Counting humps and peaks in generalized Motzkin paths

Let us call a lattice path in Z x 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 UH'D, i >= 0. This generalizes recent results concerning the cases when k = 1 and a = 1 or a = infinity. A similar relation may be given involving peaks (consecutive steps of the form UD). (C) 2013 Elsevier By. All rights reserved.