Abstract: A partial Motzkin path of length n is a lattice path from (0,0) to (n,k), which run through the integer points, consisting of up steps U=(1,1), down steps D=(1,－1) and horizontal steps H=(1,0), and it never goes below the x-axis. The number of Motzkin paths from (0,0) to(n,0) is called the n-th Motzkin number. The generating function of Motzkin numbers and the representation of the Riordan matrix of the number of partial Motzkin paths are obtained by using the kernel method. Finally, the generating functions of partial Motzkin paths with restricted height are given by using recurrence relations and the linear algebraic method. Some relevant examples are presented here.

Key words: Motzkin path, partial Motzkin path, Motzkin number, generating function, kernel method