Some Enumerations On Lukasiewicz Paths

A Lukasiewicz path is a lattice path with steps (1,1),(1,0)and =(1,-), = 1,2,3....In this thesis,we study some enumerations on Lukasiewicz paths.Using the weighted Lukasiewicz paths,we give a special combinatorial interpretation to the half of a Riordan array.We decompose the Lukasiewicz paths according to the first return.This allows us to use the symbolic method to deal with the enumerations of Lukasiewicz paths without hills,peaks,valleys,corners,respectively.Further enumeration results are obtained when Lukasiewicz paths have a fix number of returns,peaks,valleys,respectively.Let ? be the set of all Lukasiewicz paths and be the set of all Dyck paths.We provide a combinatorial proof for this relationship by introducing a bijection between some subset of ? and some subset of .