本书为“经典原版书库”丛书之一。该书是系统阐述组合数学基础、理论、方法和实例的优秀教材。主要介绍了组合数学的概念和思想。包括鸽巢原理、计数技术、排列组合、Polya计数法、二项式系数、容斥原理、生成函数和递推关系以及组合结构(匹配、实验设计、图)等。

本书是系统阐述组合数学基础、理论、方法和实例的优秀教材，出版30多年来多次改版，被MIT、哥伦比亚大学、UIUC、威斯康星大学等众多国外高校采用，对国内外组合数学教学产生了较大影响，也是相关学科的主要参考文献之一。

本书侧重于组合数学的概念和思想。包括鸽巢原理、计数技术、排列组合、Polya计数法、二项式系数、容斥原理、生成函数和递推关系以及组合结构(匹配、实验设计、图)等。深入浅出地表达了作者对该领域全面和深刻的理解。除包含第4版中的内容外，本版又进行了更新，增加了有限概率、匹配数等内容。此外，各章均包含大量练习题，并在书末给出了参考答案与提示。

Preface

1 What Is Combinatorics?

 1.1 Example: Perfect Covers of Chessboards

 1.2 Example: Magic Squares

 1.3 Example: The Four-Color Problem

 1.4 Example: The Problem of the 36 OfFicers

 1.5 Example: Shortest-Route Problem

 1.6 Example: Mutually Overlapping Circles

 1.7 Example: The Game of Nim

 1.8 Exercises

2 Permutations and Combinations

 2.1 Four Basic Counting Principles

 2.2 Permutations of Sets

 2.3 Combinations (Subsets) of Sets

 2.4 Permutations of Multisets

 2.5 Combinations of Multisets

 2.6 Finite Probability

 2.7 Exercises

3 The Pigeonhole Principle

 3.1 Pigeonhole Principle: Simple Form

 3.2 Pigeonhole Principle: Strong Form

 3.3 A Theorem of Ramsey

 3.4 Exercises

4 Generating Permutations and Combinations

 4.1 Generating Permutations

 4.2 Inversions in Permutations

 4.3 Generating Combinations

 4.4 Generating r-Subsets

 4.5 Partial Orders and Equivalence Relations

 4.6 Exercises

5 The Binomial Coefficients

 5.1 Pascal's Triangle

 5.2 The Binomial Theorem

 5.3 Unimodality of Binomial Coefficients

 5.4 The Multinomial Theorem

 5.5 Newton's Binomial Theorem

 5.6 More on Partially Ordered Sets

 5.7 Exercises

6 The Inclusion-Exclusion Principle and Applications

 6.1 The Inclusion-Exclusion Principle

 6.2 Combinations with Repetition

 6.3 Derangements

 6.4 Permutations with Forbidden Positions

 6.5 Another Forbidden Position Problem

 6.6 M6bius Inversion

 6.7 Exercises

7 Recurrence Relations and Generating Functions

 7.1 Some Number Sequences

 7.2 Generating Functions

 7.3 Exponential Generating Functions

 7.4 Solving Linear Homogeneous Recurrence Relations

 7.5 Nonhomogeneous Recurrence Relations

 7.6 A Geometry Example

 7.7 Exercises

8 Special Counting Sequences

 8.1 Catalan Numbers

 8.2 Difference Sequences and Stirling Numbers

 8.3 Partition Numbers

 8.4 A Geometric Problem

 8.5 Lattice Paths and Schr6der Numbers

 8.6 Exercises

9 Systems of Distinct Representatives

 9.1 General Problem Formulation

 9.2 Existence of SDRs

 9.3 Stable Marriages

 9.4 Exercises

10 Combinatorial .Designs

 10.1 Modular Arithmetic

 10.2 Block Designs

 10.3 Steiner Triple Systems

 10.4 Latin Squares

 10.5 Exercises

11 Introduction to Graph Theory

 11.1 Basic Properties

 11.2 Eulerian Trails

 11.3 Hamilton Paths and Cycles

 11.4 Bipartite Multigraphs

 11.5 Trees

 11.6 The Shannon Switching Game

 11.7 More on Trees

 11.8 Exercises

12 More on Graph Theory

 12.1 Chromat,ic Number

 12.2 Plane and Planar Graphs

 12.3 A Five-Color Theorem

 12.4 Independence Number and Clique Number

 12.5 Matching Number

 12.6 Connectivity

 12.7 Exercises

13 Digraphs and Networks

 13.1 Digraphs

 13.2 Networks

 13.3 Matchings in Bipartite Graphs Revisited

 13.4 Exercises

14 Polya Counting

 14.1 Permutation and Symmetry Groups

 14.2 Burnside's Theorem

 14.3 Polya's Counting Formula

 14.4 Exercises

Answers and Hints to Exercises

Bibliography

Index