On factor-free Dyck words with half-integer slope

TL;DR

This paper explores a special class of rational Dyck paths with half-integer slopes, providing algebraic, combinatorial, and enumeration insights into factor-free Dyck words and their related languages.

Contribution

It introduces a new algebraic and combinatorial framework for factor-free Dyck words with specific slopes, including explicit enumeration formulas.

Findings

Lattice path description of the language of factor-free Dyck words

Explicit enumeration formula using partial Bell polynomials

New formulas for counting factor-free generalized Dyck words

Abstract

We study a class of rational Dyck paths with slope (2m+1)/2 corresponding to factor-free Dyck words, as introduced by P. Duchon. We show that, for the slopes considered in this paper, the language of factor-free Dyck words is generated by an auxiliary language that we examine from the algebraic and combinatorial points of view. We provide a lattice path description of this language, and give an explicit enumeration formula in terms of partial Bell polynomials. As a corollary, we obtain new formulas for the number of associated factor-free generalized Dyck words.