Some Statistics on Generalized Motzkin Paths with Vertical Steps

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 for short, that is from (0, 0) to (n, in the first quadrant XY-plane consist up $${\textbf{u}}=(1, 1)$$ , down $${\textbf{d}}=(1, -1)$$ horizontal $${\textbf{h}}=(1, 0)$$ and $${\textbf{v}}=(0, . The main purpose paper count number length n given $${\textbf{z}}$$ -steps $${\textbf{z}}\in \{{\textbf{u}}, {\textbf{h}}, {\textbf{v}}, {\textbf{d}}\}$$ enumerate statistics “number -steps” at level Some explicit formulas combinatorial identities are by bijective algebraic methods, some enumerative results linked Riordan arrays according structure decompositions paths. We also discuss $${\textbf{z}}_1{\textbf{z}}_2$$ $${\textbf{z}}_1, {\textbf{z}}_2\in exact counting except $${\textbf{z}}_1{\textbf{z}}_2={\textbf{dd}}$$ obtained Lagrange inversion formula their generating functions.