An Identity for Catalan Numbers via Restricted Dyck Paths

TL;DR

This paper introduces a new combinatorial identity involving Catalan numbers by analyzing constrained Dyck paths with height and valley restrictions, expanding the understanding of their structural properties.

Contribution

It provides a novel identity for Catalan numbers derived from a detailed combinatorial analysis of restricted Dyck paths with specific height and valley constraints.

Findings

Derived enumeration formulas for constrained Dyck paths.

Established a new combinatorial identity involving Catalan numbers.

Connected path restrictions to new algebraic relations among Catalan numbers.

Abstract

Catalan numbers and their interpretations in terms of Dyck paths are widely used in different topics of applied mathematics and computer science. Here, we consider a general approach for constrained Dyck paths. In particular, we study Dyck paths of height at most $h$ with the additional restriction of having no $k-1$ consecutive valleys at height $h-1$. We give a combinatorial description of this class of paths and derive enumeration formulas using classical techniques for counting constrained lattice paths. As a consequence of this analysis, we obtain an identity involving Catalan numbers which, to the best of the authors' knowledge, does not appear in the existing literature. This identity arises naturally from the combinatorial interpretation and provides a new relation among families of Dyck paths with height and local structural constraints.