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

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