Non uniform random generation of generalized Motzkin paths

We consider in this paper the class Mkn of generalized Motzkin paths of length n, that is, lattice paths using steps (1,1), (1,−1), (k,0), where k is a fixed positive integer, starting at the origin (0,0), running above the x-axis, and ending at (n,0). The area is the region bounded by the path and the x-axis. We first establish a bijection between the area of paths in Mkn and some lattice paths of length n+1. Then, by using a rejection technique, we obtain a linear algorithm with an average time complexity (k mod 2 +1)(n+1).