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A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory (Second Edition), by Miklos Bona
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This is a textbook for an introductory combinatorics course that can take up one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course. Just as with the first edition, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible for the talented and hard-working undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings and Eulerian and Hamiltonian cycles. The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, and algorithms and complexity.As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.
- Sales Rank: #2465067 in Books
- Brand: Brand: Wspc
- Published on: 2006-10-09
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x 1.11" w x 6.22" l, 1.78 pounds
- Binding: Paperback
- 492 pages
- Used Book in Good Condition
Review
"This is a very attractive textbook on combinatorics ... A special feature of this book is the extensive list of interesting exercises with complete solutions."
"Miklos Bona's book is the best introductory combinatorics book that I have ever seen. It is extremely lively yet mathematically accurate, and the writing is lucid and very entertaining at the same time." -- Doron Zeilberger
"The strong points of the book are in particular a very inviting style of exposition, in which developments are always well motivated and well illustrated by numerous examples, and the long list of exercises at the end of each chapter, with detailed solutions ... This is very pleasant and instructive reading."
Plentiful examples .... Bona does a supreme job of walking us through combinatorics -- Choice "Choice"
From the Inside Flap
This is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.
Just as with the first two editions, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible to the talented and hardworking undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings, Eulerian and Hamiltonian cycles, and planar graphs.
The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs (new to this edition), enumeration under group action (new to this edition), generating functions of labeled and unlabeled structures and algorithms and complexity.
As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.
About the Author
Miklos Bona received his PhD at Massachusetts Institute of Technology. He is a Professor of Mathematics at the University of Florida, where he has been inducted into the Academy of Distinguished Teaching Scholars. His research has been supported by the National Science Foundation, the National Security Agency, and the Howard Hughes Medical Institute. Miklos Bona has received teaching awards at the University of Florida and at the University of Pennsylvania. He is one of the Editor-in-Chiefs of the Electronic Journal of Combinatorics.
Most helpful customer reviews
1 of 1 people found the following review helpful.
Great book for undergrad level of combinatoric
By Ha Luu
I love this book. It's very interesting through out the book. Each section goes from very basic stuff to advance stuff which makes it's easy for you to follow.
Each chapter of the book, there are about 20+ problems with detail solutions in the book (Yes, I am not kidding-SOLUTION). It's perfect for people who want to practice and check their work or finding some hint when you are stuck.
It's a great way to learn how to solve combinatoric problems.
Moreover, it also have the supplementary problems ( this one with no solution), for people who want to challenge themselves more.
It's a great book for those who are new to combinatoric and want to find their way in. And it has so many difficult levels that you won't be bored. Recommended!!!
29 of 30 people found the following review helpful.
A Stroll Through the Old and New
By Christopher Frenzen
Combinatorics often, but not always, involves finite sets, and the ideas of counting. But the subject of combinatorics has indeed become very large, and it has worked its way into many others parts of mathematics, computer science, science, and engineering. Bona's book, `A Walk Through Combinatorics', is a text designed for an introductory course in combinatorics. It covers the traditional areas of combinatorics like enumeration and graph theory, but also makes a real effort to introduce some more sophisticated ideas in combinatorics like Ramsey Theory and the probabilistic method.
The book is very exciting to read, and the author has a wonderful sense of humor: in Chapter 3 he introduces the idea of a permutation by the example of n people arriving at a dentist's office at the same time. They must decide the order in which they will be served. How many orders are possible?
The problems are a great strength of this text. Each chapter ends with a set of exercises with solutions. These tend to be very interesting and often quite challenging. A set of supplementary exercises follows. These tend to be a little easier, though not always, and make good homework assignments. The supplementary exercises do not have solutions, but a solutions manual is available to instructors.
The book walks through four parts: I. Basic Methods; II. Enumerative Combinatorics; III. Graph Theory; IV. Horizons. I particularly like the fourth part which includes Ramsey Theory, subsequence conditions on permutations, the probabilistic method, and partial orders and lattices. A glimpse of these subjects can whet the walker's appetite for more challenging terrain.
I would have liked to give this book 5 stars, but it suffers from a lack of clarity in some places. For example, the discussion of example 2.2 in Chapter 2 on induction just does not read clearly or make sense as it is written. Though an instructor can figure out what is missing, it would be much harder for a student to do so. And figure 13.1 on the colors of the edge of a triangle in Chapter 13 on Ramsey Theory is mislabeled. Again, this could steer an unwary student off the path of understanding. But these defects are minor compared to the riches contained in this text. The author has chosen his subjects carefully, illustrated them well and provided a wealth of wonderful exercises. And he has given the reader a glimpse of some of the less traditional and newer areas of combinatorics at the end of the book.
6 of 7 people found the following review helpful.
Concise, but not thorough
By Jon Torodash
If looking to improve your combinatorial problem solving or better grasp the scope of the field, "A Walk Through Combinatorics" is a slim, suitable solution. It is not, however, a full primer. Although it acquaints the reader with several topics, it seems little concerned with their unified development. For example, partitions and generating functions share an important connection not discussed. Induction is presented without elucidating strong induction or the well-ordered principle.
The book does not use a sequential approach, equally to its credit as its criticism as a learning tool. Of course, some chapters cohere out of necessity (e.g. binomial identities and compositions come after counting principles) but you can largely jump in wherever you wish, which is good for quickly gaining a working definition with an sample application or two. But there are also numerous unexpected forward references, which is hard to forgive in a mathematics textbook. The closing chapter section on computability however, is an appreciated capstone.
Practice problems lie mostly at what the intended audience would probably call the "challenging" level. This is fine, because Bona has meticulously worked out solutions for many of the problems in each chapter. I have not taken a precise count (ironically) but solutions appear to comprise at least half of the book's pages. If you can understand the explanatory parts of the chapter and the brief examples given, taking the time to work through the practice sets will reinforce the foundation well.
One thing I will never understand is why discrete math textbooks insist on a probability unit, Bayesian or otherwise.
Take the title seriously. This is a "walk" through the subject, and you don't get a chance to sit and analyze in one spot on a gentle pedestrian stroll: the major landmarks are tagged with introductory information, and you get to see a lot of them.
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