Potential polynomials and Motzkin paths

A Motzkin path   of length nn is a lattice path from (0,0)(0,0) to (n,0)(n,0) in the plane integer lattice Z×ZZ×Z consisting of horizontal-steps (1,0)(1,0), up-steps (1,1)(1,1), and down-steps (1,−1)(1,−1), which never passes below the xx-axis. A uu-segment   (resp. hh-segment  ) of a Motzkin path is a maximal sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: “number of uu-segments” and “number of hh-segments”. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the two 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.