Counting Restricted Dyck Paths Through Random Walks

TL;DR

This paper explores the relationship between Dyck paths with height restrictions and random walks, providing a probabilistic approach to Chebyshev polynomials and revealing new insights into their combinatorial structure.

Contribution

It establishes a novel connection between restricted Dyck paths and random walks, offering a new probabilistic perspective on Chebyshev polynomials.

Findings

Established a correspondence between restricted Dyck paths and random walks

Developed a probabilistic framework for analyzing Chebyshev polynomials

Provided new combinatorial insights into Dyck paths with bounded peaks

Abstract

We show connection between Dyck paths with peaks of bounded height and random walks. The correspondence between a certain class of random walks and such Dyck paths allows us to develop a probabilistic perspective on Chebyshev polynomials.