Riordan Arrays And Enumeration Of Dyck Paths

Riordan array theory has important applications in algebraic Combinatorics.By using Riordan matrix,we can characterize many combinatorial problems and also prove a large number of combinatorial identities.There are many relations among Catalan numbers,Motzkin numbers,Schr(?)der numbers as common combination sequences.In this paper,we study the problem of counting of Dyck paths and Schr(?)der paths,prove several identities related to Catalan numbers,Motzkin numbers and Schr(?)der numbers,and testify the Chung-Feller Theorem in the super 3-Dyck paths.In Chapter 1,we briefly introduce research backgrounds,basic concepts of Riordan array and common definitions of Lattice paths.In Chapter 2,based on the original Dyck paths,several kinds of combinatorial matrices related to Catalan numbers are obtained through the enumeration of different end points.The inversion of these matrices,the row sums sequence and diagonal sequence of these matrices are all obtained.Some combinatorial identities are proved by using the enumerations of lattice paths.In Chapter 3,we generalize the Chung-Feller Theorem,and obtain a new Chung-Feller type Theorem of super 3-Dyck paths and super k-Dyck paths.In Chapter 4,we study the generalized Schr(?)der paths.The large Schr(?)der matrix and the small Schr(?)der matrix are obtained by using the Riordan array.