On rational Dyck paths and the enumeration of factor-free Dyck words

TL;DR

This paper explores rational Dyck paths and factor-free Dyck words, establishing bijections, new statistics, and formulas for enumeration, advancing combinatorial understanding of these structures.

Contribution

It introduces a bijection between rational and regular Dyck paths with colored ascents, and derives formulas for counting factor-free words and related sequences.

Findings

Established a bijection between rational and regular Dyck paths with colored ascents

Derived a formula for enumerating factor-free Dyck words

Provided alternative formulas for related combinatorial sequences

Abstract

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.