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# Combinatorics

By Unknown

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Book Description

Table of Contents

- 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

- Chapter 2. Basic Counting Techniques
- 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

- Chapter 11. Basics of Graph Theory
- 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

- Chapter 16. Latin squares

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