Potential polynomials and Motzkin paths

Abstract. AMotzkin path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (1, 0), up-steps (1, 1), and down-steps (1,−1), which never passes below the x-axis. A u-segment (resp. h-segment ) of a Motzkin path is a maximum sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: ”number of u-segments” and ”number of h-segments”. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.