Riordan Arrays And Their Applications In Enumerations Of Lattice Paths

Combinatorics is a very important branch in modern mathematics, it mainly studies the existence, enumerations, structures and optimization of discrete objects.The enumerations of lattice paths are important problems in combinatorics.In this thesis, using Riordan arrays, we mainly study the enumerations of two kinds of restricted lattice paths: the enumerations of the generalized Motzkin paths,the enumerations of m-Dyck paths.In Chapter 1, we briefly introduce the research background of this paper, give the definitions of lattice paths and Riordan arrays, which is the theoretical foundation of the following two chapters.In Chapter 2, by means of Riordan arrays, the enumerations of generalized Motzkin paths are studied, and a new class of enumerative arrays, i.e., generalized Motzkin arrays, are introduced. Meanwhile, the Riordan array expressions of these arrays are given, and the counting formulas also obtained. It turns out that Catalan array, Schrder array and Motzkin array are all the special cases of the generalized Motzkin arrays.In Chapter 3, we recall the basic knowledge of m-Dyck paths briefly. Then by enumerating m-Dyck paths, we obtain the ECO arrays of these paths, and give the combinatorial interpretations of the coefficients in Taylor expansions of the m-Catalan numbers. Furthermore, we deduce some identities related to m-Catalan numbers. By enumerating a new kind of(i, j)-balance m-Dyck paths we obtain some new Chung-Feller properties.