Dyck Paths Enumerated by the Q-bonacci Numbers
Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani
TL;DR
The paper investigates Dyck paths of height at most two with valley-based constraints parametrized by a positive rational $q\in\mathbb{Q}^+$ and shows these restricted paths are enumerated by Kirgizov’s $\mathbb{Q}$-bonacci numbers, extending the classical $q$-generalized Fibonacci counts. It develops a constructive framework: for $q=1/s$ it yields a simple recurrence $w_n= w_{n-1}+w_{n-s-1}$ with explicit initial conditions, and for general $q=r/s$ it introduces a more elaborate constraint on blocks of consecutive $1$-peaks, leading to recurrences involving $w_{n-p-\lceil p s / r \rceil}$ terms that reproduce the $\mathbb{Q}$-bonacci sequence. The results connect restricted Dyck-path enumerations with rational $q$-bonacci numbers, offering explicit combinatorial constructions and recurrences that generalize existing integer-parameter cases. This advances understanding of how combinatorial restrictions on Dyck paths map to generalized Fibonacci-type sequences and their rational extensions.
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.