Product 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 There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results.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 (6)

Excellent!
I am what one of the reviewers called an arm-chair mathematician..

While I do not believe, that this book is suitable as a first introduction to combinatorics, it is a great book for anyone, who is interested in the subject and has had some prior exposure.
The formal mathematical prerequisites are quite minimal. The proofs are such, that after some thinking one can understand them - and they are always rigorous.
If one some occasions, the author would have given a short hint, instead of simply saying "it is easily seen" this would have made the book even more readable (but even then, after enough thinking one does see, albeit maybe not easily...)

All in all, very recommendable
(I am referring to volume 1 only, I did not read volume 2)

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. ... Read more

Product 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
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 (11)

Neat problems
Combinatorics is one of those subjects that you get good at by doing lots of problems (to build creative muscle). See, there's a book called "Combinatorcs Through Guided Discovery" by Kenneth Bogart. Basically, the author attempts to have the reader discover combinatorics through first principles. My combinatorics professor used that book for our combinatorics class. Now while I learned a lot from doing those problems (and learned what it is like to be genuinely stressed), the exam questions my professor gave were necessarily more application problems that used the principles of combinatorics rather than problems designed to teach the fundamental principles (so the exam questions were trickier but shorter), so to prepare for exams and to apply what I learned, I needed a bunch of problems that allowed me to do this. This book is *filled* with such problems and I honed my combinatorial problem instinct by just assigning myself problem sets from this book and doing them with (and without) time limits.

So that basically sums this book up: it has a TON of really good problems and if you really put the work in (and assuming you are reasonably mathematically talented), you'll walk away with much better problem solving skills than you did before you worked through the book. There's no branch of mathematics that really caters to the "problem solver" type of mentality than combinatorics and this book is just excellent to satisfy the need to just solve tricky problems.

I do reccomend that if you're buying this for self-study/supplement that you do NOT buy this 5th edition and that you go buy a really cheap 3rd or 4th edition (what I did). I skimmed my prof's copy of the 5th edition and saw that 95% of the problems are the same, with a few added/reworded.

Jaime's review
it was prompt and in the condition stated.
Those were my expectations, so good job.

Positives and Negatives
This book was assigned for a class in applied combinatorics, and in many instances I had to ask "why?"

The Positives:This book has the simplest introduction to building generating function models that I've ever come across, in regards to ordinary generating functions.The examples in this section really shine, and if you spend time on this section (Chapter 6.1) you shouldn't have too much trouble at all as you progress through the rest of the chapter.I cannot stress that enough!Master 6.1;the rest falls into place.

The intro to graph theory is just that:Tucker doesn't spend too much time on any major results, in fact treats Euler's polyhedral formula almost as an afterthought.I mean, I realize this is enumeration, but the fact that Euler's proof was really combinatorial in the first place is an excellent place to tie in a branch of mathematical study.The emphasis though are on graph problems and it gives an excellent study of two algorithms for solving the traveling salesman problem.

Binomial identities (the ultimate goal of chapter 5) aren't covered quite as comprehensively as I would have hoped.(see the book "Art of Combinatorial Proof")But the writing here is good and I only had to consult outside material on some of the problems.

The Negatives:

The treatment of recurrence relations are a joke.I mean, seriously!It starts out assuming you've spent time on Diff EQ (which I hadn't) and uses language such as 'obviously,' a damn dangerous word to anyone who studies math seriously!

In Homgeneous recurrence relations they introduce the general solutionburied within the discussion and make the additional mistake of using the * symbol in the notation.Organization is my biggest gripe here, they should treat this solution like a theorem or do SOMETHING to make it stand out.I finally just complied my own chart for my notes because I got tired of flipping pages back and forth and scanning for what I wanted.Tucker also misses the fact that an explicit treatment of notation when multiple or complex roots should be in order.

For solving recurrence relations I went back to my trusty "Discrete and Combinatorial Mathematics" text from Grimaldi.Real theory, and plainly spoken.For an undergrad text I expect my hand to be held at least a little, and Tucker obviously thinks recurrence relations are no big deal.

They aren't once you know them. I'm in a class of mostly math majors and you should have seen their faces when in an Inhomogeneous solution the teacher finished with "...can be solved with partial fraction decomposition."

The final gripe is the lack of any solutions manual.Maybe that's the point.I'm supposed to look at a solution and reverse engineer it on my own;its just that many times I'm not getting the point in the first place and I need to see more solutions to different kinds of problems.

In conclusion, I will keep this book on my shelf.The problems are incredibly challenging, and once you can solve them with one method, you can go back and solve them with another, that's the fun part about combinatorics.

For other clear treatments, Grimaldi, Harris, and Bona.

Harris's book is at the upper end of the undergrad spectrum, but its brevity on some topics is especially excellent as its derivation of generating function models sped up my understanding of this crucial area.

Docked 1 star for retail price for such a thin tome, and 1 star for assorted problems.

For self-study, I'd recommend the Grimaldi text before you tackle this one, but this one is good if you just want a source of problems.Many of these books I suggest should be available in your school's library, to avoid racking up high book costs.

Meandering Approach Leaves Me Frustrated
Very briefly, Tucker loses you through examples rather than developing his theoretical discussions.

I've noticed that some people have said that this text is clearly written, to me, nothing could be further from the truth.He seems to have a great knack for over-complicating simple ideas.This may be a personality thing -- some people really identify with his approach, I do not.I think he gets bogged down in smoothing out the details of his examples and definitions and ends up obfuscating points where simple brevity not only would have sufficed, but would have been more illuminating.

Also, except by inference on the reader's part, it can be difficult to distinguish the important from the trivial . . . this is a very poor text for self-study.I think I'm going to check out what selections Springer offers covering these topics, I've had pretty good luck finding well-written texts with their UTM series.

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. ... Read more

Product Description
Combinatorics is a subject of increasing importance because of its links with computer science, statistics, and algebra. This textbook stresses common techniques (such as generating functions and recursive construction) that underlie the great variety of subject matter, and the fact that a constructive or algorithmic proof is more valuable than an existence proof. The author emphasizes techniques as well as topics and includes many algorithms described in simple terms. The text should provide essential background for students in all parts of discrete mathematics. ... 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

Product Description
These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline. ... Read more

Customer Reviews (10)

Look it over carefully
I would exercise some caution before purchasing or adopting this text.While many reviewers find the style exuberant and humorous, I find it a bit breezy and even flip, at the expense of clarity.Here's an example, the definition of an SDR:

Given some family of sets X, a system of distinct representatives, or SDR, for
the sets in X can be thought of as a "representative" collection of distinct
elements from the sets of X.For instance, ....[What follows is a collection
of 5 sets and an SDR for them, along with a subcollection of 4 sets that doesn't
have an SDR.]

"Can be thought of"?Such tentative language is not helpful, and I'm not sure that the example would nail it down for the uninitiated.

The third chapter of the book (there are just three), "Infinite combinatorics and graphs", is what initially caught my attention, but the primary emphasis here is symbolic logic and abstract set theory.Very interesting topics, but the connection to combinatorics is a bit thin.

succinct and eloquent
I find the book to explain exactly what it intends to, providing pertinent examples where useful. I wish there were more examples, actually, but there is something to be said for being concise. The problems are well-organized and good problems. Also, it is a nice, sturdy hardcover version with non-glossy pages, which makes it easy to carry around without getting it beat up and easy on the eyes under fluorescent lights.

A highlight of my undergraduate math education
This is by far the best math book I have ever read. The authors present the material in a clear but also incredibly engaging way. The problems are interesting and spot-on in terms of difficulty for the sophomore to junior level introductory course. Those looking for a more classical definition-theorem-proof style textbook should look elsewhere. This is not to say that the textbook lacks rigor (the proofs are very precise), but that it reads more like a narrative, so that it might not serve as the best reference. For the student, however, there really isn't much more you could ask for in a math book.

Not bad, but not the best
The book does an ok job of explaining things.However, there are very few examples.

Doesn't Give Away the Store...
My background:I am an MIS major that discovered too late that he had an intense love for the mathematics behind the magic of computer science.I had previously only taken business calc(!) and Discrete Math (for CS majors).The book assigned was Tucker's book which does a great job on generating functions, but loses brevity completely when entering the field of recursive relations.

This book's explanations dealing with poker hands did what Tucker's and Grimaldi's books left me hanging on.Treatment on the binomial theorem and its related applications was also very thorough and at an acceptable level.The beauty of this book however is that the exercises rapidly increase in punch, and I still return to it from time to time to tease out new relationships.

It's introduction to graph theory is also very stellar... and it decides to introduce it before the combinatorial arguments, which if I'd had a little stronger comp sci background before taking the class, I would have found a much more gradual introduction to the general theories.

I'm still raising in mathematical ability, and I plan on tackling this book when I've gotten a little more maturity under my belt.

Excellent book.Hands down. ... Read more

Product 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. ... Read more

Customer Reviews (5)

Excellent Text Book
This is great textbook. Like all the Schaum Outlines, the focus is on problem solving. The author (Prof. V.K. Balakrishnan) does an absolutely marvelous job in leading the reader to an understanding of basic combinatorics via a seemingly endless series of problems. The problems are clearly stated and the solutions are well done. The general quality of the book is very high.
If you need more exposition, I would suggest something like Notes on Introductory Combinatorics by Polya, Tarjan and Woods, but as I say, I think the book by Balakrishnan is just fine as it is.

Not ideal "self-study" book for undergrads
The book does a poor job at laying out the basics for the reader. In the first chapter only 2 pages out of 33 are dedicated to describing the theory, the rest of the chapter consists of word problems. Some of those word problems go beyond the scope of the theory outlined for you, to which I thought was not only unfair but a bit intimidating to the curious math student. Sadly, the majority of this book follows the same routine.

From this layout it was clear that this book wasn't written with the complete novice in mind. The Discrete Math [also a Schaum's outline] book I bought alongside this one happens to cover the same topics and does a much better job at explaining the basics with reasonable problems to practice from.

As I said before, I can't recommend this book to anyone who is looking to break way into Combinatorics.

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

Product Description
For courses in undergraduate Combinatorics for juniors or seniors.This carefully crafted text emphasizes applications and problem solving. It is divided into 4 parts. Part I introduces basic tools of combinatorics, Part II discusses advanced tools, Part III covers the existence problem, and Part IV deals with combinatorial optimization. ... Read more

Customer Reviews (11)

Combinatorics
Seems like a reasonable book. Problems are reasonably hard and the literature is accurate. The only problem i have is there are no solutions to any prblems so you dont know if you are doing them correctly.

A very good text on graphs
The first two thirds of the book is pretty standard, but the k\last third goes into detail on some of the modern theory including Hadamard matrices,
Euler paths on graphs and Latin squares.
It is good enough that I'm going to buy my own copy after reading a library copy. It doesn't mention Vega triangle free graphs, but it gives one a better chance at modern graph theory than three other books that I have.
The close relationship of graph theory to group theory, geometry and Cartan
lie algebras isn't covered either.

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... ... Read more

Product Description
Written for graduate students in mathematics or non-specialist mathematicians who wish to learn the basics about some of the most important current research in the field, this book provides an intensive, yet accessible, introduction to the subject of algebraic combinatorics. After recalling basic notions of combinatorics, representation theory, and some commutative algebra, the main material provides links between the study of coinvariant or diagonally coinvariant spaces and the study of Macdonald polynomials and related operators. This gives rise to a large number of combinatorial questions relating to objects counted by familiar numbers such as the factorials, Catalan numbers, and the number of Cayley trees or parking functions. The author offers ideas for extending the theory to other families of finite Coxeter groups, besides permutation groups. ... Read more

Product Description
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. ... Read more

Customer Reviews (7)

Great book for undergrad level of combinatoric
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!!!

Can't say it enough - I wish I had this book when I was beginning to learn combinatorics
Wow, what an awesome book it is (even with so many good introductory books on combinatorics).I really like the fact that (i) the author engaged the reader on solving the problems early [combinatorics is as much about problem solving as theory building]; (ii) the great number of problems + solutions; and (iii) the selection of topics.

I cannot help but repeat here (foreword by Richard Stanley) - "I only wish that when I was a student beginning to learn combinatorics there was a textbook available as attractive as Bona's."

A good book on combinatorics
This is a good book on combinatorics. The problems constitute the real meat of this text. They range from the elementary to the very challenging. The first set of exercises has solutions. A set of supplementary exercises (without solutions) follows. This is a suitable book for self-study. Another combinatorics book worth looking at is Martin's Counting:The Art of Enumerative Combinatorics.

A Walk Through Combinatorics
I wish that all math books should be written the way this book offers with great examples w/solutions for students to try out first and then a set of supplemental problems that follow through for more practices.

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. ... Read more

Product 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 text 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

Product Description
A textbook suitable for undergraduate courses, the materials in this book are presented very explicitly so that students will find it easy to read and also find a wide range of examples. A number of combinatorial problems taken from mathematical competitions and exercises are also included. ... Read more

Customer Reviews (4)

DO NOT let the price nor the names of the authors deter you...
Oh how I wish I had found this book at the beginning of my semester of combinatorics instead of at its end...

When I found this tome here on Amazon it was at the behest of a reviewer that stated the strength of this book in the explanation of Recurrence Relations.I can't agree further.The text assigned for my class was Tucker's Applied Combinatorics book, and to be frank, it's a fairly decent text that does have an incredibly strong introduction to Generating Functions that goes down as one of the finest I have encountered.However, this book also discusses sequence creation using generating functions, something that Tucker leaves out.This book is a *true* introduction to combinatorics, explicitly detailing every step of every proof--something direly missing in most other texts of this type.Most people taking this class have only had a rudimentary sampling of proof techniques (comp-sci majors usually take combinatorics) and this book helps fill in the missing gaps left in slightly higher-flyers such as Tucker's.

But the reason this one is such a gem in recurrence relations is that it goes in depth in teaching you HOW TO MODEL with this tool.Tucker assumes you can do that already.His chapter on modeling is light on problems and doesn't explain the examples as clearly as this one does.This book also shows some incredibly creative problem solutions that crafty high-schoolers have devised (being olympiad trainers) that help you think about other implications in things such as Pascal's triangle.It does a great job of improving mathematical thinking and if I didn't enjoy Tucker's chapter on generating functions so much I would sell that thing in a heartbeat.

In studying for my combinatorics final I have also found that its plain explanations of other material from earlier chapters would have saved me (a whole lot) of 'head against the wall bashing.'

This book underlines the difference in how people who trained in education write books vs. people who typically write college textbooks.I know its typical as a native english speaker to think twice when buying a book written by a foreign sounding name, but trust me, you get the thing you always hope for in a math textbook.(Clarity, blessed, beautiful, sweet sweet clarity!)

If you're a student who enjoys--but struggles with--higher math, this is the book to get, hands down.

A useful book on combinatorics
This is a useful book on combinatorics. Concepts are explained clearly and simply without excessive elaboration. Especially valuable are the problems, some of which are of IMO standard. This book can be used by students preparing for maths competitions. A solutions manual would complement this book nicely.

excellent for competition training
A very clearly written classic of an important topic, with lots of problems meant for mathematical competition training. Highly recommended.

Rigorous and Lucid
I wish I had this book on hand during my course in combinatorics. It is a must-have introduction for math undergrads and the general public alike. (Never let the price of a mathematics book fool you. Like a classical music CD, the value of the a math book is entirely determined by its content.) It is lucidly written, which is critical for this ostensibly simple field. Fuzzy explanations can lead readers far astray; poor examples can obscure key concepts. Michael Tucker's "Applied Combinatorics" suffers from this very problem. However, this is not so with Chen and Koh's volume. They are kind enough to provide multiple combinatorial proofs for a considerable number of problems, and they do so in the familar AMS format. There are an abundance of problems at the end of each chapter. Their chapters on Inclusion-Exclusion and recurrence relations shine brightly. Pick it up now! ... Read more

Product Description

This book presents methods of solving problems in three areas of elementary combinatorial mathematics: classical combinatorics, combinatorial arithmetic, and combinatorial geometry. Brief theoretical discussions are immediately followed by carefully worked-out examples of increasing degrees of difficulty and by exercises that range from routine to rather challenging. The book features approximately 310 examples and 650 exercises.

Customer Reviews (1)

A good book on combinatorics-lots of interesting examples and solved exercises
This is a good book on combinatorics-lots of interesting examples and solved exercises. Generally the exercises are arranged in order of difficulty, progressing from the elementary to the more demanding. It is not just for students who are preparing for maths competitions e.g.,IMO, but also for the interested student who wish to do self-study. ... Read more

Product Description
Useful guide covers two major subdivisions of combinatorics--enumeration and graph theory--with emphasis on conceptual needs of computer science. Each part is divided into a "basic concepts" chapter emphasizing intuitive needs of the subject, followed by four "topics" chapters that explore these ideas in depth. Invaluable practical resource for graduate students, advanced undergraduates, and professionals with an interest in algorithm design and other aspects of computer science and combinatorics. Unabridged republication of original 1985 edition. References for Linear Order & for Graphs, Trees, and Recursions. 219 figures.
... Read more

Customer Reviews (2)

Suitable for a graduate level course
While most computer science departments require their students to take calculus, with few exceptions it is a skill that they will not use. The mathematics used by computer scientists is almost exclusively discrete in nature. In fact, assumptions concerning continuity can often lead to subtle errors in programming, the laws of algebra do not universally apply in computing. This book presents most of the key ideas of discrete mathematics applied to computing and the coverage is thorough and detailed.
It is split into two parts, linear order and graphs, trees and recursion. The emphasis is on detailed problem solving rather than explanations of the foundations. For example, there is a detailed example of a lexicographic bucket sort on page 16. The material is presented using higher level mathematical notation, so it is best suited for graduate level courses. A large number of exercises are included, but no solutions are given.
If you are looking for a text to be used in classes that have significant discrete mathematics prerequisites, then this book is certainly appropriate. It would be best suited for students with a great deal of programming experience.

Excellent reference for data structure algorithm complexity.
This excellent reference has numerous examples, definitions and exercises covering trees, graphs, linked lists etc.This book provides a great companion to any data structures text. ... Read more

Product Description

Trees are a fundamental object in graph theory and combinatorics as well as a basic object for data structures and algorithms in computer science. During the last years research related to (random) trees has been constantly increasing and several asymptotic and probabilistic techniques have been developed in order to describe characteristics of interest of large trees in different settings.

The aim of this book is to provide a thorough introduction into various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. It should serve as a reference book as well as a basis for future research. One major conceptual aspect is to bridge combinatorial and probabilistic methods that range from counting techniques (generating functions, bijections) over asymptotic methods (saddle point techniques, singularity analysis) to various sophisticated techniques in asymptotic probability (martingales, convergence of stochastic processes, concentration inequalities).

Product Description
A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. They are scattered in research papers or outlined in surveys, and they often use topological notions not commonly known among combinatorialists or computer scientists.

This bookis the first textbook treatment of a significant part of such results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level (for example, homology theory and homotopy groups are completely avoided). No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.

At the same time, many substantial combinatorial results are covered, sometimes with some of the most important results, such as Kneser's conjecture, showing them from various points of view.

The history of the presented material, references, related results, and more advanced methods are surveyed in separate subsections. The text is accompanied by numerous exercises, of varying difficulty. Many of the exercises actually outline additional results that did not fit in the main text. The book is richly illustrated, and it has a detailed index and an extensive bibliography.

This text started with a one-semester graduate course the author taught in fall 1993 in Prague. The transcripts of the lectures by the participants served as a basis of the first version. Some years later, a course partially based on that text was taught by G\"unter M. Ziegler in Berlin, who made book is based on a thoroughly rewritten version prepared during a pre-doctoral course I taught at the ETH Zurich in fall 2001.

Most of the material was covered in the course: Chapter 1 was assigned as an introductory reading text, and the other chapters were presented in approximately 30 hours of teaching (by 45 minutes), with some omissions throughout and with only a sketchy presentation of the last chapter. ... Read more

Customer Reviews (1)

Insight Pedagogy and Honesty
While allowing you to learn new concepts and definitions in a classical sense , this book shows what mathematics is about. It makes connections and gives you insight.
If you think that mathematics is made out of connections and is an "Art of variations" you will be rewarded. Those interested in the "frontier" between discrete and continuous will be interested too.
Definitely pedagogical and moreover intellectually honest.
... Read more

Product Description
This book provides an introduction to discrete mathematics. At the end of the book the reader should be able to answer counting questions such as: How many ways are there to stack n poker chips, each of which can be red, white, blue, or green, such that each red chip is adjacent to at least 1 green chip? The book can be used as a textbook for a semester course at the sophomore level. The first five chapters can also serve as a basis for a graduate course for in-service teachers. ... Read more

Customer Reviews (2)

Interesting book on combinatorics
This is a charming text on elementary combinatorics. Counting, or enumerative combinatorics, is-as the writer says-hard, but after you have gone through this very readable book, it becomes less hard and more interesting-so much so that you will want more of it. The reader is very much helped by the Back of the Book section which provides answers and, in many cases, solutions as well.

Great Introduction on Counting
This book is a good introduction to counting (combinatoric type counting).The book has answers to most of it's problems and recommends you try the problems before looking at the answers. Most of the problems are simple, and hit hard on the idea the section of the book is trying to get across.

It covers simple counting, groups, generating functions, recurrence relations and mathematical induction. The book concludes with graph theory. Some chapter sections get a little hard to understand, hence the 4 star and not 5 star rating (2 stars is what I'd give a decent book, so this one is a shining star). Most of the book is clear cut. ... Read more

Product Description
This unique approach to combinatorics is centered around challenging examples, fully-worked solutions, and hundreds of problems---many from Olympiads and other competitions, and many original to the authors. Each chapter highlights a particular aspect of the subject and casts combinatorial concepts in the guise of questions, illustrations, and exercises that are designed to encourage creativity, improve problem-solving techniques, and widen the reader's mathematical horizons.

Topics encompass permutations and combinations, binomial coefficients and their applications, recursion, bijections, inclusions and exclusions, and generating functions. The work is replete with a broad range of useful methods and results, such as Sperner's Theorem, Catalan paths, integer partitions and Young's diagrams, and Lucas' and Kummer's Theorems on divisibility. Strong emphasis is placed on connections between combinatorial and graph-theoretic reasoning and on links between algebra and geometry.

The authors' previous text, 102 Combinatorial Problems, makes a fine companion volume to the present work, which is ideal for Olympiad participants and coaches, advanced high school students, undergraduates, and college instructors. The book's unusual problems and examples will stimulate seasoned mathematicians as well. A Path to Combinatorics for Undergraduates is a lively introduction not only to combinatorics, but to mathematical ingenuity, rigor, and the joy of solving puzzles. ... Read more

Customer Reviews (1)

Interesting and Clear
The book is written very clearly and presents a lot of combinatorial subjects by very interesting examples.
... Read more

Product Description

The study of random graphs was begun in the 1960s and now has a comprehensive literature. This excellent book by one of the top researchers in the field now joins the study of random graphs (and other random discrete objects) with mathematical logic. The methodologies involve probability, discrete structures and logic, with an emphasis on discrete structures.

Customer Reviews (1)

stuff not in some of the other books
Two things were new to me:
1) the Fermat matrix
2) an Umbral Calculus expansion of the Eulerian numbers
I can't give the book fives stars because "the number of graphs or order k on n"
just doesn't seem to be something you can compute using his text.
For 1974 this was a very advanced text and he talks of a sequel
that I'm going to look for now!
This text is maybe one that Dover Books should reproduce
for students. ... Read more

Customer Reviews (1)

Excellent approachment
Unlike other textbook in the prosperous combinatorics , this introduction book takes a very different pace.It's paradigm is "SHOW me the proof". From the very beginning to the last page ,authors us that wecan make a proof clear by write out directly the algorithm or just make aapparent bijection. The book contains 4 chapters, the first 2 stress onbasic enumeration objects and posets, the last 2 on bijection andinvolution. With authour's carefully-selected topic and examples, this bookis self-contained. this book shows us the splendid new concepts ofcombinatorics. I must say that I'm very happy and shocked that ,in such afew pages ,by using combinatorical method developed here we can EASILYprove Cayley's theorem, Vandemonde determinent, Roger-Ramanujan's partitionformula. etc. The exercises are excellent too. Very many good seed ideaswaiting to be developed. ... Read more