组合数学

Preface

1WhatIsCombinatorics?

1.1Example:PerfectCoversofChessboards

1.2Example:MagicSquares

1.3Example:TheFour-ColorProblem

1.4Example:TheProblemofthe36OfFicers

1.5Example:Shortest-RouteProblem

1.6Example:MutuallyOverlappingCircles

1.7Example:TheGameofNim

1.8Exercises

2PermutationsandCombinations

2.1FourBasicCountingPrinciples

2.2PermutationsofSets

2.3Combinations(Subsets)ofSets

2.4PermutationsofMultisets

2.5CombinationsofMultisets

2.6FiniteProbability

2.7Exercises

3ThePigeonholePrinciple

3.1PigeonholePrinciple:SimpleForm

3.2PigeonholePrinciple:StrongForm

3.3ATheoremofRamsey

3.4Exercises

4GeneratingPermutationsandCombinations

4.1GeneratingPermutations

4.2InversionsinPermutations

4.3GeneratingCombinations

4.4Generatingr-Subsets

4.5PartialOrdersandEquivalenceRelations

4.6Exercises

5TheBinomialCoefficients

5.1PascalsTriangle

5.2TheBinomialTheorem

5.3UnimodalityofBinomialCoefficients

5.4TheMultinomialTheorem

5.5NewtonsBinomialTheorem

5.6MoreonPartiallyOrderedSets

5.7Exercises

6TheInclusion-ExclusionPrincipleandApplications

6.1TheInclusion-ExclusionPrinciple

6.2CombinationswithRepetition

6.3Derangements

6.4PermutationswithForbiddenPositions

6.5AnotherForbiddenPositionProblem

6.6M6biusInversion

6.7Exercises

7RecurrenceRelationsandGeneratingFunctions

7.1SomeNumberSequences

7.2GeneratingFunctions

7.3ExponentialGeneratingFunctions

7.4SolvingLinearHomogeneousRecurrenceRelations

7.5NonhomogeneousRecurrenceRelations

7.6AGeometryExample

7.7Exercises

8SpecialCountingSequences

8.1CatalanNumbers

8.2DifferenceSequencesandStirlingNumbers

8.3PartitionNumbers

8.4AGeometricProblem

8.5LatticePathsandSchr6derNumbers

8.6Exercises

9SystemsofDistinctRepresentatives

9.1GeneralProblemFormulation

9.2ExistenceofSDRs

9.3StableMarriages

9.4Exercises

10Combinatorial.Designs

10.1ModularArithmetic

10.2BlockDesigns

10.3SteinerTripleSystems

10.4LatinSquares

10.5Exercises

11IntroductiontoGraphTheory

11.1BasicProperties

11.2EulerianTrails

11.3HamiltonPathsandCycles

11.4BipartiteMultigraphs

11.5Trees

11.6TheShannonSwitchingGame

11.7MoreonTrees

11.8Exercises

12MoreonGraphTheory

12.1Chromat,icNumber

12.2PlaneandPlanarGraphs

12.3AFive-ColorTheorem

12.4IndependenceNumberandCliqueNumber

12.5MatchingNumber

12.6Connectivity

12.7Exercises

13DigraphsandNetworks

13.1Digraphs

13.2Networks

13.3MatchingsinBipartiteGraphsRevisited

13.4Exercises

14PolyaCounting

14.1PermutationandSymmetryGroups

14.2BurnsidesTheorem

14.3PolyasCountingFormula

14.4Exercises

AnswersandHintstoExercises

Bibliography

Index

　　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。【作者简介】　　RichardA．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。