Book Description
Updated with new material, this Fifth Edition of the most widely used book in combinatorial problems explains how to reason and model combinatorically.  It also stresses the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity. Combinatorical reasoning underlies all analysis of computer systems. It plays a similar role in discrete operations research problems and in finite probability. This book seeks to develop proficiency in basic discrete math problem solving in the way that a calculus text develops proficiency in basic analysis problem solving. ... Read more

Customer Reviews (9)

not impressed
I am a math grad student taking graph theory this semester.As a math major, I understand that one should have the ability to "fill in the holes". You can overdo anything, however.This book is ruined by its lack of examples.Also, it is like the author is in a hurry when he is talking about key ideas.Definitions are often stated in a rushed way that confuses me.Yet, he rambles on when discussing less relevant things.In short, this is a very hard text to read.Easy exercises seem hard because little foundation has been laid.I only paid $17 for this book on amazon.I could not imagine paying $100 for it.

I will admit that this book is a good source for exercises.Also, the proofs are fairly readable, provided you can grasp the "under-explained" key concepts that are less than readable.

Haphazard Applied Combinatorics is more like it...
Mr. Tucker has, in his mundane brilliance, decided that all college professors would be able enough to fill in the gaps that he has blatantly left out in this book. The examples assume that the reader has actually been well versed on the subject prior to picking up this bound misfit, nor do they offer very detailed explanations on how he gets from point A to point B. It's the cut and dry "Here's the start, then you do this, and here's the answer" approach is very annoying, especially when your instructor is not the greatest. Furthermore, the exercises listed in some chapters have little or no relevance to the examples the author presents prior. This is a poor author and a poor choice of colleges to choose this book to teach from.

Do not be mislead by the positive reviews; this book is mediocre
This book covers basically two topics: Graph Theory and Enumeration.

The things I liked about this book were challenging problems.This book will certainly be a great SOURCE of problems for an upper-level undergraduate course in graph theory or combinatorics.

However, there are too many shortcomings.The book does not cover topics in depth, and the definitions and theorems it gives are stated very precisely and not explained.Unless you have had an introductory course in graph theory or combinatorics, these definitions will take a lot of time to sink in and make intuitive sense.Several useful theorems are not presented at all, are subtly stated in the text, or are presented in some problem.

The next problem is that this book is riddled with errors.And these are more than just errors in the Answers section, of which there are many, but errors in the actual problems!Sometimes even errors in the proofs.Usually these are typographical errors or sometimes just flat out wrong answers.You can find an errata list on the author's site, but it is far from complete.

I assume the other reviewers did not thoroughly work through this book and did not notice the errors.It is inexcusable for a math textbook to have this many errors.It almost seems as though this book wasn't edited at all.It is truly poor.

Excellent for applications
The book covers the fundamentals of graph theory and combinatorics (enumeration) and is designed for first courses for undergraduates.

The material is presented in a clear, friendly manner. The sections are short and specific and the emphasis is on problem-solving. Many examples are provided and constitute the majority of the book's volume. Each section ends with 20-30 exercises with answers (not full solutions) at the end of the book.

The book is excellent for computer science and applied math majors looking for a clear, application-based introduction to combinatorics and graph theory. It is also excellent for self-study.

The book's main flaw is that the proofs are not rigorous and are sometimes more intuitive than mathematical. For pure math students looking to explore graph theory and combinatorics in a more rigorous manner, other books (e.g. Diestel, "Graph Theory") will serve that purpose better.

An almost ideal introduction book to combinatorics
There have been wonderfully written reviews of this book, but since this is really an excellent textbook, I am urged to praise again. Fully recommended.

This book is easily and clearly written; covers almost every important basic concept and technic in graph theory and enumerative combinatorics, with neatly selected and wonderfully organised exercises.

And I highly suggest the author give the references to those last exercises in every section, since each of them does lead into a theory. ... Read more

Book Description
This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions.The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm.Also covered are connections between symmetric functions and representation theory.An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference. ... Read more

Customer Reviews (5)

Very challenging, very deep
This is an excellent book on combinatorics, but it is quite difficult to understand--written for experts, not novices.The author often chooses a more general framework in which to present things, and this can make the material quite difficult to follow.But the rewards for the diligent reader are great.Occasionally I question how Stanley chooses to present a certain topic, but usually if I look closely enough, I see that there are deep reasons for his choice of notation or presentation.

Some of the material in this book is easier than others; some of it depends on earlier chapters, but some stands on its own.People interested in partially ordered sets and lattices may want to jump ahead to that chapter--much of this chapter stands on its own, and it is an excellent exposition of that topic, and I think somewhat easier to understand than the rest of the book.

The most precious thing about this book is that the author manages to provide several comprehensive frameworks for solving large classes of enumeration problems.Combinatorics seems a hodge-podge subject to many mathematicians, but Stanley manages to see it as a unified subject with a number of general theories and common techniques.This book is truly the only text I have ever read that has this perspective on the subject.

I would recommend this book only to someone who has a strong background in mathematics and wants a challenging text that can take them to a deeper level of understanding.Students of combinatorics may want to take this book out of the library and read the introductory pages; there are some particularly useful comments right at the beginning.As a final note, the exercises in this book are also helpful and of diverse difficulty levels--and Stanley classifies the exercises by their difficulty level.People who find this book difficult to follow may want still benefit from some of the easier exercises.Students wanting an easier-to-follow text might want to check out Cameron's "Combinatorics", or Wilf's "Generatingfunctionology".As a final note I would like to remark that this book is very reasonably priced, especially when you consider the wealth of material it contains.

A Masterpiece on Enumerative Combinatorics
I agree with the other reviewers.The book is a masterpiece on enumerative combinatorics.However, I am not so sure that it is a good book for a beginner.If you are a beginner, then you should read another book first, like John Riordan's book on "Combinatorial Analysis."Stanley's book is best suited for an advanced student who has a high level of mathematical mental maturity.The reason I say this is that in a few places Stanley's formalism, which is entirely appropriate for professional exposition, actually obscures the underlying simplicity of the mathematical ideas.We have all seen this in research papers, where a mathematician takes a trivial idea and "obsures" the underlying simplicity with too much formalism.However, for an advanced student, the book has a high density of important ideas and methods.

This is for people who likes to COUNT
Gosh! This is for people who count, what else does a combinatorist do? Before people dismiss me as somebody who don't know hoot about math: I took a class with Prof. Stanley (the author) in college, and I had actually used vol 1 as a text. The material is highbrow (I agree on the 'hardcore' math observation) but the main theme of the book is how to 'count' -- needless to say not in the sense of everyday counting, but in the sense that 'topology' is 'coffee-to-donut transformation' and 'analysis' is 'honors calculus'. You have to know how to count, and comfortable with combinatorial proof to actually learn from this. I like the fact that Prof. Stanley asks for combinatorial proof to some known results, marking them as unsolved -- he really elevates the status of combinatorial proof, a method many dismiss as 'handwaving'. There is a number given to each exercise, according to the level of difficulty: [1] for trivial, [5] unsolved. I saw a professor who worked in differential topology for 40 years refer to this book -- and first year undergrads thumbing through the pages for exercises marked [1] and [2] to solve in spare time. This is a book for all levels of mathematicians: I am sure even the armchair amateur mathematicians can grasp some of the materials after a hard day's thought. I dont see this book as any less than a definitive text on enumerative combinatiorics.

People who like to COUNT?!? People who like hard-core math.
There was an earier review that claimed this book is for "people who like to count." That's a little silly. This book is a rigorous math text. And it's glorious. It's probably my favorite text. But it's not light reading at all.

I spent a semester actively reading and working on this book with my advisor. I read this book and worked on research, 50/50 split on my time. I got through 2.5 of the 4 chapters, and I'm damn proud of myself. It's a great book, but if you didn't know that 'enumerative' was for "people who like to count", you probably want a different text.

A Classic!
This book is a must for anyone who likes how to count. In addition to the superb exposition of deep and important mathematics, it contains so many intriguing problems, some of them even puzzle-like. Read this bookcover-to-cover or open it at a random page. Either way you would love it! ... Read more

Book Description

Combinatorics deals with the enumeration, existence, analysis, and optimization of discrete structures. With this study guide, students can master this growing field--with applications in several physical and social sciences, including chemistry, computer science, operations research, and statistics. Includes hundreds of problems with detailed solutions.

Customer Reviews (3)

kind of good
Not bad for those that have become accustomed to extensive math language within a text. The underlying concepts are explained well, however the density of material does take something away. Graph and group theory explanations should be more comprehensive. Considering the complexity of the various topics being presented this book is kind of good.

Nice job
Combinatorics is an area of mathematics that is frequently looked on as one that is reserved for a small minority of mathematicians: die-hard individualists who shun the limelight and take on problems that most would find boring. In addition, it has been viewed as a part of mathematics that has not followed the trend toward axiomatization that has dominated mathematics in the last 150 years. It is however also a field that has taken on enormous importance in recent years do its applicability in network engineering, combinatorial optimization, coding theory, cryptography, integer programming, constraint satisfaction, and computational biology. In the study of toric varieties in algebraic geometry, combinatorics has had a tremendous influence. Indeed combinatorial constructions have helped give a wide variety of concrete examples of algebraic varieties in algebraic geometry, giving beginning students in this area much needed intuition and understanding.It is the the advent of the computer though that has had the greatest influence on combinatorics, and vice versa.The consideration of NP complete problems typically involves enumerative problems in graph theory, one example being the existance of a Hamiltonian cycle in a graph. The use of the computer as a tool for proof in combinatorics, such as the 4-color problem, is now legendary.In addition, several good software packages, such as GAP and Combinatorica, have recently appeared that are explicitly designed to do combinatorics. One fact that is most interesting to me about combinatorics is that it gave the first explicit example of a mathematical statement that is unprovable in Peano arithmetic. Before coming across this, I used to think the unprovable statements of Godel had no direct relevance for mathematics, but were only interesting from the standpoint of its foundations.

This book is an introduction to combinatorics for the undergraduate mathematics student and for those working in applications of combinatorics. As with all the other guides in the Schaums series on mathematics, this one has a plethora of many interesting examples and serves its purpose well. Readers who need a more in-depth view can move on to more advanced works after reading this one. The author dedicates this book to the famous mathematician Paul Erdos, who is considered the father of modern combinatorics, and is considered one of most prolific of modern mathematicians, with over 1500 papers to his credit.

The author defines combinatorics as the branch of mathematics that attempts to answer enumeration questions without considering all possible cases. The latter is possible by the use of two fundamental rules, namely the sum rule and the product rule. The practical implementation of these rules involves the determination of permutations and combinations, which are discussed in the first chapter, along with the famous pigeonhole principle. Most of this chapter can be read by someone with a background in a typical college algebra course. The author considers some interesting problems in the "Solved Problems" section, for example one- and two-dimensional binomial random walks, and problems dealing with Ramsey, Catalan, and Stirling numbers. The consideration of Ramsey numbers will lead the reader to several very difficult open problems in combinatorics involving their explicit values.

Generalized permutations and combinations are considered in chapter two, along with selections and the inclusion-exclusion principle. The author proves the Sieve formula and the Phillip Hall Marriage Theorem. In the "Solved Problems" section, the duality principle of distribution, familiar from integer programming is proved, and the author works several problems in combinatorial number theory. A reader working in the field of dynamical systems will appreciate the discussion of the Moebius function in this section. Particularly interesting in this section is the discussion on rook and hit polynomials.

The consideration of generating functions and recurrence relations dominates chapter 3, wherein the author considers the partition problem for positive integers. The first and second identities of Euler are proved in the "Solved Problems" section, and Bernoulli numbers, so important in physics, are discussed in terms of their exponential generating functions. The physicist reader working in statistical physics will appreciate the discussion on Vandermonde determinants. Applications to group theory appear in the discussion on the Young tableaux, preparing the reader for the next chapter.

A more detailed discussion of group theory in combinatorics is given in chapter 4, the last chapter of the book. The author proves the Burnside-Frobenius, the Polya enumeration theorems, and Cayley's theorem in the "Solved Problems" section. Readers without a background in group theory can still read this chapter since the author reviews in detail the basic constructions in group theory, both in the main text and in the "Solved Problems" section. Combinatorial techniques had a large role to play in the problem of the classification of finite simple groups, the eventual classification proof taking over 15,000 journal pages and involving a large collaboration of mathematicians. Combinatorics also made its presence known in the work of Richard Borchers on the "monstrous moonshine" that brought together ideas from mathematical physics and the largest simple group, called the monster simple group.

The author devotes an appendix to graph theory, which is good considering the enormous power of combinatorics to problems in graph theory and computational geometry. Even though the discussion is brief, he does a good job of summarizing the main results, including a graph-theoretic version of Dilworth's theorem. Combinatorial/graph-theoretic considerations are extremely important in network routing design and many of the techniques discussed in this appendix find their way into these kinds of applications. The author asks the reader to prove that Dilworths' theorem, the Ford-Fulkerson theorem, Hall's marriage theorem, Konig's theorem, and Menger's theorem are equivalent. A very useful glossary of the important definitions and concepts used in the book is inserted at the end of the book.

excellent book !
In its usual way schaum's series gives out another book which is both helpful yet concise. This book gives the essential grounding for combinatorics and graph theory without being overly gargantuan encyclopedia..ample problems set the tone for a future mathematician. theycould've done better though..hence not the perfect 5 ! ... Read more

Book Description
This is a textbook for an introductory combinatorics course that can take up one or two semesters. An extensive list of exercises, ranging in difficulty from "routine" to "worthy of independent publication", is included. In each section, there are also exercises that contain material not explicitly discussed in the text before, so as to provide instructors with extra choices if they want to shift the emphasis of their course.

It goes without saying that the text covers the classic areas, i.e. combinatorial choice problems and graph theory. What is unusual, for an undergraduate textbook, is that the author has included a number of more elaborate concepts, such as Ramsey theory, the probabilistic method and - probably the first of its kind - pattern avoidance. While the reader can only skim the surface of these areas, the author believes that they are interesting enough to catch the attention of some students. 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. ... Read more

Customer Reviews (3)

Encompassing and Very Clear
This book goes step by step on the elementary subjects of Combinatorics, contains many of examples and solved exercises, such that the reader or any autodidact student can experience a meaningful studying experience.

Well structured book
The best thing I like about this book, is that it has carefully selected subjects and rich set of exercises with detailed solutions. For the first several chapters, there are even more pages devoted to exercises+answers than the text. I think it is better to learn math by doing exercises than memorizing lots of theorems.

I would have given it 5 stars if there were not so many typos. It is annoying because a lot of times when I puzzled about something, it turns out be a typo. There are several versions of errata online, and none of them is complete. :) You can find them here:

http://www-math.mit.edu/~apost/courses/18.314/

I hope the author will correct all those typos then this would be the very best textbook!

A Stroll Through the Old and New
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. ... Read more

Book Description
Including many algorithms described in simple terms, this book stresses common techniques (such as generating functions and recursive construction) that underlie the great variety of subject matter. ... Read more

Customer Reviews (7)

Excellent textbook for researchers
The book is an excellent source of combinatorial insights and techniques for researchers, especially those who are not mathematicians. The book is comprehensive but not too dense. Puritans would complain that it skips details, but details can always be found by referring the bibliography. An excellent source of problems, with solutions for earlier versions provided by the author on his web-page. Should turn out to be a classic if not already one.

Sigmas all over the place
This isn't your usual "urn-has-3-red-balls-and-5-white-balls" sort of combinatorics book. It's sigma notation all over the place, if you know what I mean.
The first part can be used for undergraduates and the second part is more advanced. The book is broad in scope because, as the author explains, so is the subject matter.
The chapters have "techniques" and "algorithms." It's not a book that has a slew of examples of combinatorial problems (like so many), but leans toward mathematical sophistication in formalizing the techniques. This is either a feature or a bug, depending on what you needs are. For instance, it's not very often that introductory books present derrangements next to Fibonacci numbers. Or explain how calculate the average number of comparisons that Hoare's Quicksort does with a differential equation for the recurrence relation in the context of finite fields. It sounds scary, I know, but if you look at the explanation, you'll see you should have been born a nephew to this author.
In case you like Knuth's Concrete Mathematics you will like this book too (there's some overlap, because both are concerned with the analysis of algorithms). Knuth's book works more on skill-building, and I think Cameron's book is better for theoretical explanation.
Disclaimer: I haven't worked with the whole book (because of a lack of time - "Ars long, vita brevis", as they say).

Excellent book...very clear, well-organized
This is a graduate level text that presents advanced material and yet is easier to understand than most high school texts and could probably be used without trouble at the undergraduate level.The writing is vibrant and lucid; it is a pleasure to read.I could come up with a few minor complaints about the presentation of this or that but these comments would be silly and not very relevant.

The book contains an absolute wealth of topics.There is an interesting combinatorial approach to groups, and the book's presentation of certain topics, such as matroids and quasigroups, is among the best I have found; many books make these structures appear painfully abstract and difficult to grasp.The book is organized so that it's fairly easy to skip around, but I actually like the order in which the topics are presented.

This text makes an excellent addition to the collection of anyone interested in combinatorics, and if someone were to buy only one book on the subject, I would recommend this book.I think this would make an excellent textbook--it was used as such in one of my graduate courses, and would probably be suitable for an undergraduate course as well.

Greatbook for Computer Scientist
I am M.Sc.Computer Science student and work for software company. I needed a book covering aspects in Combinatorics and this is the book.

Very helpful
Combinatorics is a bit of an oddity. Although a few principles (like pigeonholing) apply in many cases, every combinatorial problem has unique features. Attacking a new situation is almost like starting all over again, unless you can recognize an old problem in your new one.

This book gives a number brief case studies. Its 18 chapters (not counting intro and closing) span a variety of interesting topics. Cameron doesn't write down to the reader - it takes serious thought and some mathematical background to get full value from the reading. The examples are nowhere near as concrete as you'd expect in a popularized version. Still, the author avoids opaque references to specialist terms, and keeps the text approachable.

I have personal reason to like this book more than it's high quality warrants. I was thumbing through it in a store, and skimmed a page that described Kirkman's schoolgirls (a two-level problem in selecting subsets). Quite abruptly, I realized that those charming young ladies exactly represented a problem I had in connecting the parts of a multiprocessor. One or two references later, I had a practical way out of a potentially ugly quandry. This material is not just fun for its own intellectual challenge, it has application to real engineering, too. ... Read more

Book Description
This book emphasizes combinatorial ideas including the pigeon-hole principle, counting techniques, permutations and combinations, Pólya counting, binomial coefficients, inclusion-exclusion principle, generating functions and recurrence relations, and combinatortial structures (matchings, designs, graphs).The volume provides a complete examination of combinatorial ideas and techniques.For individuals interested in combinatorial concepts. ... Read more

Customer Reviews (2)

nice accessable text
Most (but not all) of the copious errors in earlier editions have been fixed.
(Brualdi maintains an errata list on his website.)I like this book a lot,
it has a nice, relaxed style of exposition and the choice of topics is good
for an introductory course.

Interesting Problems, Too Many Mistakes
I used the book to guide me through a Combinatorics class I took in the summer of 1998. The author has presented some very interesting problems like prove that of any 10 points chosen withen an equlateral triangle ofside length 1, there are 2 whose distance apart is at most 1/3 that usesome interesting techniques such as the pigeonhole principal. The book,however contained too many mistakes. My professor said on average there isone mistake per page and he wasn't exagerating either.Luckily with hishelp, we corrected the many mistakes and then were successfully able to usethe book.I notice that the author has written a new edition.I hope mostof the mistakes have been corrected because when I pay a good sum of moneyfor a book I expect it to be good book without errors. ... Read more

Book Description
Combinatorics, a subject dealing with ways of arranging and distributing objects, involves ideas from geometry, algebra, and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become an essential tool in many scientific fields. In this second edition the authors have made the text as comprehensive as possible, dealing in a unified manner with such topics as graph theory, extremal problems, designs, colorings, and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. It is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level, and working mathematicians and scientists will also find it a valuable introduction and reference. ... Read more

Customer Reviews (5)

A real math book
I am a lover of combinatorics, and I have read quite a few on the topic. This one is as good as any. Lucidly written, you can pretty much dive into any chapter, reading, scribbling, racking your brain, and come away with a deep sense of satisfaction and pride and vanity:). Price is so resonable with regard for its extensive content. You get a feel that the author really wants to share with readers his love and joy for the subject and not just to make some money. Thank you, my dear professors!

Excellent book, but organized in a unorthodox and inconvenient manner
I think this is an excellent book but I have a few concerns about its organization.

The writing is very clear and there is a lot of explanation.Exercises are mixed in with the text, which I like very much; it makes them seem more natural, and it makes the book well-suited for self-study.I would say the difficulty level of this book is a bit inconsistent--but this is more a function of the material than of the writing style.The authors make everything as clear as possible, but they choose to include some difficult topics which require more thought.

My main criticism of this book is about the order of topics, which is not only unorthodox but can be inconvenient as well.Many concepts which are often presented earlier in combinatorics texts, such as binomial coefficients and stirling numbers, are relegated to later chapters, where their presentation depends on results from earlier chapters.I find it difficult to skip around in this book--if you do not read it from the beginning, in order, it will be hard to follow the arguments in some of the chapters.This sort of dependency is something I can accept in a more advanced text but I think is inappropriate for a text at this level.

I think this is an excellent book to add to your collection, but if you're going to grab only one or two books in combinatorics I would recommend other books.The organization issues I mentioned could make this book hard to use as a standalone text for a course if you did not wish to follow the same course of development chosen by the authors.Cameron's book is written at a similar level and covers a similar amount of material (although it has a very different style of presentation), and it is much easier to skip around in.Stanley's "Enumerative Combinatorics" is a denser, more advanced text that most will find more difficult to follow than this book, but it is still easier to skip around in as well.

A nice tour of combinatorics
The first word that comes to my mind when I think of this text is "encyclopedic". It contains around 40 chapters, hitting most of the high points of combinatorics that a graduate student should see. The exposition is generally good with nice examples. The one thing that I fault it for is the number of statements that the authors claim are "obvious". In a way, this is good, because it makes you pay attention and understand the material, but sometimes the statement isn't obvious until you've thought about it for an hour and written out a lengthy proof. At that point, it does become completely obvious and you can't believe that you ever thought it wasn't, so I can understand why van Lint and Wilson fell into the trap so often. (In fact, I've heard that Wilson even stumbles over some of those points in lectures.) This is a great book to have on your shelf if you need somewhere to look up combinatorial ideas.

A gentle introduction to combinatorics
This book was the text for a graduate-level course I took.The presentation is very laid-back, much like the lecturing style of one of the authors (Wilson), and so it was quite readable (unlike many other mathbooks which you have to stop every few pages and pick apart everythingbefore it sinks in).

Combinatorics is a relatively recent development inmathematics, one which is generally easy to explain, but with manydifficult open questions.Van Lint and Wilson do an excellent jobexplaining, but there are a few places where the reader needs to know somebackground to place the particular problem in the appropriate mathematicalcontext.Understandably, if the authors were to include all themathematical machinery needed, the book would be huge!Instead, they havechosen to describe as many facets of the field as possible, and thereforehave written a broad, well-balanced book which approaches the topic in anon-threatening way.

My one criticism, then, is that there is a lack ofdepth in several areas of the book, with further discussion of advancedtopics or open problems.But even so, I can appreciate the omission forthe sake of accessibility.

To fully appreciate the subject, the authorsare correct in mentioning that the book is written with the graduatestudent in mind.But by no means does the reader require such a backgroundto appreciate the remarkable concepts and the exciting questions revealedin this book.

Proof that you can't judge a book by its cover
The cover says, "...ideally suited for use as a text...at the advanced undergraduate or beginning graduate level." WRONG!!I'm a sixteen year old-- far from graduate school-- and I am reading,understanding, and LOVING this book.I cannot think of a greaterintroduction to combinatorics-- it has examples and problems to test yourcomprehension, and logical flow from one subject to another.This book isa rare find-- clear explanations and definitions at a fast pace thatdoesn't bore you.I would recommend this book unconditionally to ANYBODYinterested in mathematics. ... Read more

Product Description
Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. This volume is a compilation of expository articles at the interface between combinatorics and geometry, based on a three-week program of lectures at the Institute for Advanced Study/Park City Math Institute (IAS/PCMI) summer program on Geometric Combinatorics. The topics covered include posets, graphs, hyperplane arrangements, discrete Morse theory, and more. These objects are considered from multiple perspectives, such as in enumerative or topological contexts, or in the presence of discrete or continuous group actions.Most of the exposition is aimed at graduate students or researchers learning the material for the first time. Many of the articles include substantial numbers of exercises, and all include numerous examples. The reader is led quickly to the state of the art and current active research by worldwide authorities on their respective subjects.Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price. ... Read more

Book Description
The book is a concise, self-contained and up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called gems of the theory. A wide spectrum of most powerful combinatorial tools is presented: methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A throughout discussion of some recent applications to computer science motivates the liveliness and inherent usefulness of these methods to approach problems outside combinatorics. No special combinatorial or algebraic background is assumed. All necessary elements of linear algebra and discrete probability are introduced before their combinatorial applications. Aimed primarily as an introductory text for graduates, it provides also a compact source of modern extremal combinatorics for researchers in computer science and other fields of discrete mathematics. ... Read more

Book Description
Combinatorics on words, or finite sequences, is a field that grew from the disparate mathematics branches of group theory and probability. In recent times, it has gained recognition as an independent theory and has found substantial applications in computer science automata theory and linguistics. This volume is the first to present a thorough treatment of this theory and includes discussions of Thue's square free words, Van der Waerden's theorem, and Ramsey's theorem. This volume is an accessible text for undergraduate and graduate level students in mathematics and computer science as well as specialists in all branches of applied mathematics. ... Read more

Book Description

Customer Reviews (2)

certainly better than decent
This book provides excellent coverage of sperners theorem including multiple proofs ,like the original one by sperner and more concise proofs using closely related concepts. The various proofs of sperners theoremprovides a firm understanding of its connections with many other fundamental topics in finite combinatorial mathematics.Great book for those that have a good grasp on algebraic concepts.

An excellent and unique perspective on combinatorics
When one thinks of combinatorics of finite sets, he or she might first think of codes and designs. But this book introduced me to an area of combinatorics which I knew very little about, namely extremal set problems and their solutions which fall under famous Theorems by famous mathematicians: Erdos-Ko-Rado, Sperner, and Kruskal-Katona to name a few. I found these topics fascinating and fun to think about, which is in large part due to the author's coherent style, organization, explanation, and expertise of the subject-matter. Moreover, the author provided solutions to *every* one of the 150+ problems!!! How many math books can boast such a claim? Aside from may be a rough presentation of Lemma 4.3.2 the rest of the book is a masterpiece which I hope will gain more recognition within the next twenty years.

I highly recommend this book to both mathematicians and computer scientists. Although the book has very few "algorithms" in it, the thinking and reasoning about discrete structures (e.g. families of finite sets and multisets) will do wonders in developing the mind of a computer scientist, whether advanced or undergraduate. Yet it is quite sad that many cs departments (and math for that matter) invest little if any curriculum in discrete mathematics. Hopefully this will change at least to the point where the cs major will take two or three semesters of discrete math instead of two or three of calculus. For, as this book demonstrates, calculus is not a prerequisite for engaging one's mind in some quite fascinating mathematical problems related to finite sets.

Finally, it should be noted that Bela Bollobas also has an interesting book titled "Combinatorics: Set Systems, etc...." which significantly intersects with this book, but not to the degree where the reader should think they are interchangeable. I recommend both, and to read Anderson's book first; as I believe this book lays a better foundation than the latter. ... Read more

Customer Reviews (9)

Great Combi book.
Recommended by one of my professors.Fairly clear and useful when I need to clarify something gone over in class.

Good Text
This text is an excellent introduction to the subject, as it will give the reader a solid base of the main ideas in combinatorics.However, I rated it with four stars for two reasons.One is that the authors sometimes use nonconventional notation, another is that there are occasional typos.Other than that I highly recommend this book.

Hope this is a 1st edition error...
looking at the sample pages, on page 3, it said 5! = 60... I may not know anything about combinatorics, but I do know my factorials...

Superb textbook
We used this text for our introductory class in Combinatorics in graduate Mathematics at Louisville. Very well written with fantastic examples.

Excellent Book for a Graph Theory and Combinatorics Course
Simply stated, this is an excellent book. I used this book to teach an upper division undergraduate course in Graph Theory and Combinatorics. This book is by far the best book I have come across for teaching these topics.Incredibly the book contains very few errors (I've only found two errors(typos) ). In addition, the book has applications of the concepts for awide range of disciplines.

The author has a wide range of problems at theend of each section. Almost all of the problems are well written with cleardirections.

Every Computer Science/Mathematics major should have thisbook in their library. It's great! ... Read more

Product Description
This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the state polytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Gröbner bases of toric ideals and other methods from commutative algebra.The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes. ... Read more

Book Description
Combinatorics research, the branch of mathematics that deals with the study of discrete, usually finite, structures, covers a wide range of problems not only in mathematics but also in the biological sciences, engineering, and computer science. The Handbook of Combinatorics brings together almost every aspect of this enormous field and is destined to become a classic. Ronald L. Graham, Martin Grötschel, and László Lovász, three of the world's leading combinatorialists, have compiled a selection of articles that cover combinatorics in graph theory, theoretical computer science, optimization, and convexity theory, plus applications in operations research, electrical engineering, statistical mechanics, chemistry, molecular biology, pure mathematics, and computer science.

The 20 articles in Volume 1 deal with structures while the 24 articles in Volume 2 focus on aspects, tools, applications, and horizons. ... Read more

Customer Reviews (1)

Get your library to get this.
This is a collection of survey articles by various top class mathematicians about combinatorics, and the links the subject has with other branches of knowledge, from topology to biology. I have only read acouple of chapters in any detail and they were very useful. If you are incombinatorics, you must have access to this. The expense is worth it. ... Read more

Book Description
Focusing on the core material of value to students in a wide variety of fields, this book presents a broad comprehensive survey of modern combinatorics at an introductory level. The author begins with an introduction of concepts fundamental to all branches of combinatorics in the context of combinatorial enumeration. Chapter 2 is devoted to enumeration problems that involve counting the number of equivalence classes of an equivalence relation. Chapter 3 discusses somewhat less direct methods of enumeration, the principle of inclusion and exclusion and generating functions. The remainder of the book is devoted to a study of combinatorial structures. ... Read more

Customer Reviews (2)

Reasonable, but not exceptional text
Bogart has done a decent job with this text. It does a good job of covering a broad selection of topics and is generally readable. Beware the typographical errors, as they are often very subtle and leave you wondering if it was an error or if you just don't understand. Don't let Chapter 1 give you the wrong impression of the text, as it belabors things much more than the rest of the text does. It initally left a bad taste in my mouth, but the remainder of the book provides the right amount of exposition, except for a few points in Chapter 6 (Combinatorial Designs) where Bogart leaves too many proofs as exercises without giving a sufficient idea on how to complete the proof.

One of the most complete combinatorics books around...
I was amazed at the quality of the descriptions in this book.It has easy to understand examples and does a very good job of connecting combinatorics and graph theory.It even has integer partitions and Stirling numbers. ... Read more

Book Description
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate level textbook will allow students and researchers easy entry into this fascinating field. Here, for the first time, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results. ... Read more

Book Description
With examples of all 450 functions in action plus tutorial text on the mathematics, this book is the definitive guide to Experimenting with Combinatorica, a widely used software package for teaching and research in discrete mathematics. Three interesting classes of exercises are provided--theorem/proof, programming exercises, and experimental explorations--ensuring great flexibility in teaching and learning the material.The Combinatorica user community ranges from students to engineers, researchers in mathematics, computer science, physics, economics, and the humanities. Recipient of the EDUCOM Higher Education Software Award, Combinatorica is included with every copy of the popular computer algebra system Mathematica. ... Read more

Book Description

Customer Reviews (17)

Excellent book!
This book is very good. However, it's dense, so you'll have to parse it carefully and never in a hurry.

Well written
I bought this book because I wanted to have theory on linear programming including duality, integer linear programming, typical graph algorithms and matroid theory in one book. Up to now I have read only most of the chapter on matroids and I would like to say a big thanks to the author.

Although you will not solve the world's problems with greedy algorithms, my mathematical part of the heart was pleased and satisfied by the theory which explained the very nice relation between matroids and greedy algorithms.

Maybe I will tell you more in a few months

Combinatorial Optimization: Algorithms and Complexity
The book's state is very good, so I am satisfied with it.

A classic...
I won't lie to you: this book is well written but relatively hard to read. The subject is inherently difficult, after all! I highly suggest it, though, because the author is a recognized expert on the field and the price is relatively low. It's worth it even if you enjoy a few pages...

Mmm, algorithms....
This is a very nice, self-contained introduction to linear programming, algorithm design and analysis, and computational complexity. The contents are as follows:

Chap. 1 Optimization Problems 1.1 Introduction; 1.2 Optimization Problems;1.3 Neighborhoods; 1.4 Local and Global Optima; 1.5 Convex Sets and Functions; 1.6 Convex Programming Problems

Chap. 2 The Simplex Algorithm 2.1 Forms of the Linear Programming Problem; 2.2 Basic Feasible Solutions; 2.3 The Geometry of Linear Programs; 2.3.1 Linear and Affine Spaces; 2.3.2 Convex Polytopes; 2.3.3 Polytopes and LP; 2.4 Moving from bfs to bfs; 2.5 Organization of a Tableau; 2.6 Choosing a Profitable Column; 2.7 Degeneracy and Bland's Anticycling Algorithm; 2.8 Beginning the Simplex Algorithm; 2.9 Geometric Aspects of Pivoting

Chap. 3 Duality 3.1 The Dual of a Linear Program in General Form; 3.2 Complementary Slackness; 3.3 Farkas' Lemma; 3.4 The Shortest-Path Problem and Its Dual; 3.5 Dual Information in the Tableau; 3.6 The Dual Simplex Algorithm; 3.7 Interpretation of the Dual Simplex Algorithm

Chap. 4 Computational Considerations for the Simplex Algorithm 4.1 The Revised Simplex Algorithm; 4.2 Compuational Implications of the Revised Simplex Algorithm; 4.3 The Max-Flow Problem and Its Solution by the Revised Method; 4.4 Dantzig-Wolfe Decomposition

Chap. 5 The Primal-Dual Algorithm 5.1 Introduction; 5.2 The Primal-Dual Algorithm; 5.3 Comments on the Primal-Dual Algorithm; 5.4 The Primal-Dual Method Applied to the Shortest-Path Problem; 5.5 Comments on Methodology; 5.6 The Primal-Dual Method Applied to Max-Flow

Chap. 6 Primal-Dual Algorithms for Max-Flow and Shortest Path: Ford-Fulkerson and Dijkstra 6.1 The Max-Flow, Min-Cut Theorem; 6.2 The Ford and Fulkerson Labeling Algorithm; 6.3 The Question of Finiteness of the Labeling Algorithm; 6.4 Dijkstra's Algorithm; 6.5 The Floyd-Warshall Algorithm

Chap. 7 Primal-Dual Algorithms for Min-Cost Flow 7.1 The Min-Cost Flow Problem; 7.2 Combinatorializing the Capacities--Algorithm Cycle; 7.3 Combinatorializing the Cost--Algorithm Buildup; 7.4 An Explicit Primal-Dual Algorithm for the Hitchcock Problem--Algorithm Alphabeta; 7.5 A Transformation of Min-Cost Flow to Hitchcock; 7.6 Conclusion

Chap. 8 Algorithms and Complexity 8.1 Computability; 8.2 Time Bounds; 8.3 The Size of an Instance; 8.4 Analysis of Algorithms; 8.5 Polynomial-Time Algorithms; 8.6 Simplex Is Not a Polynomial-Time Algorithm; 8.7 The Ellipsoid Algorithm; 8.7.1 LP, LI, and LSI; 8.7.2 Affine Transformations and Ellipsoids; 8.7.3 The Algorithm; 8.7.4 Arithmetic Precision

Chap. 9 Efficient Algorithms for the Max-Flow Problem 9.1 Graph Search; 9.2 What Is Wrong With the Labeling Algorithm; 9.3 Network Labeling and Digraph Search; 9.4 An O(|V|²) Max-Flow Algorithm; 9.5 The Case of Unit Capacities

Chap. 10 Algorithms For Matching 10.1 The Matching Problem; 10.2 A Bipartite Matching Algorithm; 10.3 Bipartite Matching and Network Flow; 10.4 Nonbipartite Matching: Blossoms; 10.5 Nonbipartite Matching: An Algorithm

Chap. 11 Weighted Matching 11.1 Introduction; 11.2 The Hungarian Method for the Assignment Problem; 11.3 The Nonbipartite Weighted Matching Problem; 11.4 Conclusions

Chap. 12 Spanning Trees and Matroids 12.1 The Minimum Spanning Tree Problem; 12.2 An O(|E|log|V|) Algorithm for the Minimum Spanning Tree Problem; 12.3 The Greedy Algorithm; 12.4 Matroids; 12.5 The Intersection of Two Matroids; 12.6 On Certain Extensions of the Matroid Intersection Problem; 12.6.1 Weighted Matroid Intersection; 12.6.2 Matroid Parity; 12.6.3 The Intersection of Three Matroids

Chap. 13 Interger Linear Programming 13.1 Introduction; 13.2 Total Unimodularity; 13.3 Upper Bounds for Solutions of ILPs

Chap. 14 A Cutting-Plane Algorithm for Integer Linear Programs 14.1 Gomory Cuts; 14.2 Lexicography; 14.3 Finiteness of the Fractional Dual Algorithm; 14.4 Other Cutting-Plane Algorithms

Chap. 15 NP-Complete Problems 15.1 Introduction; 15.2 An Optimization Problem Is Three Problems; 15.3 The Classes P and NP; 15.4 Polynomial-Time Reductions; 15.5 Cook's Theorem; 15.6 Some Other NP-Complete Problems: Clique and the TSP; 15.7 More NP-Complete Problems: Matching, Covering, and Partitioning

Chap. 16 More About NP-Completeness 16.1 The Class co-NP; 16.2 Pseudo-Polynomial Algorithms and "Strong" NP-Complete Problems; 16.3 Special Cases and Generalizations of NP-Complete Problems; 16.3.1 NP-Completeness By Restriction; 16.3.2 Easy Special Cases of NP-Complete Problems; 16.3.3 Hard Special Cases of NP-Complete Problems; 16.4 A Glossary of Related Concepts; 16.4.1 Polynomial-Time Reductions; 16.4.2 NP-Hard problems; 16.4.3 Nondeterministic Turing Machines; 16.4.4 Polynomial-Space Complete Problems; 16.5 Epilogue

Chap. 17 Approximation Algorithms 17.1 Heuristics for Node Cover: An Example; 17.2 Approximation Algorithm for the Traveling Salesman Problem; 17.3 Approximation Schemes; 17.4 Negative Results

Chap. 18 Branch-and-Bound and Dynamic Programming 18.1 Branch-and-Bound for Integer Linear Programming; 18.2 Branch-and-Bound in a General Context; 18.3 Dominance Relations; 18.4 Branch-and-Bound Strategies; 18.5 Application to a Flowshop Scheduling Problem; 18.6 Dynamic Programming

Chap. 19 Local Search 19.1 Introduction; 19.2 Problem 1: The TSP; 19.3 Problem 2: Minimum-Cost Survivable Networks; 19.4 Problem 3: Topology of Offshore Natural Gas Pipeline Systems; 19.5 Problem 4: Uniform Graph Partitioning; 19.6 General Issues in Local Search; 19.7 The Geometry of Local Search; 19.8 An Example of a Large Minimal Exact Neighborhood; 19.9 The Complexity of Exact Local Search for the TSP

All chapters have problem sets and notes and references.

As can be seen, this book has a mighty amount of information, and it is amazingly well-explained. Of course, you need a firm grasp of your linear algebra, and some knowledge of very elementary calc./real analysis and graph theory (although most of the graph theory needed, technically speaking, is supplied in an appendix). You don't even really need to know a programming language, since the authors use a "pidgin algol," explained in yet another appendix, for most of the algorithm stuff; all it takes is an orderly thought process to follow it.

Despite the book's age, it mostly holds up very well in terms of topics and presentation. In the preface to the Dover edition, the authors briefly discuss some more current topics not dealt with in the text and make some (probably also out of date!) referrals for those wishing to "catch up." All in all, this book is a great value both as a text and a reference. ... Read more

Book Description
The first chapter of this book provides a brief treatment ofthe basics of the subject. The other chapters deal with the variousdecompositions of non-negative matrices, Birkhoff type theorems, thestudy of the powers of non-negative matrices, applications of matrixmethods to other combinatorial problems, and applications ofcombinatorial methods to matrix problems and linear algebra problems.
The coverage of prerequisites has been kept to a minimum.Nevertheless, the book is basically self-contained (an Appendixprovides the necessary background in linear algebra, graph theory andcombinatorics). There are many exercises, all of which are accompaniedby sketched solutions.
Audience: The book is suitable for a graduate course as well asbeing an excellent reference and a valuable resource formathematicians working in the area of combinatorial matrix theory. ... Read more