Lattice and Schroder paths with periodic boundaries

TL;DR

This paper proves that the generating function for lattice paths with periodic boundaries and specific steps is algebraic, using combinatorial decomposition and umbral calculus techniques.

Contribution

It introduces a novel combinatorial approach to show algebraicity of generating functions for paths with periodic boundaries and mixed steps.

Findings

The generating function is algebraic when the boundary is periodic with slope at most b/a.

A new method converts umbral generating functions into ordinary ones explicitly.

The approach combines combinatorial decomposition with solving polynomial equations.

Abstract

We consider paths in the plane with $(1,0),$ $(0,1),$ and $(a,b)$-steps that start at the origin, end at height $n,$ and stay to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most $b/a,$ then the ordinary generating function for the number of such paths ending at height $n$ is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or ``umbral'' generating function, in which the power $z^n$ is replaced by a power series of the form $z^n \phi_n(z),$ where $\phi_n(0) = 1.$ Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic.