On the Number of Generalized Dyck Paths

Abstract

It is known that the number of Dyck paths is given by a Catalan number. Dyck paths are represented as plane lattice paths which start at the origin \(O\) and end at the point \(P_n = (n,n)\) repeating \((1,0)\) or \((0,1)\) steps without going above the diagonal line \(OP_n\). Therefore, it is reasonable to ask of any positive integers \(a\) and \(b\) what number of lattice paths start at \(O\) and end at point \(A = (a, b)\) repeating the same steps without going above the diagonal line \(OA\). In this article, we show a formula to represent the number of such generalized Dyck paths.