of $\sigma$ if $\sigma(u)>\sigma(v)$. Foata and Zeilberger introduced a generating function $A_D(t)$ for labelings of $D$ weighted by descents, which simultaneously generalizes both Eulerian polynomials and Mahonian polynomials. Motivated by work of Kalai, we look at problems related to $-1$ evaluations of $A_D(t)$. In particular, we give a combinatorial interpretation of $|A_D(-1)|$ in terms of "generalized alternating permutations" whenever the underlying graph of $D$ is bipartite.