Counting humps and peaks in generalized Motzkin paths

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


Abstract Let us call a lattice path in ℤ × ℤ from (0,0) to (n,0) using U=(1,k), D=(1,-1), and H=(a,0) steps and never going below the x-axis, a (k,a)-path (of order n). A super (k,a)-path is a (k,a)-path which is permitted to go below the x-axis. We relate the total number of humps in all of the (k,a)-paths of order n to the number of super (k,a)-paths, where a hump is defined to be a sequence of steps of the form UHiD, i≥0. This generalizes recent results concerning the cases when k=1 and a=1 or a=∞. A similar relation may be given involving peaks (consecutive steps of the form UD).

Original languageEnglish
Pages (from-to)2213-2216
Number of pages4
JournalDiscrete Applied Mathematics
Issue number13-14
StatePublished - Sep 2013


  • Dyck paths
  • Humps Peaks
  • Motzkin paths

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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