Combinatorics

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Contents

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Chapter 1. What is Combinatorics?

1A. Enumeration

1B. Graph Theory

1C. Ramsey Theory

1D. Design Theory

1E. Coding Theory

Summary

Part I. Enumeration

Chapter 2. Basic Counting Techniques

2A. The product rule

2B. The sum rule

2C. Putting them together

2D. Summing up

Summary

Chapter 3. Permutations, Combinations, and the Binomial Theorem

3A. Permutations

3B. Combinations

3C. The Binomial Theorem

Summary

Chapter 4. Bijections and Combinatorial Proofs

4A. Counting via bijections

4B. Combinatorial proofs

Summary

Chapter 5. Counting with Repetitions

5A. Unlimited repetition

5B. Sorting a set that contains repetition

Summary

Chapter 6. Induction and Recursion

6A. Recursively-defined sequences

6B. Basic induction

6C. More advanced induction

Summary

Chapter 7. Generating Functions

7A. What is a generating function?

7B. The Generalised Binomial Theorem

7C. Using generating functions to count things

Summary

Chapter 8. Generating Functions and Recursion

8A. Partial fractions

8B. Factoring polynomials

8C. Using generating functions to solve recursively-defined sequences

Summary

Chapter 9. Some Important Recursively-Defined Sequences

9A. Derangements

9B. Catalan numbers

9C. Bell numbers and exponential generating functions

Summary

Chapter 10. Other Basic Counting Techniques

10A. The Pigeonhole Principle

10B. Inclusion-Exclusion

Summary

Part II. Graph Theory

Chapter 11. Basics of Graph Theory

11A. Background

11B. Basic definitions, terminology, and notation

11C. Deletion, complete graphs, and the Handshaking Lemma

11D. Graph isomorphisms

Summary

Chapter 12. Moving through graphs

12A. Directed graphs

12B. Walks and connectedness

12C. Paths and cycles

12D. Trees

Summary

Chapter 13. Euler and Hamilton

13A. Euler tours and trails

13B. Hamilton paths and cycles

Summary

Chapter 14. Graph Colouring

14A. Edge colouring

14B. Ramsey Theory

14C. Vertex colouring

Summary

Chapter 15. Planar graphs

15A. Planar graphs

15B. Euler's Formula

15C. Map colouring

Summary

Part III. Design Theory

Chapter 16. Latin squares

16A. Latin squares and Sudokus

16B. Mutually orthogonal Latin squares (MOLS)

16C. Systems of distinct representatives

Summary

Chapter 17. Designs

17A. Balanced Incomplete Block Designs (BIBD)

17B. Constructing designs, and existence of designs

17C. Fisher's Inequality

Summary

Chapter 18. More designs

18A. Steiner and Kirkman triple systems

18B. t-designs

18C. Affine planes

18D. Projective planes

Summary

Chapter 19. Designs and Codes

19A. Introduction

19B. Error-correcting codes

19C. Using the generator matrix for encoding

19D. Using the parity-check matrix for decoding

19E. Codes from designs

Summary

Index

List of Notation

Appendix A. Solutions to selected exercises

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