Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Chapter 1: Basic Counting Methods

1.1 The multiplication principle

1.2 Permutations

1.3 Combinations

1.4 Binomial coefficient identities

1.5 Distributions

1.6 The principle of inclusion and exclusion

1.7 Fibonacci numbers

1.8 Linear recurrence relations

1.9 Special recurrence relations

1.10 Counting and number theory

Notes

Chapter 2: Generating Functions

2.1 Rational generating functions

2.2 Special generating functions

2.3 Partition numbers

2.4 Labeled and unlabeled sets

2.5 Counting with symmetry

2.6 Cycle indexes

2.7 Pólya’s theorem

2.8 The number of graphs

2.9 Symmetries in domain and range

2.10 Asymmetric graphs

Notes

Chapter 3: The Pigeonhole Principle

3.1 The principle

3.2 The lattice point problem and SET®

3.3 Graphs

3.4 Colorings of the plane

3.5 Sequences and partial orders

3.6 Subsets

Notes

Chapter 4: Ramsey Theory

4.1 Ramsey’s theorem

4.2 Generalizations of Ramsey’s theorem

4.3 Ramsey numbers, bounds, and asymptotics

4.4 The probabilistic method

4.5 Schur’s theorem

4.6 Van der Waerden’s theorem

Notes

Chapter 5: Error-Correcting Codes

5.1 Binary codes

5.2 Perfect codes

5.3 Hamming codes

5.4 The Fano Configuration

Notes

Chapter 6: Combinatorial Designs

6.1 t-designs

6.2 Block designs

6.3 Projective planes

6.4 Latin squares

6.5 MOLS and OODs

6.6 Hadamard matrices

6.7 The Golay code and S(5, 8, 24)

6.8 Lattices and sphere packings

6.9 Leech’s lattice

Notes

Appendix A: Web Resources

Appendix B: Notation

Exercise Solutions

References

Index

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