Dyck Path And Its Application In Combinatorial Structures

Combinatorics is an important branch of applied mathematics,and combinatorial structure is the core of research on combinatorics.As a special combinatorial structure,Dyck Path has attracted wide interests,and it has been studied by a lot of famous experts and scholars,including Academician Yongchuan Chen and R.P.Stanley in recent years.In this thesis,we will discuss Dyck Path and its application in combinatorial structures.Firstly,we introduce the research background and significance of Dyck Path.Moreover,the structure of this paper is given.Secondly,several combinatorial numbers and structures,which related to Dyck Path in combinatorics,are discussed,such as the planar tree,the Catalan number,the Motzkin number and the Narayana number.There exist bijections between these combinatorial numbers and Dyck Path.Hence,combinatorical interpretations are presented by using Dyck Path.Finally,applying the restricted combinatorical structure of the Dyck Paths,we give combinatorial interpretations of four identities related to the k-Catalan number,which are proposed by E.H.M.Brietzke in the journal of “Discrete Mathematics”.