Potential polynomials and Motzkin paths

A Motzkin 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 maximal 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 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.