Weighted Partial Motzkin Paths And Riordan Arrays

The research object of combinatorial mathematics is the mathematical problems arranging some discrete things according to some certain rules. It is an ancient and new branch of mathematics. Combinatorial mathematics began in the game.now, whether in pure or applied science, combinatorial mathematics plays its important role. One of the most basic of combinatorial mathematics is the counting problem. There are a lot of classical sequences, such as binomial coefficients, Fibonacci sequence, Catalan sequence, Motzkin sequence and so on. In this paper, starting from Motzkin path, combined with Riordan array, research the important identities of some classic sequence.The main contents of this discourse are listed as follows:Chapter1introduces the background of combinatorial mathematics, lattice and the preliminary knowledge for Motzkin path and Riordan array. A partial Motzkin path is a path from(0,0) to (n,k) in the XOY-plane that does not go below the X-axis and consists of up steps U=(1,1), down steps D=(1,-1)and horizontal steps H=(1,0). A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by1, the horizontal steps are endowed with a weight x if they are lying on X-axis, and endowed with a weight y if they are not lying on X-axis.Chapter2introduces how to get a new matrix according to the Pascal triangle. Select certain elements of the Pascal triangle, getting elements of the new matrix by the method of determinant calculation. The new matrix has a lot of interesting properties. At the end of this chapter, enumerates several simple examples.Chapter3denote by M,nk (x,y) to be the weight function of all weighted partial Motzkin paths from(0,0)to(n,k),and M=(Mn,k(x,y))n≥k≥0to be the infinite lower triangular matrices. Then, prove out it is Riordan array. Chapter4obtain a lot of interesting determinant identities related to M, which are proved by bijections using weighted partial Motzkin paths. When the weight parameters (x,y)are specialized, several new identities are obtained related to some classial sequences involving Catalan numbers.Chapter5consider aboat the alternating cases related to M. Using the creative telescoping algorithm, we obtain some new explicit formulas for Catalan numbers.