Description
Book SynopsisHow many ways do exist to mix different ingredients, how many chances to win a gambling game, how many possible paths going from one place to another in a network ? To this kind of questions Mathematics applied to computer gives a stimulating and exhaustive answer. This text, presented in three parts (Combinatorics, Probability, Graphs) addresses all those who wish to acquire basic or advanced knowledge in combinatorial theories. It is actually also used as a textbook.
Basic and advanced theoretical elements are presented through simple applications like the Sudoku game, search engine algorithm and other easy to grasp applications. Through the progression from simple to complex, the teacher acquires knowledge of the state of the art of combinatorial theory. The non conventional simultaneous presentation of algorithms, programs and theory permits a powerful mixture of theory and practice.
All in all, the originality of this approach gives a refreshing view on combinatorial theory.
Trade Review"On the other hand if you are looking for an approach to combinatorics that is routed in applications and with lots of exercises then this is the book for you. Yes, dare I say it, it's fun." (I Programmer, 21 January 2011)
Table of ContentsGeneral Introduction xxiii
Chapter 1. Some Historical Elements 1
PART 1. COMBINATORICS 17
Part 1. Introduction 19
Chapter 2. Arrangements and Combinations 21
Chapter 3. Enumerations in Alphabetical Order 43
Chapter 4. Enumeration by Tree Structures 63
Chapter 5. Languages, Generating Functions and Recurrences 85
Chapter 6. Routes in a Square Grid 105
Chapter 7. Arrangements and Combinations with Repetitions 119
Chapter 8. Sieve Formula 137
Chapter 9. Mountain Ranges or Parenthesis Words: Catalan Numbers 165
Chapter 10. Other Mountain Ranges 197
Chapter 11. Some Applications of Catalan Numbers and Parenthesis Words 215
Chapter 12. Burnside’s Formula 227
Chapter 13. Matrices and Circulation on a Graph 253
Chapter 14. Parts and Partitions of a Set 275
Chapter 15. Partitions of a Number 289
Chapter 16. Flags 305
Chapter 17. Walls and Stacks 315
Chapter 18. Tiling of Rectangular Surfaces using Simple Shapes 331
Chapter 19. Permutations 345
PART 2. PROBABILITY 387
Part 2. Introduction 389
Chapter 20. Reminders about Discrete Probabilities 395
Chapter 21. Chance and the Computer 427
Chapter 22. Discrete and Continuous 447
Chapter 23. Generating Function Associated with a Discrete Random Variable in a Game 469
Chapter 24. Graphs and Matrices for Dealing with Probability Problems 497
Chapter 25. Repeated Games of Heads or Tails 509
Chapter 26. Random Routes on a Graph 535
Chapter 27. Repetitive Draws until the Outcome of a Certain Pattern 565
Chapter 28. Probability Exercises 597
PART 3. GRAPHS 637
Part 3. Introduction 639
Chapter 29. Graphs and Routes 643
Chapter 30. Explorations in Graphs 661
Chapter 31. Trees with Numbered Nodes, Cayley’s Theorem and Prüfer Code 705
Chapter 32. Binary Trees 723
Chapter 33. Weighted Graphs: Shortest Paths and Minimum Spanning Tree 737
Chapter 34. Eulerian Paths and Cycles, Spanning Trees of a Graph 759
Chapter 35. Enumeration of Spanning Trees of an Undirected Graph 779
Chapter 36. Enumeration of Eulerian Paths in Undirected Graphs 799
Chapter 37. Hamiltonian Paths and Circuits 835
APPENDICES 867
Appendix 1. Matrices 869
Appendix 2. Determinants and Route Combinatorics 885
Bibliography 907
Index 911
