Fibonacci and Catalan Numbers – An Introduction

Grimaldi, R
Annet format / 2011 / Engelsk

Produktdetaljer

ISBN
9781118159743
Publisert
2011-12-28
Utgiver
Vendor
John Wiley & Sons Inc
Vekt
666 gr
Høyde
250 mm
Bredde
150 mm
Dybde
15 mm
Aldersnivå
P, 06
Språk
Product language
Engelsk
Format
Product format
Annet format
Antall sider
380

Forfatter
Grimaldi, R

Biographical note

RALPH P. GRIMALDI, PhD, is Professor of Mathematics at Rose-Hulman Institute of Technology. With more than forty years of experience in academia, Dr. Grimaldi has published numerous articles in discrete mathematics, combinatorics, and graph theory. Over the past twenty years, he has developed and led mini-courses and workshops examining the Fibonacci and the Catalan numbers.

Fibonacci and Catalan Numbers – An Introduction

Grimaldi, R
Annet format / 2011 / Engelsk
Grimaldi, R
Annet format / 2011 / Engelsk
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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.
Les mer
The material has been extensively class-tested for over ten years at both the author's own university and other institutions. The book is uniquely organized into two main sections, one on Fibonacci Numbers and one on Catalan Numbers,each containing subsections that explore related topics in intricate detail.
Les mer
Preface xi Part One. The Fibonacci Numbers 1. Historical Background 3 2. The Problem of the Rabbits 5 3. The Recursive Definition 7 4. Properties of the Fibonacci Numbers 8 5. Some Introductory Examples 13 6. Composition and Palindromes 23 7. Tilings: Divisibility Properties of the Fibonacci Numbers 33 8. Chess Pieces on Chessboards 40 9. Optics, Botany, and the Fibonacci Numbers 46 10. Solving Linear Recurrence Relations: The Binet Form for F n 51 11. More on alpha and beta: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65 12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79 13. The Lucas Numbers: Further Properties and Examples 100 14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113 15. The ged Property for the Fibonacci Numbers 121 16. Alternate Fibonacci Numbers 126 17. One Final Example? 140 Part Two. The Catalan Numbers 18. Historical Background 147 19. A First Example: A Formula for the Catalan Numbers 150 20. Some Further Initial Examples 159 21. Dyck Paths, Peaks, and Valleys 169 22. Young Tableaux, Compositions, and Vertices and Ares 183 23. Triangulating the Interior of a Convex Polygon 192 24. Some Examples from Graph Theory 195 25. Partial Orders, Total Orders, and Topological Sorting 205 26. Sequences and a Generating Tree 211 27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 219 28. The Catalan Numbers at Sporting Events 226 29. A Recurrence Relation for the Catalan Numbers 231 30. Triangulating the Interior of a Convex Polygon for the Second Time 236 31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 238 32. Staircases, Arrangements of Coins, Handshaking Problem, and Noncrossing Partitions 250 33. The Narayana Numbers 268 34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schroder Numbers 282 35. Generalized Catalan Numbers 290 36. One Final Example? 296 Solutions for the Odd-Numbered Exercises 301 Index 355
Les mer

Relaterte produkter

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.
Les mer
The material has been extensively class-tested for over ten years at both the author's own university and other institutions. The book is uniquely organized into two main sections, one on Fibonacci Numbers and one on Catalan Numbers,each containing subsections that explore related topics in intricate detail.
Les mer
Preface xi Part One. The Fibonacci Numbers 1. Historical Background 3 2. The Problem of the Rabbits 5 3. The Recursive Definition 7 4. Properties of the Fibonacci Numbers 8 5. Some Introductory Examples 13 6. Composition and Palindromes 23 7. Tilings: Divisibility Properties of the Fibonacci Numbers 33 8. Chess Pieces on Chessboards 40 9. Optics, Botany, and the Fibonacci Numbers 46 10. Solving Linear Recurrence Relations: The Binet Form for F n 51 11. More on alpha and beta: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65 12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79 13. The Lucas Numbers: Further Properties and Examples 100 14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113 15. The ged Property for the Fibonacci Numbers 121 16. Alternate Fibonacci Numbers 126 17. One Final Example? 140 Part Two. The Catalan Numbers 18. Historical Background 147 19. A First Example: A Formula for the Catalan Numbers 150 20. Some Further Initial Examples 159 21. Dyck Paths, Peaks, and Valleys 169 22. Young Tableaux, Compositions, and Vertices and Ares 183 23. Triangulating the Interior of a Convex Polygon 192 24. Some Examples from Graph Theory 195 25. Partial Orders, Total Orders, and Topological Sorting 205 26. Sequences and a Generating Tree 211 27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 219 28. The Catalan Numbers at Sporting Events 226 29. A Recurrence Relation for the Catalan Numbers 231 30. Triangulating the Interior of a Convex Polygon for the Second Time 236 31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 238 32. Staircases, Arrangements of Coins, Handshaking Problem, and Noncrossing Partitions 250 33. The Narayana Numbers 268 34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schroder Numbers 282 35. Generalized Catalan Numbers 290 36. One Final Example? 296 Solutions for the Odd-Numbered Exercises 301 Index 355
Les mer

Produktdetaljer

ISBN
9781118159743
Publisert
2011-12-28
Utgiver
Vendor
John Wiley & Sons Inc
Vekt
666 gr
Høyde
250 mm
Bredde
150 mm
Dybde
15 mm
Aldersnivå
P, 06
Språk
Product language
Engelsk
Format
Product format
Annet format
Antall sider
380

Forfatter
Grimaldi, R

Biographical note

RALPH P. GRIMALDI, PhD, is Professor of Mathematics at Rose-Hulman Institute of Technology. With more than forty years of experience in academia, Dr. Grimaldi has published numerous articles in discrete mathematics, combinatorics, and graph theory. Over the past twenty years, he has developed and led mini-courses and workshops examining the Fibonacci and the Catalan numbers.

Relaterte produkter