A {em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. A {em $u$-segment {rm (resp.} $h$-segment {rm)}} of a Motzkin path is a maximum sequence of consecutive up-steps ({rm 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.
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