New formulas for Dyck paths in a rectangle

Abstract

We consider the problem of counting the set of Da b of Dyck pathsinscribedinarectangleofsize a x b. They are a natural generalization of the classical Dyck words enumerated by the Catalan numbers. By using Ferrers diagrams associated to Dyck paths, we derive formulas for the enumeration of Da b with a and b non relatively prime, in terms of Catalan numbers.