Dyck Paths Enumerated by the Q-bonacci Numbers

TL;DR

This paper introduces a new combinatorial enumeration of constrained Dyck paths, showing they are counted by Q-bonacci numbers, a generalization of q-bonacci numbers, expanding understanding of path enumeration with specific height and valley constraints.

Contribution

It establishes a novel link between constrained Dyck paths and Q-bonacci numbers, extending the combinatorial applications of these generalized sequences.

Findings

Dyck paths with height at most two are enumerated by Q-bonacci numbers.

The Q-bonacci numbers generalize classical q-bonacci numbers for rational q.

The enumeration provides new insights into constrained lattice path combinatorics.

Abstract

We consider Dyck paths having height at most two with some constraints on the number of consecutive valleys at height one which must be followed by a suitable number of valleys at height zero. We prove that they are enumerated by so-called Q-bonacci numbers (recently introduced by Kirgizov) which generalize the classical q-bonacci numbers in the case where q is a positive rational.