Parking functions of length $n$ are well known to be in correspondence with both labelled trees on $n+1$ vertices and factorizations of the full cycle ${\sigma }_n=(01\dots n)$ into $n$ transpositions. In fact, these correspondences can be refined: Kreweras equated the area enumerator of parking functions with the inversion enumerator of labelled trees, while an elegant bijection of Stanley maps the area of parking functions to a natural statistic on factorizations of ${\sigma }_n$. We extend these relationships in two principal ways. First, we introduce a bivariate refinement of the inversion enumerator of trees and show that it matches a similarly refined enumerator for factorizations. Secondly, we characterize all full cycles $\sigma$ such that Stanley’s function remains a bijection when the canonical cycle ${\sigma }_n$ is replaced by $\sigma$. We also exhibit a connection between our refined inversion enumerator and Haglund’s bounce statistic on parking functions.

停车长度 $ñ$ 众所周知，它们与 $ñ+\mathrm{1个}$ 整个周期的顶点和因式分解 ${\sigma }_{ñ}=（0\mathrm{1个}\dots ñ）$ 进入 $ñ$换位。实际上，可以对这些对应关系进行细化：Kreweras将停车功能的区域枚举数与带标签的树木的倒置枚举数等价，而Stanley的优雅双射则将停车功能的面积映射为自然分解的${\sigma }_{ñ}$。我们以两种主要方式扩展这些关系。首先，我们对树的反演枚举器进行了双变量细化，并表明它与用于分解的相似细化的枚举器匹配。其次，我们表征所有完整周期$\sigma$ 这样当规范循环时斯坦利函数仍然是双射 ${\sigma }_{ñ}$ 被替换为 $\sigma$。我们还展示了改进的倒数枚举器与Haglund的停车功能反弹统计之间的联系。