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Fibonacci and Catalan Numbers: An Introduction

Book Description

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.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Dedication
  5. Preface
  6. Part One: The Fibonacci Numbers
    1. Chapter 1: Historical Background
    2. Chapter 2: The Problem of the Rabbits
    3. Chapter 3: The Recursive Definition
    4. Chapter 4: Properties of the Fibonacci Numbers
    5. Chapter 5: Some Introductory Examples
    6. Chapter 6: Compositions and Palindromes
    7. Chapter 7: Tilings: Divisibility Properties of the Fibonacci Numbers
    8. Chapter 8: Chess Pieces on Chessboards
    9. Chapter 9: Optics, Botany, and the Fibonacci Numbers
      1. Exercise for Chapter 9
    10. Chapter 10: Solving Linear Recurrence Relations: The Binet Form for Fn
    11. Chapter 11: More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science
    12. Chapter 12: Examples from Graph Theory: An Introduction to the Lucas Numbers
    13. Chapter 13: The Lucas Numbers: Further Properties and Examples
    14. Chapter 14: Matrices, The Inverse Tangent Function, and an Infinite Sum
    15. Chapter 15: The gcd Property for the Fibonacci Numbers
    16. Chapter 16: Alternate Fibonacci Numbers
    17. Chapter 17: One Final Example?
      1. References
  7. Part Two: The Catalan Numbers
    1. Chapter 18: Historical Background
    2. Chapter 19: A First Example: A Formula for the Catalan Numbers
    3. Chapter 20: Some Further Initial Examples
    4. Chapter 21: Dyck Paths, Peaks, and Valleys
    5. Chapter 22: Young Tableaux, Compositions, and Vertices and Arcs
    6. Chapter 23: Triangulating the Interior of a Convex Polygon
    7. Chapter 24: Some Examples from Graph Theory
    8. Chapter 25: Partial Orders, Total Orders, and Topological Sorting
    9. Chapter 26: Sequences and a Generating Tree
    10. Chapter 27: Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem
    11. Chapter 28: The Catalan Numbers at Sporting Events
    12. Chapter 29: A Recurrence Relation for the Catalan Numbers
    13. Chapter 30: Triangulating the Interior of a Convex Polygon for the Second Time
    14. Chapter 31: Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures
    15. Chapter 32: Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions
    16. Chapter 33: The Narayana Numbers
    17. Chapter 34: Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers
    18. Chapter 35: Generalized Catalan Numbers
    19. Chapter 36: One Final Example?
      1. References
  8. Solutions for the Odd-Numbered Exercises
    1. Part One: The Fibonacci Numbers
    2. Part Two: The Catalan Numbers
  9. Index