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**Counting Peaks at Height ***k* in a Dyck Path

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Toufik Mansour

LaBRI

Université Bordeaux 1

351, cours de la Libération

33405 Talence Cedex, France

**Abstract:**
A Dyck path is a lattice path in the plane integer lattice
Z x 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 for the generating function for the number of Dyck
paths starting at (0,0) and ending at (2*n*,0) with exactly *r*
peaks at height *k*. This allows us to express this function via
Chebyshev polynomials of the second kind and the generating
function for the Catalan numbers.

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(Mentions sequence
A000108
.)

Received March 21, 2002;
revised version received April 14, 2002.
Published in Journal of Integer Sequences May 1, 2002.

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