Combinatorics
Free

Combinatorics

By Unknown
Free
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
  • 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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