An Introduction to Catalan Numbers

ISBN-13: 9783319221434 / Angielski / Twarda / 2015 / 121 str.

Steven Roman

Opis

This textbook provides an introduction to the Catalan numbers and their remarkable properties, along with their various applications in combinatorics. Intended to be accessible to students new to the subject, the book begins with more elementary topics before progressing to more mathematically sophisticated topics. Each chapter focuses on a specific combinatorial object counted by these numbers, including paths, trees, tilings of a staircase, null sums in Z_n+1, interval structures, partitions, permutations, semiorders, and more. Exercises are included at the end of book, along with hints and solutions, to help students obtain a better grasp of the material. The text is ideal for undergraduate students studying combinatorics, but will also appeal to anyone with a mathematical background who has an interest in learning about the Catalan numbers."Roman does an admirable job of providing an introduction to Catalan numbers of a different nature from the previous ones. He has made an excellent choice of topics in order to convey the flavor of Catalan combinatorics. Readers] will acquire a good feeling for why so many mathematicians are enthralled by the remarkable ubiquity and elegance of Catalan numbers." - From the foreword by Richard Stanley

Recenzje

"The pace of this book is of an introductory nature, the coverage of Catalan numbers is rigorous and will provide the reader with a firm grasp of many of the properties of these numbers. ... a scholarly work and one that number theorists will find well worth reading." (James Van Speybroeck, Computing Reviews, April, 2016)

"The book is supplemented by a set of exercises for those who want to go further. The reader is helped by 70 carefully designed figures throughout the book. The 24 enumeration problems are selected carefully to show Catalan numbers from very different viewpoints. Several of these problems likely have their most readable write-up in this book. A must for anyone, who wants to understand the significance of Catalan numbers!" (László Székely, zbMATH 1342.05002, 2016)

Spis treści

Introduction.- Dyck Words.- The Catalan Numbers.- Catalan Numbers and Paths.- Catalan Numbers and Trees.- Catalan Numbers and Geometric Widgits.- Catalan Numbers and Algebraic Widgits.- Catalan Numbers and Interval Structures.- Catalan Numbers and Partitions.- Catalan Numbers and Permutations.- Catalan Numbers and Semiorders.- Exercises.- Solutions and Hints.- Appendix A: A Brief Introduction to Partially Ordered Sets.- Appendix B: A Brief Introduction to Graphs and Trees.- Index.

O autorze

Steven Roman is Professor Emeritus of Mathematics at California State University Fullerton.  He is the author of numerous other mathematics textbooks, including Field Theory (2006), Advanced Linear Algebra(2008), Fundamentals of Group Theory (2012), Introduction to the Mathematics of Finance (2012), and An Introduction to Catalan Numbers (2015).