Generating functions for plateaus in Motzkin paths

TL;DR

This paper develops generating functions for plateaus in Motzkin paths, generalizes to longer plateaus, and introduces a continued fraction approach for deriving multivariate generating functions for various path statistics.

Contribution

It introduces three forms of bivariate generating functions for plateaus, extends to longer plateaus, and presents a continued fraction method for multivariate generating functions.

Findings

Derived three forms of generating functions for plateaus

Generalized generating functions to longer plateaus

Presented a continued fraction approach for multivariate functions

Abstract

A plateau in a Motzkin path is a sequence of three steps: an up step, a horizontal step, then a down step. We find three different forms for the bivariate generating function for plateaus in Motzkin paths, then generalize to longer plateaus. We conclude by describing a further generalization: a continued fraction form from which one can easily derive new multivariate generating functions for various kinds of path statistics. Several examples of generating functions are given using this technique.