2002/03/31 by T. Mansour · 1 citation
Mathematics · #math.CO
published as Journal of Integer Sequences 5, (2002), Article 02..1.1 · 7 pages, 3 figures
arxiv created 2002/05/02 · arxiv updated 2009/11/30
A Dyck path is a lattice path in the plane integer lattice ℤ×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.