2023/09/13 by Kyle Celano, Celano, Kyle, Nicholas Sieger +3
Mathematics · #05C20 (Primary) 05A05 #05C31 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2309.07240
openalex publication_date 2023/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.