Some enumerations on grand Lukasiewicz paths

In this paper, the super-grand Lukasiewicz paths are defined as the lattice paths with step set, and the grand Lukasiewicz paths are such paths restricted above the x-axis. We first give a bijection between the set of grand Lukasiewicz paths of semilength n and the set of Dyck paths semilength n. Then, we enumerate the grand Lukasiewicz paths with respect to the number of horizontal steps. Furthermore, we give a bijective proof for the result that the number of horizontal steps in grand Lukasiewicz paths of semilength n is the number super-grand Lukasiewicz paths from (0,0) to (2n-1,1).