Some Enumerations Of Weighted Motzkin Paths

The Motzkin paths that use the steps level,up and down with assigned weighted?,? and ? are called weighted Motzkin paths.The weighted Motzkin paths without level steps on the x-axis are called weighted Riordan paths.In Chapter 1,some basic concepts and notations about the lattice paths and Riordan matrices in combinatorial mathematics are given.In Chapter 2,the weighted Motzkin and Riordan paths are mainly studied.Firstly,we give the Motzkin paths and Riordan paths by symbolization method.On this basis,the matrix representation of the weighted Motzkin and Riordan paths are considered by using the A and Z sequences of the Riordan matrix.Secondly,the expression of general entries of the matrix are obtained by means of the Lagrange inversion formula.The generating functions of the weighted Motzkin and Riordan paths are acquired by calculation to solve the related counting problems.Finally7,we give a algebraic proof of a three-term recursive identities for the weighted Motzkin number.In Chapter 3,we first give a bijection between the restricted(6,5)-Motzkin paths of length n and the 4-Schroder paths of semilength n.Moreover,we found there is also a bijection between the set of(6,5)-Motzkin paths of length n and 4-little Schroder paths of semilength n+1.Finally,we studied and generalized the first two mappings,then we acquired the bijection between the q-Schroder paths and the(q+2,q-1)-Motzkin paths.