The Riordan Arrays And Matrix Identities

Enumerative combinatorics and combinatorial identities are fundamental research branches and important parts of combinatorics. The main content of this thesis are listed as follows:In Chapter 1, we introduce the developments of the theories of combinatorial se-quences and identities, and research status of Riordan array theorem.In Chapter 2, we present the notions of generating function, the definitions and prop-erties of Dyck path, Motzkin path and Schroder path. We also give a brief introduction of Riordan array theorem and A-, Z-sequence of Riordan array.In Chapter 3, we study the generalized Motzkin paths. Let pn,j be the number of partial Motzkin paths of length n ending at y≥j, we obtain several matrix identities by using Riordan array. Based on lattice paths and number sequences, we present three com-binatorial proofs for the identities involving Motzkin number, number sequence, Catalan number and even term of Fibonacci number.In Chapter 4, based on generating function, we consider several consistent Riordan arrays involving colored Motzkin path, Schroder path and κ-path, and we also give their A-sequences, combinatorial interpretations and the corresponding weighted Lukasiewicz paths.