Counting Humps and Peaks in Generalized Dyck Paths
نویسندگان
چکیده
Let us call a lattice path in Z × Z from (0, 0) to (n, 0) using U = (1, k), D = (1,−1), and H = (a, 0) steps and never going below the x-axis a (k, a)-path (of order n). A super (k, a)-path is a (k, a)-path which is permitted to go below the x-axis. We relate the total number of humps in all of the (k, a)paths of order n to the number of super (k, a)-paths, where a hump is defined to be a sequence of steps of the form UHD, i ≥ 0. This generalizes recent results concerning the cases when k = 1 and a = 1 or a = ∞. A similar relation may be given involving peaks (consecutive steps of the form UD).
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