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　　《組閤數學（英文版）（第5版）》是係統闡述組閤數學基礎，理論、方法和實例的優秀教材。齣版30多年來多次改版。被MIT、哥倫比亞大學、UIUC、威斯康星大學等眾多國外高校采用，對國內外組閤數學教學産生瞭較大影響。也是相關學科的主要參考文獻之一。《組閤數學（英文版）（第5版）》側重於組閤數學的概念和思想。包括鴿巢原理、計數技術、排列組閤、Polya計數法、二項式係數、容斥原理、生成函數和遞推關係以及組閤結構（匹配，實驗設計、圖）等。深入淺齣地錶達瞭作者對該領域全麵和深刻的理解。除包含第4版中的內

內容簡介

　　《組閤數學（英文版）（第5版）》英文影印版由Pearson Education Asia Ltd，授權機械工業齣版社少數齣版。未經齣版者書麵許可，不得以任何方式復製或抄襲奉巾內容。僅限於中華人民共和國境內（不包括中國香港、澳門特彆行政區和中同颱灣地區）銷售發行。《組閤數學（英文版）（第5版）》封麵貼有Pearson Education（培生教育齣版集團）激光防僞標簽，無標簽者不得銷售。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．
　　For sale and distribution in the People’S Republic of China exclusively（except Taiwan，Hung Kong SAR and Macau SAR）．

作者簡介

　　Richard A.Brualdi，美國威斯康星大學麥迪遜分校數學係教授（現已退休）。曾任該係主任多年。他的研究方嚮包括組閤數學、圖論、綫性代數和矩陣理論、編碼理論等。Brualdi教授的學術活動非常豐富。擔任過多種學術期刊的主編。2000年由於“在組閤數學研究中所做齣的傑齣終身成就”而獲得組閤數學及其應用學會頒發的歐拉奬章。

內頁插圖

目錄

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

精彩書摘

　　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.

前言/序言

　　I have made some substantial changes in this new edition of Introductory Combinatorics， and they are summarized as follows:
　　In Chapter 1, a new section （Section 1.6） on mutually overlapping circles has been added to illustrate some of the counting techniques in later chapters. Previously the content of this section occured in Chapter 7.
　　The old section on cutting a cube in Chapter 1 has been deleted, but the content appears as an exercise.
　　Chapter 2 in the previous edition （The Pigeonhole Principle） has become Chapter 3. Chapter 3 in the previous edition, on permutations and combinations, is now Chapter 2. Pascals formula, which in the previous edition first appeared in Chapter 5, is now in Chapter 2. In addition, we have de-emphasized the use of the term combination as it applies to a set, using the essentially equivalent term of subset for clarity. However, in the case of multisets, we continue to use combination instead of, to our mind, the more cumbersome term submultiset.
　　Chapter 2 now contains a short section （Section 3.6） on finite probability.
　　Chapter 3 now contains a proof of Ramseys theorem in the case of pairs.
　　Some of the biggest changes occur in Chapter 7, in which generating functions and exponential generating functions have been moved to earlier in the chapter （Sections 7.2 and 7.3） and have become more central.
　　The section on partition numbers （Section 8.3） has been expanded.
　　Chapter 9 in the previous edition, on matchings in bipartite graphs, has undergone a major change. It is now an interlude chapter （Chapter 9） on systems of distinct representatives （SDRs）——the marriage and stable marriage problemsand the discussion on bipartite graphs has been removed.
　　As a result of the change in Chapter 9, in the introductory chapter on graph theory （Chapter 11）, there is no longer the assumption that bipartite graphs have been discussed previously.
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