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Title: The number of parking functions with center of a given length
Author: Duarte, Rui
Guedes de Oliveira, António
Keywords: Parking functions
Shi arrangement
Ish arrangement
Issue Date: Jun-2019
Publisher: Elsevier
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.
Peer review: yes
DOI: 10.1016/j.aam.2019.02.004
ISSN: 0196-8858
Appears in Collections:CIDMA - Artigos
AGG - Artigos

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