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 XY-plane that consist of up steps \({\textbf{u}}=(1, 1)\), down steps \({\textbf{d}}=(1, -1)\), horizontal steps \({\textbf{h}}=(1, 0)\) and vertical steps \({\textbf{v}}=(0, -1)\). The main purpose of this paper is to count the number of G-Motzkin paths of length n with given number of \({\textbf{z}}\)-steps for \({\textbf{z}}\in \{{\textbf{u}}, {\textbf{h}}, {\textbf{v}}, {\textbf{d}}\}\), and to enumerate the statistics “number of \({\textbf{z}}\)-steps” at given level in G-Motzkin paths for \({\textbf{z}}\in \{{\textbf{u}}, {\textbf{h}}, {\textbf{v}}, {\textbf{d}}\}\). 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 \({\textbf{z}}_1{\textbf{z}}_2\)-steps” in G-Motzkin paths for \({\textbf{z}}_1, {\textbf{z}}_2\in \{{\textbf{u}}, {\textbf{h}}, {\textbf{v}}, {\textbf{d}}\}\), the exact counting formulas except for \({\textbf{z}}_1{\textbf{z}}_2={\textbf{dd}}\) are obtained by the Lagrange inversion formula and their generating functions.