On rational Dyck paths and the enumeration of factor-free Dyck words
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.
Keywords
Cite
@article{arxiv.1606.02183,
title = {On rational Dyck paths and the enumeration of factor-free Dyck words},
author = {Daniel Birmajer and Juan B. Gil and Michael D. Weiner},
journal= {arXiv preprint arXiv:1606.02183},
year = {2018}
}
Comments
10 pages, 2 figures. Revised version with a shorter introduction and minor reorganization of section 2