A refinement of Dyck paths: A combinatorial approach

Local maxima and minima of a Dyck path are called peaks and valleys, respectively. A Dyck path is called restricted[Formula: see text]-Dyck if the difference between any two consecutive valleys is at least [Formula: see text] (right-hand side minus left-hand side) or if it has at most one valley. In this paper, we use several techniques to enumerate some statistics over this new family of lattice paths. For instance, we use the symbolic method, the Chomsky–Schűtzenberger methodology, Zeilberger’s creative telescoping method, recurrence relations, and bijective relations. We count, for example, the number of paths of length [Formula: see text], the number of peaks, the number of valleys, the number of peaks of a fixed height, and the area under the paths. We also give a bijection between the restricted [Formula: see text]-Dyck paths and a family of binary words.