Fibonacci and Catalan Numbers: An Introduction
John Wiley & Sons, 21 Şub 2012 - 448 sayfa
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:
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.
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Compositions and Palindromes
Tilings Divisibility Properties of the Fibonacci Numbers
Optics Botany and the Fibonacci Numbers
More on α and β Applications in Trigonometry Physics
Examples from Graph Theory An Introduction to the Lucas Numbers
The Lucas Numbers Further Properties and Examples
Matrices The Inverse Tangent Function and an Infinite Sum113
The gcd Property for the Fibonacci Numbers
Young Tableaux Compositions and Vertices and Arcs
Triangulating the Interior of a Convex Polygon
Some Examples from Graph Theory
Partial Orders Total Orders and Topological Sorting
Sequences and a Generating Tree
Maximal Cliques a Computer Science Example and the Tennis Ball Problem
The Catalan Numbers at Sporting Events
A Recurrence Relation for the Catalan Numbers
Alternate Fibonacci Numbers
One Final Example?
The Catalan Numbers
A First Example A Formula for the Catalan Numbers
Some Further Initial Examples
Dyck Paths Peaks and Valleys