组合数学

Preface1 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 Exercises2 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 Exercises3 The Pigeonhole Principle 3.1 Pigeonhole Principle: Simple Form 3.2 Pigeonhole Principle: Strong Form 3.3 A Theorem of Ramsey 3.4 Exercises4 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 Exercises5 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 Exercises6 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 Exercises7 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 Exercises8 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 Exercises9 Systems of Distinct Representatives 9.1 General Problem Formulation 9.2 Existence of SDRs 9.3 Stable Marriages 9.4 Exercises10 Combinatorial .Designs 10.1 Modular Arithmetic 10.2 Block Designs 10.3 Steiner Triple Systems 10.4 Latin Squares 10.5 Exercises11 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 Exercises12 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 Exercises13 Digraphs and Networks 13.1 Digraphs 13.2 Networks 13.3 Matchings in Bipartite Graphs Revisited 13.4 Exercises14 Polya Counting 14.1 Permutation and Symmetry Groups 14.2 Burnside's Theorem 14.3 Polya's Counting Formula 14.4 ExercisesAnswers and Hints to ExercisesBibliographyIndex

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