Prime Parking Functions on Rooted Trees

TL;DR

This paper introduces prime parking functions on rooted trees, counts their total number as (2n-2)! for n-vertex trees, and generalizes increasing parking functions with a new enumeration involving Schr"oder numbers.

Contribution

It defines prime parking functions on rooted trees, derives their total count, and extends increasing parking functions to trees with a novel enumeration formula.

Findings

Total prime parking functions on n-vertex trees are (2n-2)!

Number of increasing prime parking functions is (n-1)! times the large Schr"oder number S_{n-1}

Provides combinatorial enumeration results for parking functions on rooted trees.

Abstract

For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the first unoccupied spot along the directed path to the root. If all $n$ drivers park, the sequence of spot preferences is called a parking function. We consider the sequences, called \emph{prime} parking functions, for which each driver parks and each edge in the tree is traversed by some driver after failing to park at her preferred spot. We prove that the total number of prime parking functions on trees with $n$ vertices is $(2n-2)!$. Additionally, we generalize \emph{increasing} parking functions, those in which the drivers park with a weakly-increasing order of preference, to trees and prove that the total number of increasing prime parking functions on trees with $n$ vertices is $(n-1)!S_{n-1}$, where $\{S_i\}_{i \geq 0}$ are the large Schr\"oder numbers.