Counting generalized Dyck paths

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 $(0,0)$ to $(n,n)$ which is below the diagonal line $y=x$. 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 $(0,0)$ to $(m,n)\in {\mathbb{N}}^2$ which is below the diagonal line $y=\frac{n}{m}x$, and denote by $C(m,n)$ the number of Dyck paths from $(0,0)$ to $(m,n)$. In this paper, we give a formula to calculate $C(m,n)$ for arbitrary $m$ and $n$.

15 pages, 2 figures

Cited by in corpus (3)

• Cylindrical Dyck paths and the Mazorchuk-Turowska equation
• Enumerations of humps and peaks in $(k,a)$-paths and $(n,m)$-Dyck paths via bijective proofs
• Bijections on $r$-Shi and $r$-Catalan Arrangements