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 z 1 z 2 -steps" in G-Motzkin paths for z 1 , z 2 ∈ {u, h, v, d}, the exact counting formulas except for z 1 z 2 = dd are obtained by the Lagrange inversion formula and their generating functions.