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.

• Preface xiPart One The Fibonacci Numbers1. Historical Background 32. The Problem of the Rabbits 53. The Recursive Definition 74. Properties of the Fibonacci Numbers 85. Some Introductory Examples 136. Compositions and Palindromes 237. Tilings: Divisibility Properties of the Fibonacci Numbers 338. Chess Pieces on Chessboards 409. Optics, Botany, and the Fibonacci Numbers 4610. Solving Linear Recurrence Relations: The Binet Form for Fn5111. More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 6512. Examples from Graph Theory: An Introduction to the Lucas Numbers 7913. The Lucas Numbers: Further Properties and Examples 10014. Matrices, The Inverse Tangent Function, and an Infinite Sum 11315. The gcd Property for the Fibonacci Numbers 12116. Alternate Fibonacci Numbers 12617. One Final Example? 140Part Two The Catalan Numbers18. Historical Background 14719. A First Example: A Formula for the Catalan Numbers 15020. Some Further Initial Examples 15921. Dyck Paths, Peaks, and Valleys 16922. Young Tableaux, Compositions, and Vertices and Arcs 18323. Triangulating the Interior of a Convex Polygon 19224. Some Examples from Graph Theory 19525. Partial Orders, Total Orders, and Topological Sorting 20526. Sequences and a Generating Tree 21127. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 21928. The Catalan Numbers at Sporting Events 22629. A Recurrence Relation for the Catalan Numbers 23130. Triangulating the Interior of a Convex Polygon for the Second Time 23631. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 23832. Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions 25033. The Narayana Numbers 26834. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers 28235. Generalized Catalan Numbers 29036. One Final Example? 296Solutions for the Odd-Numbered Exercises 301Index 355

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