On [math]-Counting of Noncrossing Chains and Parking Functions

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 917-946, March 2024.
Abstract. For a finite Coxeter group [math], Josuat-Vergès derived a [math]-polynomial counting the maximal chains in the lattice of noncrossing partitions of [math] by weighting some of the covering relations, which we call bad edges, in these chains with a parameter [math]. We study the connection of these weighted chains with parking functions of type [math] ([math], respectively) from the perspective of the [math]-polynomial. The [math]-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 in 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 [math]-positivity of the [math]-polynomial in the case when [math] is the hyperoctahedral group.