Fibonacci and Catalan Numbers : An Introduction.
Title:
Fibonacci and Catalan Numbers : An Introduction.
Author:
Grimaldi, Ralph.
ISBN:
9781118159774
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (382 pages)
Contents:
FIBONACCI AND CATALAN NUMBERS: AN INTRODUCTION -- CONTENTS -- PREFACE -- PART ONE: THE FIBONACCI NUMBERS -- 1. Historical Background -- 2. The Problem of the Rabbits -- 3. The Recursive Definition -- 4. Properties of the Fibonacci Numbers -- 5. Some Introductory Examples -- 6. Compositions and Palindromes -- 7. Tilings: Divisibility Properties of the Fibonacci Numbers -- 8. Chess Pieces on Chessboards -- 9. Optics, Botany, and the Fibonacci Numbers -- 10. Solving Linear Recurrence Relations: The Binet Form for Fn -- 11. More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science -- 12. Examples fromGraph Theory:AnIntroduction to the Lucas Numbers -- 13. The Lucas Numbers: Further Properties and Examples -- 14. Matrices, The Inverse Tangent Function, and an Infinite Sum -- 15. The gcd Property for the Fibonacci Numbers -- 16. Alternate Fibonacci Numbers -- 17. One Final Example? -- PART TWO: THE CATALAN NUMBERS -- 18. Historical Background -- 19. A First Example: A Formula for the Catalan Numbers -- 20. Some Further Initial Examples -- 21. Dyck Paths, Peaks, and Valleys -- 22. Young Tableaux, Compositions, and Vertices and Arcs -- 23. Triangulating the Interior of a Convex Polygon -- 24. Some Examples from Graph Theory -- 25. Partial Orders, Total Orders, and Topological Sorting -- 26. Sequences and a Generating Tree -- 27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem -- 28. The Catalan Numbers at Sporting Events -- 29. A Recurrence Relation for the Catalan Numbers -- 30. Triangulating the Interior of a Convex Polygon for the Second Time -- 31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures -- 32. Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions -- 33. The Narayana Numbers.
34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers -- 35. Generalized Catalan Numbers -- 36. One Final Example? -- Solutions for the Odd-Numbered Exercises -- Index.
Abstract:
Discover the properties and real-world applications of the Fibonacci and the Catalan numbers With clear explanations and easy-to-follow examples, Fibonacci and Catalan Numbers: An Introduction offers a fascinating overview of these topics that is accessible to a broad range of readers. Beginning with a historical development of each topic, the book guides readers through the essential properties of the Fibonacci numbers, offering many introductory-level examples. The author explains the relationship of the Fibonacci numbers to compositions and palindromes, tilings, graph theory, and the Lucas numbers. The book proceeds to explore the Catalan numbers, with the author drawing from their history to provide a solid foundation of the underlying properties. The relationship of the Catalan numbers to various concepts is then presented in examples dealing with partial orders, total orders, topological sorting, graph theory, rooted-ordered binary trees, pattern avoidance, and the Narayana numbers. The book features various aids and insights that allow readers to develop a complete understanding of the presented topics, including: Real-world examples that demonstrate the application of the Fibonacci and the Catalan numbers to such fields as sports, botany, chemistry, physics, and computer science More than 300 exercises that enable readers to explore many of the presented examples in greater depth Illustrations that clarify and simplify the concepts Fibonacci and Catalan Numbers is an excellent book for courses on discrete mathematics, combinatorics, and number theory, especially at the undergraduate level. Undergraduates will find the book to be an excellent source for independent study, as well as a source of topics for research. Further, a great deal of the material can also be used for enrichment in high school courses.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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Electronic Access:
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