A natural extension of Catalan numbers

Noam Solomon, Shay Solomon*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A Dyck path is a lattice path in the plane integer lattice ℤ × ℤ consisting of steps (1,1) and (1,-1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0,0) and finish at (2n, 0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p1 and P2, for the number of Dyck paths starting at pi, ending at P2, and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.

Original languageEnglish
Article number08.3.5
JournalJournal of Integer Sequences
Issue number3
StatePublished - 7 Aug 2008
Externally publishedYes


  • Catalan numbers
  • Dyck paths


Dive into the research topics of 'A natural extension of Catalan numbers'. Together they form a unique fingerprint.

Cite this