Some statistics on generalized Motzkin paths with vertical steps
Abstract
Recently, several authors have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called G-Motzkin paths for short, that is lattice paths from (0, 0) to (n, 0) in the first quadrant of the XOY-plane that consist of up steps u=(1, 1), down steps d=(1, -1), horizontal steps h=(1, 0) and vertical steps v=(0, -1). We mainly count the number of G-Motzkin paths of length n with given number of z-steps for z∈ \u, h, v, d\, and enumerate the statistics "number of z-steps" at given level in G-Motzkin paths for z∈ \u, h, v, d\, some explicit formulas and combinatorial identities are given by bijective and algebraic methods, some enumerative results are linked with Riordan arrays according to the structure decompositions of G-Motzkin paths. We also discuss the statistics "number of z1z2-steps" in G-Motzkin paths for z1, z2∈ \u, h, v, d\, the exact counting formulas except for z1z2=dd are obtained by the Lagrange inversion formula and their generating functions.