离散数学引论

Contents

1. Counting and Binomial Coefficients

1.1 Basic Principles

1.2 Factorials

1.3 Selections

1.4 Binomial Coefficients and Pascals Triangle

1.5 Selections with- --Repetitions

1.6 AUsefulMatrixInversion

2. Recurrence

2.1 Some Examples

2.2 The Auxiliary Equation Method

2.3 Generating Fhnctions

2.4 Derangements

2.5 Sorting Algorithms

2.6 Catalan Numbers

3. Introduction to Graphs

3.1 The Concept of a Graph

3.2 Paths in Graphs

3.3 Trees

3.4 Spanning Trees

3.5 Bipartite Graphs

3.t5 Planarity

3.7 Polyhedra.

4. Travelling Round a Graph

4 1 Hamiltonian Graphs

4.2 Planarity and Hamiltonian Graphs

4.3 The Travelling Salesman Problem

4.4 Gray Codes

4.5 EulerianDigraphs

5. Partitions and Colourings

5.1 Partitions of a Set

5.2 StirlingNumbers

5.3 Counting Functions

5.4 Vertex Colourings of Graphs

5.5 Edge Colourings of Graphs

6 The Inclusion-Exclusion Principle

6.1 The Principle

6.2 Counting Surjections

6.3 Counting Labelled Trees

6.4 Scrabble.

15.5 The MSnage Problem

7. Latin Squares and Halls Theorem.

7.1 Latin-Squares and -Orthogonality

7.2 Magic Squares

7.3 Systems of Distinct Representatives

7.4 From Latin Squares to Affine Planes

8 Schedules and 1-Factorisations

8.1 The Circle Method

8.2 Bipartite Tournaments and 1-Factorisations of Kn

8.3 Tournaments from Orthogonal Latin Squares

9. Introduction to Designs.

9.1 Balanced Incomplete Block Designs

0.2 Resolvable Designs

0.3 Finite Projective Planes

0.4 Hadamard Matrices and Designs

0.15 Difference Methods

9.15 Hadamard Matrices and Codes

Appendix

Solutions

Further Reading

Bibliography

Index

《离散数学引论》以简洁和通俗的形式介绍组合数学的一些本质性内容图论的重要问题，计数方法和试验设计，其中图论约占一半篇幅。《离散数学引论》很适于和中国中学数学教材的内容相衔接，阅读《离散数学引论》所需的预备知识只是中学数学（唯一的例外是在图论中需要矩阵的描述方式，但即使没有学过线性代数，也是可以接受的）。 书中有大量习题和例题，习题附有部分解答和提示，适于自学。《离散数学引论》可用作数学、计算机科学、信息科学等专业大学本科生的组合数学教材，可在大学一年级讲授。