An Introduction to Catalan Numbers

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 Zn+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

A short, compact yet comprehensive introduction to Catalan numbers and their applications Designed to be as accessible for students as possible Includes exercises with hints and solutions to help students gain a better grasp on the material

Autorentext

Steven Roman is currently an Emeritus Professor of Mathematics at the University of California. He is a prolific Springer author; some of his books include Field Theory, Advanced Linear Algebra, Introduction to Coding and Information Theory, and most recently Fundamentals of Group Theory.

Inhalt
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.