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.

中文翻译：

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[数学]-非交叉链的计数和停车函数

SIAM 离散数学杂志，第 38 卷，第 1 期，第 917-946 页，2024 年 3 月。
摘要。对于有限 Coxeter 群 [数学]，Josuat-Vergès 导出了一个 [数学] 多项式，通过对这些链中的一些覆盖关系（我们称为坏边）进行加权，对 [数学] 的非交叉分区格中的最大链进行计数带有参数[math]。我们从[数学]多项式的角度研究这些加权链与[数学]类型（分别为[数学]）的停车函数的联系。数学多项式是停车函数（任一类型）相对于未停放在其首选停车位的汽车数量的生成函数。无论哪种情况，我们都会给出一个双射结果，该结果会给不幸的汽车带来不好的边缘，同时保留它们的相对顺序。利用这一点，我们给出了当 [math] 是超八面体群时 [math]-多项式的 [math]-正性的解释。