2008/05/28 by Sun, Yidong
#05A05 #05A15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.0805.4358
A \em 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 \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.