Pascal Rhombus And Generalized Dyck Paths

In this paper,combinatorial interpretations of the generalized Pascal rhombus and left bounded rhombus are given by the generalized Dyck paths.Relationships of Dyck rook paths,double colored Dyck paths,small(3,1)-Schr�oder paths and(5,4)-Motzkin paths are discussed.In chapter 1,some concepts about lattice paths,Riordan arrays,-matrix and symbolic method are introduced.In chapter 2,combinatorial interpretations of the 6)-generalized Pascal rhombus and left bounded rhombus are given by the 6)-generalized Dyck paths,and the Riordan arrays representation of the 6)-generalized Pascal rhombus and left bounded rhombus are given,and explicit formula for the generic elements and row sums of the 6)-generalized Pascal rhombus and left-bounded rhombus are obtained in terms of 6)-Bonacci numbers.In chapter 3,the bijection between Dyck rook paths and double colored Dyck paths is given,and the bijection between double colored Dyck paths and small(3,1)-Schr�oder paths is obtained.Furthermore,the generating function of(5,4)-Motzkin paths is given by using of the symbolic method.