Geodesic
Counting peaks at height \(k\) in a Dyck path
Journal of integer sequences, Tome 5 (2002) no. 1
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 ($2n,0$) 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.
Keywords: Dyck paths, Catalan numbers, Chebyshev polynomials 
@article{JIS_2002__5_1_a3,
     author = {Mansour,  Toufik},
     title = {Counting peaks at height \(k\) in a {Dyck} path},
     journal = {Journal of integer sequences},
     year = {2002},
     volume = {5},
     number = {1},
     zbl = {1012.05004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2002__5_1_a3/}
}
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Mansour,  Toufik. Counting peaks at height \(k\) in a Dyck path. Journal of integer sequences, Tome 5 (2002) no. 1. http://geodesic.mathdoc.fr/item/JIS_2002__5_1_a3/