# Symmetry in Tree Parking Distributions

## TL;DR

This paper investigates symmetry properties in parking distributions on caterpillar trees, introducing a $q, t$-analog of Fuss-Catalan generating functions and extending results to multiple statistics.

## Contribution

It reveals a symmetry in the joint distribution of parking statistics and develops a new $q, t$-analog of Fuss-Catalan generating functions with broader applicability.

## Key findings

- Symmetry in joint distribution of parking statistics on caterpillar trees
- Development of a $q, t$-analog of Fuss-Catalan generating function
- Extension to multiple statistics satisfying certain criteria

## Abstract

In this paper, we explore parking distributions on caterpillar trees, focusing on two primary statistics: the number of lucky cars and the frequency with which cars prefer specific parking spaces. We use first-return decomposition to reveal a symmetry in their joint distribution and develop a $q, t$-analog of the Fuss-Catalan generating function. We prove that this generating function exhibits specific symmetry and satisfies a functional equation. Additionally, we extend our findings to any m statistics that satisfy certain criteria, presenting a concrete example of such $m$ statistics to illustrate the broader applicability of our results.