Rational Dyck paths

TL;DR

This paper introduces rational Dyck paths constrained by height and peak/valley patterns, providing their generating functions and linking certain subsets to Q-bonacci numbers, advancing combinatorial enumeration methods.

Contribution

It defines a new class of rational Dyck paths, derives their generating functions, and connects specific subsets to Q-bonacci numbers, expanding combinatorial enumeration techniques.

Findings

Derived generating functions for rational Dyck paths.

Characterized subsets enumerated by Q-bonacci numbers.

Provided combinatorial constructions for these paths.

Abstract

Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational Dyck paths, and provide the associated generating function, according to their semilength, as well as the construction of such a class. Moreover, we characterize some subsets of the rational Dyck paths that are enumerated by the $\mathbb Q$-bonacci numbers.