Combinatorics Through Guided Discovery

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Kenneth Bogart, Dartmouth College

Pub Date: 2004

ISBN 13:

Publisher: Independent

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Reviewed by Heath Hart, Instructor, Virginia Tech, on 6/11/2015.

In order to comment on the comprehensiveness of the book, I first have to describe the book's unusual pedagogical structure: the author poses a list … read more

 

Table of Contents

Chapter 1: What is Combinatorics?

Chapter 2: Applications of Induction and Recursion in Combinatorics and Graph Theory

Chapter 3: Distribution Problems

Chapter 4: Generating Functions

Chapter 5: The Principle of Inclusion and Exclusion

Chapter 6: Groups Acting on Sets

About the Book

This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially but not exclusively on the part of combinatorics that mathematicians refer to as “counting.” The book consists almost entirely of problems. Some of the problems are designed to lead you to think about a concept, others are designed to help you figure out a concept and state a theorem about it, while still others ask you to prove the theorem. Other problems give you a chance to use a theorem you have proved. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works. There are problems that some people will solve quickly, and there are problems that will take days of thought for everyone. Probably the best way to use this book is to work on a problem until you feel you are not making progress and then go on to the next one. Think about the problem you couldn’t get as you do other things. The next chance you get, discuss the problem you are stymied on with other members of the class. Often you will all feel you’ve hit dead ends, but when you begin comparing notes and listening carefully to each other, you will see more than one approach to the problem and be able to make some progress. In fact, after comparing notes you may realize that there is more than one way to interpret the problem. In this case your first step should be to think together about what the problem is actually asking you to do. You may have learned in school that for every problem you are given, there is a method that has already been taught to you, and you are supposed to figure out which method applies and apply it. That is not the case here. Based on some simplified examples, you will discover the method for yourself. Later on, you may recognize a pattern that suggests you should try to use this method again.

About the Contributors

Author(s)

Kenneth P. Bogart arrived at Dartmouth in 1968 after receiving his Ph.D. at the California Institute of Technology in that year. At the time of his death in 2005, Ken was in California on a sabbatical and working to complete revisions on his books, Introductory Combinatorics and Discrete Mathematics in Computer Science, while continuing his research on graph theory and partially ordered sets. During his career, Ken published nine books and over 60 articles. His many years of service to Dartmouth were marked by a dedication to teaching, which included participation in Math Across the Curriculum and his own grant for Teaching Introductory Combinatorics by Guided Discovery.