The number of parking functions with center of a given length

Rui Duarte,António Guedes De Oliveira

doi:10.1016/j.aam.2019.02.004

Rui Duarte, António Guedes De Oliveira

Open Access

https://doi.org/10.1016/j.aam.2019.02.004

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Abstract

Let 1≤r≤n and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labeled 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 Perkinson, 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.

Highlights

Let Tn be the set of rooted labeled trees on the set of vertices {0, 1, . . . , n} with root r = 0, and let T ∈ Tn
We count the number of length n rook words with run r, which is the answer to our initial question
The bijection is defined from chambers to parking functions

Summary

Introduction

The bijection is defined from chambers (represented by permutations of [n] decorated with arcs following certain rules) to parking functions. For every n ∈ N, RPF n (t) = RRW n (t) In this case we do not consider all parking functions and all rook words at once. ALT 3 (t) = ZPF 3 (t) = RPF 3 (t) = RRW 3 (t) = 4t + 6t2 + 6t3 by classifying the corresponding trees and parking functions according to the various statistics and the corresponding bijections. Let T be the tree given by Algorithm 1 with input P = a − 1 (we know this defines a bijection from PFn to Tn by [11, Theorem 3]). Note that a belongs to PF9 , as well as Φ(a) and Ψ(a)

From parking functions to rook words

Counting rook words with a given run