Riordan Group And Some Identities

Enumerative Combinatorics is one of vitally important research branches in Com-binatorics, mainly investigates counting problems of combinatorial settings on finite set under given conditions. The main contents of this discuss are listed as follows:In the first chapter, we introduce the developments of the theories of combinatorial sequences and lattice paths, and the following two chapters are the results of the present thesis.In the second chapter, we work over a simple but important combinatorial structure——symmetric lattice paths, which is a hot research topic in recent decades. Let dn, mn, sn denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schroder paths of length 2n, respectively. We get the generating function of dn, mn, sn, and using "first return decomposition" of a symmetric path easily get the relationship of them.In the third chapter, we aim to study the matrices related to some combinatorial sequences, and there are two parts in this chapter.The first part of the third chapter obtains six identities relating dn, mn, sn and also gives two of them combinatorial proofs. We also give the relations with Catalan numbers.The second part of the third chapter, we investigate some relations satisfied by the generic element of some special Riordan arrays and get the average mid-height of sym-metric Dyck paths of length 2n.