New Formulas for Dyck Paths in a Rectangle

TL;DR

This paper derives new formulas for counting Dyck paths within rectangles, generalizing Catalan number enumeration through Ferrers diagrams, especially when rectangle dimensions are not coprime.

Contribution

It introduces formulas for enumerating Dyck paths in rectangles using Ferrers diagrams, extending classical Catalan number results to non-coprime dimensions.

Findings

Formulas for counting Dyck paths in rectangles with non-coprime dimensions

Use of Ferrers diagrams to derive enumeration formulas

Extension of Catalan number enumeration to new geometric settings

Abstract

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.