Séminaire Lotharingien de Combinatoire, 91B.40 (2024), 12 pp.
Kyle Celano, Nicholas Sieger and Sam Spiro
Eulerian Polynomials for Digraphs
Abstract.
Given an n-vertex digraph D and a labeling
σ : V(D) ->
[n], we say that an arc u -> v of D is
a descent of σ
if σ(u) > σ(v). Foata and Zeilberger introduced a
generating function AD(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 AD(t). In
particular, we give a combinatorial interpretation of |AD(-1)|
in terms of "generalized alternating permutations" whenever the
underlying graph of D is bipartite.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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