Counting Dyck paths according to the maximum distance between peaks and valleys

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of upsteps u = (1, 1) and down-steps d = (1,-1) which never passes below the x-axis. Let D n denote the set of Dyck paths of length 2n. A peak is an occurrence of ud (an upstep immediately followed by a downstep) within a Dyck path, while a valley is an occurrence of du. Here, we compute explicit formulas for the generating functions which count the members of D n according to the maximum number of steps between any two peaks, any two valleys, or a peak and a valley. In addition, we provide closed expressions for the total value of the corresponding statistics taken over all of the members of D n. Equivalent statistics on the set of 231-avoiding permutations of length n are also described.

Original languageEnglish
Article number12.1.1
JournalJournal of Integer Sequences
Issue number1
StatePublished - 15 Dec 2011


  • Dyck paths
  • Generating functions
  • Peaks
  • Statistics
  • Valleys

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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