# Trees, Parking Functions and Factorizations of Full Cycles

## TL;DR

This paper explores refined relationships between parking functions, labeled trees, and cycle factorizations, extending known correspondences and introducing new enumerative and bijective insights.

## Contribution

It introduces a bivariate refinement of the inversion enumerator and characterizes cycles where Stanley's bijection applies beyond the canonical cycle.

## Key findings

- Refined inversion enumerator matches a similarly refined factorization enumerator.
- Characterization of cycles preserving Stanley's bijection.
- Connection established between refined enumerator and Haglund's bounce statistic.

## Abstract

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 $\sigma_n=(0\,1\,\cdots\,n)$ 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.