Riordan Arrays And The Counting Problems Of Several Types Of Generalized Dyck Paths

Riordan arrays and generating functions are two important tools for studying lattice paths in combinatorics mathematics.Firstly,we introduce the(m,r,s)-halves of a Riordan array,and provide an explicit expression of(m,r,s)-halves of a Riordan array.Secondly,using Riordan array,we discuss some counting problems on the Dyck paths and the 3-Dyck paths,and obtain a family of parametric Pascal matrices and give the combination explanation by using the weighted partial Free Dyck paths.Using the(1,0,1)-half and the(1,0,2)-half of the 3-Catalan matrix,we study some counting problems on the 3-Dyck paths.Finally,we introduce the concept of the skew 3-Dyck paths,and study the counting problems on the skew 3-Dyck paths by generating functions and symbolization methods,establish the bijections between the skew 3-Dyck paths and the Dyck paths,3-Dyck paths.