2023/12/12 by Yen-Jen Cheng, Sen‐Peng Eu, Cheng, Yen-Jen +5
Mathematics · #05A19 #05E16 #20F55 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2312.07351
openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a finite Coxeter group W, Josuat-Vergès derived a q-polynomial counting the maximal chains in the lattice of noncrossing partitions of W by weighting some of the covering relations, which we call bad edges, in these chains with a parameter q. We study the connection of these weighted chains with parking functions of type A (B, respectively) from the perspective of the q-polynomial. The q-polynomial turns out to be the generating function for parking functions (of either type) with respect to the number of cars that do not park at their preferred spaces. In either case, we present a bijective result that carries bad edges to unlucky cars while preserving their relative order. Using this, we give an interpretation of the γ-positivity of the q-polynomial in the case that W is the hyperoctahedral group.