$x(1-t(x+x^{-1}))F(x;t) = x-tF(0;t)$

TL;DR

This paper introduces algebraic and combinatorial methods for enumerating lattice walks, focusing on solving functional equations related to simple walks on the half-line, and aims to illustrate diverse techniques in a non-technical manner.

Contribution

It presents a comprehensive overview of algebraic and computational approaches to solving functional equations in lattice walk enumeration, emphasizing simplicity and pedagogical clarity.

Findings

Derivation of generating functions for simple lattice walks

Application of algebraic methods to solve functional equations

Illustration of diverse techniques in walk enumeration

Abstract

The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their generating functions, and the functional equations they satisfy. We focus on algebraic methods for manipulating and solving these equations. Elementary power series algebra plays a prominent role, computer algebra too, but we repeatedly digress and present ideas and methods of different kind whenever it is appropriate. The exposition is organized around the most simple yet non-trivial problem: the enumeration of simple walks on the half-line. The intention is to illustrate different techniques without getting technical.