波利亚计数定理在研究和解决很多组合计数问题中发挥了举足轻重的作用，使组合计数理论取得了进一步发展。本文采用理论分析与典型示例相结合的方法，直观清晰的介绍了波利亚计数定理的理论内容和部分实际应用。 全文分为三章。第一章中，首先借助一个具体的计数问题提出问题-如何计算不等效构型数。接着介绍有限群论的相关知识和Burnside引理，作为后续内容的理论基础。然后结合分析示例，引入映射和置换群解决问题，提出构型计数记录和群的循环指标多项式，逐步形成并证明了波利亚计数定理。第二章主要介绍了波利亚计数定理的一个重要应用。第三章介绍了波利亚计数定理的教学设计。

Pólya’s Enumeration Theorem(for short, PET) plays a vital role in researching and solving a great many counting problems in combination problems, and drives the significant progress of the combinatorial counting theory. This paper carries out the theoretical analysis and examples, so as to introduce the mathematical theory and a part of practical applications of PET. This paper consists of three parts. In chapter 1, firstly, we pose the question-how to count the number of nonequivalent colorings by means of a specific counting problem. Then introduce some relevant content in Finite Group Theory and Burnside’s Lemma as basis knowledge of subsequent content. After that, by analyzing the typical examples, using the view of mapping and permutation to think about the problem, addressing cycle index polynomial, we build and prove PET step by step. Chapter 2 introduces one of the important use of PET. Chapter 3 introduces the teaching design of PET.