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# Combinatorics

By Unknown
Free
Book Description
• 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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