Counting peaks at height k in a Dyck path

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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 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.

Original languageEnglish
JournalJournal of Integer Sequences
Issue number1
StatePublished - 2002
Externally publishedYes


  • Catalan numbers
  • Chebyshev polynomials
  • Dyck paths

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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