Some statistics on Dyck paths
type: Item
instance of: scholarly article
title: Some statistics on Dyck paths
zbMATH Open document ID: 0998.05004
DOI: 10.1016/S0378-3758(01)00180-X
author: V. Pereyra
published in: Journal of Statistical Planning and Inference
publication date: 2002-05-15T00:00:00Z
review text: Dyck paths are underdiagonal paths in the \(Z^2\) lattice, starting at the origin, never going above the main diagonal, and making east \(=(1,0)\) and north \(=(0,1)\) steps. A basic counting tool for the characteristics of Dyck paths are Catalan numbers. It turns out that a Dyck path corresponds to a word in a language generated by a specified grammar and this allows one to apply classical methods to find proper generating functions. The authors combine this approach with the Lagrange inversion formula to obtain generating function identities and further to establish unimodality properties and asymptotics of certain characteristics of Dyck paths. The main role in their study plays the so-called trinomial statistic, counting Dyck words with semilength \(n\) (i.e. Dyck paths arriving at point \((n,n\))) having \(k\) occurrences of the string 010.
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A16
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 1741602
zbMATH Keywords: Dyck paths
zbMATH Keywords: Catalan numbers
zbMATH Keywords: generating functions
zbMATH Keywords: Lagrange inversion formula
zbMATH Keywords: trinomial statistic
zbMATH Keywords: counting
reviewed by: Ljuben Mutafchiev
MaRDI profile type: Publication
cites work: Q4013349
cites work: Motzkin numbers
cites work: Q4092015
cites work: Q3669422
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full work available at URL: s0378-3758(01)00180-x
OpenAlex ID: W2015617320
Wikidata QID: Q127595067
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description: scientific article; zbMATH DE number 1741602