New Formulas for Dyck Paths in a Rectangle

We consider the problem of counting the set of $\mathscr{D}_{a,b}$ of Dyck paths inscribed in a rectangle of size $a\times b$. 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 $\mathscr{D}_{a,b}$ with $a$ and $b$ non relatively prime, in terms of Catalan numbers.