  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.

Combinatorics