Counting peaks at height k in a Dyck path

Details Counting peaks at height k in a Dyck path Counting peaks at height k in a Dyck path

2002-03-21

arxiv.org/abs/math/0203222v2

Journal of Integer Sequences 5, (2002), Article 02..1.1

Mathematics

Combinatorics

7 pages, 3 figures

Scientific paper

A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y=k that is immediately preceded by a (1,1) step and immediately followed by a (1,-1) step. In this paper we find an explicit expression to the generating function for the number of Dyck paths starting at (0,0) and ending at (2n,0) with exactly r peaks at height k. This allows us to express this function via Chebyshev polynomials of the second kind and generating function for the Catalan numbers.

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