Parking function varieties for combinatorial tree models

TL;DR

This paper introduces new combinatorial models called tree parking functions for various types of rooted labeled trees, providing enumeration formulas and asymptotic analysis for different tree families and driver scenarios.

Contribution

It develops a unified combinatorial framework for enumerating tree parking functions across multiple tree families and driver configurations, extending to asymptotic behavior analysis.

Findings

Derived explicit generating functions for tree parking functions.

Provided asymptotic formulas for large tree sizes and driver load factors.

Extended methods to general tree parking scenarios with varying numbers of drivers.

Abstract

We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size $n$ and $n$ successful drivers, from which we obtain exact and asymptotic enumeration results. The approach can be extended to the general situation of tree parking functions for trees of size $n$ and $m<n$ drivers for which we are also able to characterize the generating functions solutions, which allow, by applying analytic combinatorics techniques, a study of the asymptotic behaviour of the total number of tree parking functions and distributions for $n \to \infty$ depending on the load factor $0 < \alpha = \frac{m}{n} < 1$.