四角形和多边形的精确求解模型
Exactly solved models of polyominoes and polygons
摘要：
本章讨论正方形格上某些类自避多边形和多角形的精确计数。我们提出了三个一般的方法，适用于许多类的polyominoes。它们的共同原理是对多项式进行递归描述，然后将其转化为生成函数所满足的函数方程。第一种方法适用于具有线性递归结构的polyominoes类，并在一个合理的生成函数的结果。第二种方法适用于具有代数递归结构的多项式类，并产生代数生成函数。第三种方法，通常称为Temperley方法，是基于向polyominoes添加新列的操作。最后，我们讨论一些开放的问题。
英文摘要：
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.
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