The number of parking functions with center of a given length

TL;DR

This paper counts specific classes of parking functions related to rooted trees and introduces new combinatorial bijections and properties, providing exact enumeration formulas for parking functions with a given run or center.

Contribution

It establishes bijections between parking functions with center, run, and rook words, and derives enumeration formulas for parking functions with specified properties.

Findings

Number of parking functions with a given run equals the number of rook words with the same run.

Established bijections linking parking functions, rook words, and tree properties.

Derived explicit counts for parking functions with specified center and run sizes.

Abstract

Let $1\leq r\leq n$ and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labelled tree on $n+1$ vertices, exactly $r$ vertices are visited before backtracking. Let $R$ be the set of trees with this property. We count the number of elements of $R$. For this purpose, we first consider a bijection, due to Parkinson, Yang and Yu, that maps $R$ onto the set of parking function with center (defined by the authors in a previous article) of size $r$. A second bijection maps this set onto the set of parking functions with run $r$, a property that we introduce here. We then prove that the number of length $n$ parking functions with a given run is the number of length $n$ rook words (defined by Leven, Rhoades and Wilson) with the same run. This is done by counting related lattice paths in a ladder-shaped region. We finally count the number of length $n$ rook words with run $r$, which is the answer to our initial question.