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Combinatorics is an area of mathematics involving an impressive breadth of ideas, and it encompasses topics ranging from codes and circuit design to algorithmic complexity and algebraic graph theory. In a highly distinguished career Béla Bollobás has made, and continues to make, many significant contributions to combinatorics, and this volume reflects the wide range of topics on which his work has had a major influence. It arises from a conference organized to mark his 60th birthday and the thirty-one articles contained here are of the highest calibre. That so many excellent mathematicians have contributed is testament to the very high regard in which Béla Bollobás is held. Students and researchers across combinatorics and related fields will find that this volume provides a wealth of insight to the state of the art. ... Read more

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Combinatorics is a book whose main theme is the study of subsets of a finite set. It gives a thorough grounding in the theories of set systems and hypergraphs, while providing an introduction to matroids, designs, combinatorial probability and Ramsey theory for infinite sets. The gems of the theory are emphasized: beautiful results with elegant proofs. The book developed from a course at Louisiana State University and combines a careful presentation with the informal style of those lectures. It should be an ideal text for senior undergraduates and beginning graduates. ... Read more

Customer Reviews (2)

A Classic in Combinatorics
This is the book from which I learnt Combinatorics theErdosian way. One of the most pleasing qualities of the book is the bite-sized chapters. Perhaps this is one of the first books to start this trend.

An earlier reviewer mentions the Ramsey theory chapter which I too would recommend. The chapters on Intersecting hypergraphs, Kruskal-Katona theorem, Littlewood--Offord problem, Four--functions theorem are other gems which come to mind readily. The proofs are crisp and the way the theorems are motivated means that they are not too demanding for a beginning graduate.

This is the best and fastest introduction to the way Hungarian Combinatorists operate. Anindispensable addition to any aspiring Combinatorist's book shelf.

The Gems of Combinatorics
This book provides an excellent introduction to a wide variety of extremely active areas ofcombinatorics. It focuses on the "gems" of the field -- proofs and results which aresimultaneously deep, central, and beautiful. One particular feature is thesection on infinte Ramsey theory which presents this beautiful but taxingarea in an exceptionally clear fashion. There are many good exercises. ... Read more

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One of the most active areas in mathematics today is the rapidly emerging new topic of "additive combinatorics". Building on Gowers' use of the Freiman-Ruzsa theorem in harmonic analysis (in particular, his proof of Szemerédi's theorem), Green and Tao famously proved that there are arbitrarily long arithmetic progressions of primes, and Bourgain and his co-authors have given non-trivial estimates for hitherto untouchably short exponential sums. There are further important consequences in group theory and in complexity theory and compelling questions in ergodic theory, discrete geometry and many other disciplines. The basis of the subject is not too difficult: it can be best described as the theory of adding together sets of numbers; in particular, understanding the structure of the two original sets if their sum is small. This book brings together key researchers from all of these different areas, sharing their insights in articles meant to inspire mathematicians coming from all sorts of different backgrounds.Titles in this series are co-published with the Centre de Recherches Mathématiques. ... Read more

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Emphasizing bijective methods, this introductory text details the tools needed to solve problems in enumerative combinatorics. These tools can be used to analyze many structures including words, permutations, functions, graphs, trees, and more. Later chapters delve into algebraic aspects, including treatments of formal power series, symmetric groups and polynomials, determinants, and the combinatorial calculus of tableaux. The text requires only a familiarity with basic logic, set theory, and proof techniques. Nearly 1000 exercises, from routine verifications to unsolved problems, are included, many with solutions or hints.

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Enumerative Combinatorics presents elaborate and systematic coverage of the theory of enumeration. The first seven chapters provide the necessary background, including basic counting principles and techniques, elementary enumerative topics, and an extended presentation of generating functions and recurrence relations. The remaining seven chapters focus on more advanced topics, including, Stirling numbers, partitions of integers, partition polynomials, Eulerian numbers and Polya's counting theorem.Extensively classroom tested, this text was designed for introductory- and intermediate-level courses in enumerative combinatorics, but the far-reaching applications of the subject also make the book useful to those in operational research, the physical and social science, and anyone who uses combinatorial methods. Remarks, discussions, tables, and numerous examples support the text, and a wealth of exercises-with hints and answers provided in an appendix--further illustrate the subject's concepts, theorems, and applications. ... Read more

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Combinatorics of Finite Geometries is an introductory text on the combinatorial theory of finite geometry.Assuming only a basic knowledge of set theory and analysis, it provides a thorough review of the topic and leads the student to results at the frontiers of research. This book begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes.Later, it addresses polar spaces, partial geometries, and generalized quadrangles.The revised edition contains an entirely new chapter on blocking sets in linear spaces,which highlights some of the most important applications of blocking sets--from the initial game-theoretic setting to their very recent use in cryptography.Extensive exercises at the end of each chapter insure the usefulness of this book for senior undergraduate and beginning graduate students. ... Read more

Customer Reviews (1)

Combinatorics of Finite Geometries
This is my favorite finite geometry text.The treatment is a mixture of basic and complex and is hence suitable for a wise variety of readers, probably best for undergraduate/beginning graduate courses, but works well for self-study.I am excited about the generalized quadrangles sections. ... Read more

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Coherent treatment provides comprehensive view of basic methods and results of the combinatorial study of finite set systems. The Clements-Lindstrom extension of the Kruskal-Katona theorem to multisets is explored, as is the Greene-Kleitman result concerning k-saturated chain partitions of general partially ordered sets. Connections with Dilworth's theorem, the marriage problem, and probability are also discussed. Each chapter ends with a helpful series of exercises and outline solutions appear at the end. "An excellent text for a topics course in discrete mathematics."--Bulletin of the American Mathematical Society.
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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

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GMAT problems based on combinatorics and probability constitute some of the most challenging quantitative questions you will face. Math scores have risen around the world, so more of these question types have appeared in recent years to make the GMAT more difficult. To better equip you on test day, Veritas Prep has written the only GMAT book devoted exclusively to combinatorics and probability. ... Read more

Customer Reviews (3)

A "MUST-HAVE" FOR GMAT PREP
Veritas's Combinatorics and Probability is a "must-have" for anyone aiming for a high quant score on the GMAT. Of the too many test-prep manuals flooding the market, I have found none that prepare GMAT takers in the area of combinatorics and probability as thoroughly as does this study guide from Veritas.

That most students have had limited exposure to these areas of mathematics, which involves figuring out the number of possible arrangements and groupings and the likelihood of a particular event occurring, is why many find these problems perplexing.Veritas clearly and succinctly presents these topics from basic to more advanced principles.There are plenty of problems to reinforce the various concepts, and I found the explanations thorough and easy to comprehend.

As a result of my study, I had the confidence to do battle with any of these problem types that I encountered on the GMAT.

Combinatorics & Prob
Combinatorics confuse many GMAT students who had little past exposure to these concepts, or simply forgot the concepts altogether.If you're doing well on the GMAT Math section, you will encounter a few harder questions from this subject area, and VP teaches you how to solve them very quickly and consistently.VP's strategies teach you reliable counting methods, formulas to memorize, and techniques to conquer these question types.

I was already proficient with probability concepts, but reviewed this section of the VP books anyway.GMAT preparation for probability concepts - including robust practice problems and explanations - just doesn't get any better than VP.In this section on the book, you learn quick formulas to memorize, how to spot "decoy" answers, and how to shortcut much of the written work in solving problems.I would recommend this book to anyone.

Very thorough book on GMAT Combinatorics and Probability!
I self studied and scored a 770 on the GMAT.Here are my thoughts on the Veritas Prep Combinatorics & Probability book:

===== OVERVIEW =====

If there's one quant topic that scares the vast majority of GMAT test takers, it's combinatorics. This is because very few of us have been exposed to these special counting problems, often made even more difficult by mixing in bits of probability theory. The fact that they are rarely featured in depth in most GMAT prep guides is also a reason why combinatorics and probability questions are feared by students. The Veritas Prep Combinatorics and Probability book is however an exception, since it provides a detailed theoretical review of the topics and consistent practice.

The book starts out with the most basic concepts about counting principles and builds up to more complex topics such as circle arrangements. It also contains 13 drills and 82 GMAT-style problems, out of which 13 are Data Sufficiency. This is really important especially for test takers seeking a high score in quant (48+), who often complain about the scarce availability of high quality combinatorics problems.

===== PROS =====

* In depth coverage of a topic considered advanced. What I liked the most about this book is that it really explains combinatorics starting from a basic level and up to more difficult subtopics.

* After each new formula or strategy, the authors offer one or two examples illustrating what you've learned. This is a really efficient method of study since it allows you to firmly grasp new concepts.

* Hefty amount of practice, which makes this books especially suitable for GMAT hopefuls aiming for higher quant scores. 15 questions are also designed to be the toughest of the tough and are labeled as "challenges."

* Some problems test more than one concept, a fact that makes them all the more interesting. The explanations are also of consistently high quality to help you understand no matter your level.

===== CONS =====

* More Data Sufficiency would have been a plus. While it does contain the most combinatorics and probability DS (13 questions) in all the guides I have reviewed so far, there's always room for improvement. This is because on test day, over 40% of the questions in the quant section will be DS and my opinion is that this percentage should be mirrored in the amount of practice available.

* The authors sometimes make the recommendation to simply write down all the favorable cases for a given question. I am personally not a big fan of this strategy and advise against it most of time because under the pressure of a timed exam you might lose too much time or miss a case or two which would lead to a wrong answer.

===== BOTTOM LINE =====

This is really one of the most thorough books I've seen on the subject of combinatorics and probability. As such, it is an excellent investment no matter your current level in quant, since you'll undoubtedly see at least a few problems involving combinatorics and/or probability. The mix of theory and practice in the Veritas Prep Combinatorics and Probability book will help you tackle these difficult question types with ease.

About me: Dana Jinaru, 770 GMAT scorer, expert at Beat The GMAT - a community serving 1 million+ MBA applicants each year ... 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

Customer Reviews (1)

R3
Unfortunately, I could not get it.
Please recheck the process.

Best wishes

Sang-Eon Han ... Read more

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The study of determinantal ideals and of classical determinantal rings is an old topic of commutative algebra. As in most of the cases, the theory evolved from algebraic geometry, and soon became an important topic in commutative algebra. Looking back, one can say that it is the merit of Eagon and Northcott to be the first who brought to the attention of algebraists the determinantal ideals and investigated them by the methods of commutative and homological algebra. Later on, Buchsbaum and Eisenbud, in a long series of articles, went further along the way of homological investigation of determinantal ideals, while Eagon and Hochster studied them using methods of commutative algebra in order to prove that the classical determinantal rings are normal Cohen-Macaulay domains. As shown later by C. DeConcini, D. Eisenbud, and C. Procesi the appropriate framework including all three types of rings is that of algebras with straightening law, and the standard monomial theory on which these algebras are based yields computationally effective results. A coherent treatment of determinantal ideals from this point of view was given by Bruns and Vetter in their seminal book.The author's book aims to a thorough treatment of all three types of determinantal rings in the light of the achievements of the last fifteen years since the publication of Bruns and Vetter's book. They implicitly assume that the reader is familiar with the basics of commutative algebra. However, the authors include some of the main notions and results from Bruns and Vetter's book for the sake of completeness, but without proofs. The authors recommend the reader to first look at the book of Bruns and Vetter in order to get a feel for the flavour of this field. The author's book is meant to be a reference text for the current state of research in the theory of determinantal rings. It was structured in such a way that it can be used as textbook for a one semester graduate course in advanced topics in Algebra, and at the PhD level. ... Read more

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The Nagoya 2000 International Workshop gathered together a group of scientists actively working in combinatorics, representation theory, special functions, number theory and mathematical physics, to acquaint the participants with some basic results in these fields and to discuss existing and possible interactions between the mentioned subjects. This volume constitutes the proceedings of the workshop. ... Read more

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This volume contains nine refereed research papers in various areas from combinatorics to dynamical systems, with computer algebra as an underlying and unifying theme.

Topics covered include irregular connections, rank reduction and summability of solutions of differential systems, asymptotic behaviour of divergent series, integrability of Hamiltonian systems, multiple zeta values, quasi-polynomial formalism, Padé approximants related to analytic integrability, hybrid systems.

The interactions between computer algebra, dynamical systems and combinatorics discussed in this volume should be useful for both mathematicians and theoretical physicists who are interested in effective computation. ... Read more

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Written by one of the leading authors and researchers in the field, this comprehensive modern text offers a strong focus on enumeration, a vitally important area in introductory combinatorics crucial for further study in the field.Miklós Bóna's text fills the gap between introductory textbooks in discrete mathematics and advanced graduate textbooks in enumerative combinatorics, and is one of the very first intermediate-level books to focus on enumerative combinatorics.The text can be used for an advanced undergraduate course by thoroughly covering the chapters in Part I on basic enumeration and by selecting a few special topics, or for an introductory graduate course by concentrating on the main areas of enumeration discussed in Part II.The special topics of Part III make the book suitable for a reading course.

This text is part of the Walter Rudin Student Series in Advanced Mathematics. ... Read more

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Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools and environments necessary to make connections, activities to extend and generalize combinatoric learning, and even explore implications of this learning on the undergraduate level.

This volume underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and discusses the conditions for supporting student reasoning.

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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

Customer Reviews (3)

useful for who want to learn discrete mathematics with mathematica of Wolfram research software
though the program created more than 6 years ago, it's useful reference for learning discrete mathematics with mathematica software

Way too specialized to be a stand alone discrete math text
I think the best parts of this book was the illustration overleafs for each chapter. There are some pretty cool images created using Combinatorica and Mathematica. I got this book to help me with my Discrete math course and with that in mind it wasn't as helpful as I thought it would be. I think the reason is that it's really a book about Combinatorica and how to use it with Mathematica and how the latest version is faster and how the functions were written. So in order to use it effectively you have to already be quite familiar with discrete math and graph theory to use this. There are plenty of examples to try and play with though if you can get it to run. In Mathematica7 and I would get alot of this command is obsolete warnings or errors. The book website says they have an updated version of Combinatorica but I didn't try that. Also, there are some exercises but no answers provided for any of them. There is also a complete listing of all the Combinatorica in the book but no clear indication if it's now included with the latest version of Mathematica which would be helpful. I don't see how this book could be used as a primary text for a discrete math course unless you had an excellent instructor to add all the material the book is missing and to walk you though putting the example in the book to best use. This book will appeal most to those that use Mathematica with discrete math and find the documentation lacking or want to find out how the Combinatorica functions were programmed. If you are using it just to supplement a discrete math course as I did you probably won't benefit as much.
I must admit it was kind fun seeing Mathematica choke up on trying to spit out a 20x20 gridgraph! Never seen it do that before on my quadcore pc.

It fills a Mathematica niche to perfection
This book has two primary uses, as a reference for the Combinatorica extension to Mathematica or as a textbook in a course in combinatorics and graph theory where Mathematica is extensively used. In both cases it is the best resource on the market. Combinatorica is an extremely powerful package that allows for very sophisticated operations to be performed in combinatorics and on graphs. However, there is no coverage of other areas of discrete mathematics.
The structure of the book is that it is split up into sections. For each section, there is a short textual description of the operation, a section of code illustrating the appropriate function(s) and the output from that code. The output is sometimes in the form of a textual listing and at other times an actual diagram. A complete listing of the functions and their actions is given in an appendix.
While this book is not one that could be used in a general course in discrete mathematics, it is an exceptional example of one that fills a niche to perfection. If you have any interest in doing combinatoric or graphing operations with Mathematica, then you have to look no further.

Published in Journal of Recreational Mathematics, reprinted with permission.
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Includes a rich variety of exercises to accompany the exposition of Coxeter groups

Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

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This collaborative volume presents recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Topics include numeration systems, word complexity function, morphic words, Rauzy tilings and substitutive dynamical systems, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic properties of digital functions, decidability issues for D0L systems, matrix products and joint spectral radius. Topics are presented in a way that links them to the three main themes, but also extends them to dynamical systems and ergodic theory, fractals, tilings and spectral properties of matrices. Graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, fractals, tilings and stringology will find much of interest in this book. ... Read more