The Three Kinds Of Statistics Of Lattice Paths

One of the basic research directions of combinatorial mathematics is the counting problem,while path-counting is the most popular one.Path counting refers to the study of counting formula for certain statistics of lattice paths under different constraints,it's better to establish one-to-onecorrespondencebetween such structures and other equivalent ones.In this paper,by recursion relations,generation function,Riordan array and the Lagrange inversion formula,we consider three kinds of statistics for Dyck path,Motzkin path,Schroder path ect.,and then give some counting results.Themainworkofthispaperaresummarizedasfollows.In the first chapter,we briefly introduce the research background of combinatorial mathematics as well as the research status of lattice paths.Then we give the concepts and properties of the three types oflattice paths including the three kinds of statistics,introduce the basic theory of the Riordan array and the method of the Lagrange inversion,which will be involved in the later sections.In the second chapter,we mainly count Dyck paths according to three kinds of statistics("highs","edges" and "points").First,we give the recursive relation of Dyck's path with respect to the three kinds of statistics.Then the corresponding Riordan array can be obtained by solving its generationfunctions.Finally,the Lagrange inversion formula is used to obtain its general element according to its Riordan array andthe related properties of the general element are also obtained.In the last chapter,analogousto the research in ChapterTwo for Dyck paths,we study the corresponding problems for the Motzkin paths and Schroder paths according to the three kinds of statistics "highs","edges" and "points" and then attain their general elements by the method of Rirodan arrays and the Lagrange inversion formula.