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arXiv · 2309.07240

Eulerian Polynomials for Digraphs

Kyle Celano·Nicholas Sieger·Sam Spiro
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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.

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Kyle Celano, Nicholas Sieger, Sam Spiro. 2023-09-13. Eulerian Polynomials for Digraphs. https://arxiv.org/abs/2309.07240

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