Constraint-sensitive Catalan path statistics having the Narayana distribution
type: Item
instance of: scholarly article
title: Constraint-sensitive Catalan path statistics having the Narayana distribution
zbMATH Open document ID: 0936.05004
DOI: 10.1016/S0012-365X(98)00382-3
author: Robert A. Sulanke
published in: Discrete Mathematics
publication date: 2000-05-18T00:00:00Z
review text: This paper investigates certain statistics on the set \(C(n)\) of Catalan paths, represented as length \(2n\) sequences of 0s and 1s, corresponding to vertical and horizontal steps respectively. The statistics of interest all have the Narayana distribution; that is, if \(\Theta\) is such a statistic, then \[ \#\{P\in C(n):\Theta(P)= k\}= {1\over n} {n\choose k}{n\choose k+1}. \] Moreover, the statistics studied measure certain path behaviors near the constraining line \(y= x\). The main result is a matrix encoding scheme which gives all (50) Narayana distributed statistics satisfying the author's ``constraint even-odd-even-odd'' condition. Each of the statistics is related (sometimes bijectively) to one of four ``parent'' statistics, each of which records a particular type of behavior near the constraint. The proof uses a straightforward computer program to produce candidates. Verification is accomplished using bijections and generating function techniques.
reviewed by: Jennifer R.Galovich
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 1331483
zbMATH Keywords: Catalan paths
zbMATH Keywords: Narayana distribution
zbMATH Keywords: matrix encoding scheme
zbMATH Keywords: generating function
MaRDI profile type: Publication
cites work: A new bijection between ordered trees and legal bracketings
cites work: Algebraic languages and polyominoes enumeration
cites work: A bijection on Dyck paths and its consequences
cites work: Q5252016
cites work: Counting pairs of nonintersecting lattice paths with respect to weighted turns
cites work: A new enumerative property of the Narayana numbers
cites work: Q5555362
cites work: A symmetric variation of a distribution of Kreweras and Poupard
cites work: Catalan path statistics having the Narayana distribution
cites work: Six etudes in generating functions
description: scientific article; zbMATH DE number 1331483