English

Counting generalized Dyck paths

Combinatorics 2013-04-23 v1

Abstract

The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0,0)(0,0) to (n,n)(n,n) which is below the diagonal line y=xy=x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0,0)(0,0) to (m,n)N2(m,n) \in \mathbb{N}^2 which is below the diagonal line y=nmxy=\frac{n}{m}x, and denote by C(m,n)C(m,n) the number of Dyck paths from (0,0)(0,0) to (m,n)(m,n). In this paper, we give a formula to calculate C(m,n)C(m,n) for arbitrary mm and nn.

Keywords

Cite

@article{arxiv.1304.5595,
  title  = {Counting generalized Dyck paths},
  author = {Yukiko Fukukawa},
  journal= {arXiv preprint arXiv:1304.5595},
  year   = {2013}
}

Comments

15 pages, 2 figures

R2 v1 2026-06-22T00:03:23.516Z