Prime Parking Functions on Rooted Trees

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.