Enumeration of partial Łukasiewicz paths

International audienceLukasiewicz paths are lattice paths in N 2 starting at the origin, ending on the x-axis, and consisting of steps in the set {(1, k), k ≥ −1}. We give bivariate generating functions and exact values for the number of n-length prefixes (resp. suffixes) of these paths ending (resp. starting) at height k ≥ 0 with a given type of step. We make a similar study for paths of bounded height, and we prove that the average height of n-length paths ending at a fixed height behaves as √ πn when n → ∞. Finally, we study prefixes of alternate Lukasiewicz paths, i.e., Lukasiewicz paths that do not contain two consecutive steps in the same direction