Exactly solved models of polyominoes and polygons

TL;DR

This chapter presents three recursive methods for exactly enumerating self-avoiding polygons and polyominoes on the square lattice, resulting in rational or algebraic generating functions, and discusses open questions in the field.

Contribution

It introduces three general recursive approaches for exact enumeration of polyominoes, expanding the toolkit for combinatorial enumeration on the lattice.

Findings

First approach yields rational generating functions for linearly recursive classes.

Second approach produces algebraic generating functions for algebraically recursive classes.

The Temperley method involves adding columns to polyominoes, facilitating enumeration.

Abstract

This chapter deals with the exact enumeration of certain classes of self-avoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of them is a recursive description of the polyominoes which then translates into a functional equation satisfied by the generating function. The first approach applies to classes of polyominoes having a linear recursive structure and results in a rational generating function. The second approach applies to classes of polyominoes having an algebraic recursive structure and results in an algebraic generating function. The third approach, commonly called the Temperley method, is based on the action of adding a new column to the polyominoes. We conclude by discussing some open questions.