  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.
