Richard A.Brualdi，美国威斯康星大学麦迪逊分校数学系教授（现已退休）。曾任该系主任多年。他的研究方向包括组合数学、图论、线性代数和矩阵理论、编码理论等。Brualdi教授的学术活动非常丰富。担任过多种学术期刊的主编。2000年由于“在组合数学研究中所做出的杰出终身成就”而获得组合数学及其应用学会颁发的欧拉奖章。

Chapter 3
The Pigeonhole Principle
We consider in this chapter an important， but elementary, combinatorial principle that can be used to solve a variety of interesting problems, often with surprising conclusions. This principle is known under a variety of names, the most common of which are the pigeonhole principle, the Dirichlet drawer principle, and the shoebox principle.1 Formulated as a principle about pigeonholes, it says roughly that if a lot of pigeons fly into not too many pigeonholes, then at least one pigeonhole will be occupied by two or more pigeons. A more precise statement is given below.
3.1 Pigeonhole Principle: Simple FormThe simplest form of the pigeonhole principle is tile following fairly obvious assertion.Theorem 3.1.1 If n+1 objects are distributed into n boxes, then at least one box contains two or more of the objects.
Proof. The proof is by contradiction. If each of the n boxes contains at most one of the objects, then the total number of objects is at most 1 + 1 + ... +1（n ls） = n.Since we distribute n + 1 objects, some box contains at least two of the objects.
Notice that neither the pigeonhole principle nor its proof gives any help in finding a box that contains two or more of the objects. They simply assert that if we examine each of the boxes, we will come upon a box that contains more than one object. The pigeonhole principle merely guarantees the existence of such a box. Thus, whenever the pigeonhole principle is applied to prove the existence of an arrangement or some phenomenon, it will give no indication of how to construct the arrangement or find an instance of the phenomenon other than to examine all possibilities.

1 What Is Combinatorics?
1.1 Example：Perfect Covers of Chessboards
1.2 Example：Magic Squares
1.3 Example：The Fou r-CoIor Problem
1.4 Example：The Problem of the 36 C)fficers
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 ofMUltisets
2.5 Cornblnations of Multisets
2.6 Finite Probability
2.7 Exercises

3 The Pigeonhole Principle
3.1 Pigeonhole Principle：Simple Form
3.2 Pigeon hole Principle：Strong Form
3.3 A Theorem of Ramsey
3.4 Exercises

4 Generating Permutations and Cornbinations
4.1 Generating Permutations
4.2 Inversions in Permutations
4.3 Generating Combinations
4.4 Generating r-Subsets
4.5 PortiaI Orders and Equivalence Relations
4.6 Exercises

5 The Binomiaf Coefficients
5.1 Pascals Triangle
5.2 The BinomiaI Theorem
5.3 Ueimodality of BinomiaI Coefficients
5.4 The Multinomial Theorem
5.5 Newtons Binomial Theorem
5.6 More on Pa rtially Ordered Sets
5.7 Exercises

6 The Inclusion-Exclusion P rinciple and Applications
6.1 The In Clusion-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 lnverslon
6.7 Exe rcises

7 Recurrence Relations and Generating Functions
7.1 Some Number Sequences
7.2 Gene rating 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 Sti rling Numbers
8.3 Partition Numbers
8.4 A Geometric Problem
8.5 Lattice Paths and Sch rSder Numbers
8.6 Exercises Systems of Distinct ReDresentatives

9.1 GeneraI Problem Formulation
9.2 Existence of SDRs
9.3 Stable Marriages
9.4 Exercises

10 CombinatoriaI Designs
10.1 Modular Arithmetic
10.2 Block Designs
10.3 SteinerTriple Systems
10.4 Latin Squares
10.5 Exercises

11 fntroduction 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 Chromatic 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 Bu rnsides Theorem
14.3 Polas Counting Formula
14.4 Exercises
Answers and Hints to Exercises

《组合数学（英文版）（第5版）》英文影印版由Pearson Education Asia Ltd，授权机械工业出版社少数出版。未经出版者书面许可，不得以任何方式复制或抄袭奉巾内容。仅限于中华人民共和国境内（不包括中国香港、澳门特别行政区和中同台湾地区）销售发行。《组合数学（英文版）（第5版）》封面贴有Pearson Education（培生教育出版集团）激光防伪标签，无标签者不得销售。English reprint edition copyright@2009 by Pearson Education Asia Limited and China Machine Press．
Original English language title：Introductory Combinatorics，Fifth Edition（ISBN978—0—1 3-602040-0）by Richard A．Brualdi，Copyright@2010，2004，1999，1992，1977 by Pearson Education，lnc． All rights reserved．
Published by arrangement with the original publisher，Pearson Education，Inc．publishing as Prentice Hall．
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《组合数学（英文版）（第5版）》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。《组合数学（英文版）（第5版）》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