| 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. |
