The Narayana distribution

Robert A. Sulanke

doi:10.1016/S0378-3758(01)00192-6

The Narayana distribution

Robert A. Sulanke

Mathematics Department

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

Abstract

The Narayana distribution is N_n,k = 1/n(_k-1ⁿ)(_kⁿ) for 1 ≤ k ≤ n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the nth Narayana polynomial as N_n(w) = ∑_1≤k≤n N_n,kW^k, for n ≥ 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schröder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schröder paths.

Original language	English
Pages (from-to)	311-326
Number of pages	16
Journal	Journal of Statistical Planning and Inference
Volume	101
Issue number	1-2
DOIs	https://doi.org/10.1016/S0378-3758(01)00192-6
State	Published - 15 Feb 2002

Keywords

Catalan numbers
Lattice paths
Schröder numbers

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10.1016/S0378-3758(01)00192-6

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title = "The Narayana distribution",

abstract = "The Narayana distribution is Nn,k = 1/n(k-1n)(kn) for 1 ≤ k ≤ n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the nth Narayana polynomial as Nn(w) = ∑1≤k≤n Nn,kWk, for n ≥ 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schr{\"o}der numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schr{\"o}der paths.",

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AU - Sulanke, Robert A.

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N2 - The Narayana distribution is Nn,k = 1/n(k-1n)(kn) for 1 ≤ k ≤ n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the nth Narayana polynomial as Nn(w) = ∑1≤k≤n Nn,kWk, for n ≥ 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schröder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schröder paths.

AB - The Narayana distribution is Nn,k = 1/n(k-1n)(kn) for 1 ≤ k ≤ n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the nth Narayana polynomial as Nn(w) = ∑1≤k≤n Nn,kWk, for n ≥ 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schröder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schröder paths.

KW - Catalan numbers

KW - Lattice paths

KW - Schröder numbers

UR - https://www.scopus.com/pages/publications/0037082891

U2 - 10.1016/S0378-3758(01)00192-6

DO - 10.1016/S0378-3758(01)00192-6

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AN - SCOPUS:0037082891

SN - 0378-3758

VL - 101

SP - 311

EP - 326

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

IS - 1-2

ER -

The Narayana distribution

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Keywords

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