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Accessible to undergraduate students, Introduction to Combinatorics presents approaches for solving counting and structural questions. It looks at how many ways a selection or arrangement can be chosen with a specific set of properties and determines if a selection or arrangement of objects exists that has a particular set of properties.

To give students a better idea of what the subject covers, the authors first discuss several examples of typical combinatorial problems. They also provide basic information on sets, proof techniques, enumeration, and graph theory—topics that appear frequently throughout the book. The next few chapters explore enumerative ideas, including the pigeonhole principle and inclusion/exclusion. The text then covers enumerative functions and the relations between them. It describes generating functions and recurrences, important families of functions, and the theorems of Pólya and Redfield. The authors also present introductions to computer algebra and group theory, before considering structures of particular interest in combinatorics: graphs, codes, Latin squares, and experimental designs. The last chapter further illustrates the interaction between linear algebra and combinatorics. Exercises and problems of varying levels of difficulty are included at the end of each chapter.

Ideal for undergraduate students in mathematics taking an introductory course in combinatorics, this text explores the different ways of arranging objects and selecting objects from a set. It clearly explains how to solve the various problems that arise in this branch of mathematics.

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This trusted best-seller 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, combinatortial structures (matchings, designs, graphs), and flows in networks. The Fifth Edition clarifies the exposition throughout and adds a wealth of new exercises. Appropriate for one- or two-semester, junior- to senior-level combinatorics courses.

Customer Reviews (4)

Terrific!
It came faster than expected! Thank you so much! I would definitely buy from this user again! My book came exactly as promised, and it was wrapped with extreme care. Thanks again!

Unusable
I'm very surprised that this book is in its 5th edition and it still has a many errors.There are even errors in statements of theorems, which is really scary, and many errors in the back of the book where you are supposed to be able to check your answers.

I have very little faith in this textbook.If it weren't in use for my course (and I'm not sure why it is, because I have yet to find its good qualities) I would never have bought it.

Hopefully the publishers hire a math grad student or two for editing the sixth edition.

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

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Discrete Mathematics combines a balance of theory and applications with mathematical rigor and an accessible writing style. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Throughout the text, engaging anecdotes and topics of interest inform as well as motivate learners. The text is ideal for one- or two-semester courses and for students who are typically mathematics, mathematics education, or computer science majors. Part I teaches student how to write proofs; Part II focuses on computation and problem solving. The second half of the book may also be suitable for introductory courses in combinatorics and graph theory. ... Read more

Customer Reviews (1)

Good Math Book
It's a good math book about logic. It has good examples. The answers in the back of the book are sometimes just sketches and don't always help you figure out the proof, but that's when the proffessor helps out. ... Read more

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Emphasizes a Problem Solving Approach
A first course in combinatorics

Completely revised, How to Count: An Introduction to Combinatorics, Second Edition shows how to solve numerous classic and other interesting combinatorial problems. The authors take an easily accessible approach that introduces problems before leading into the theory involved. Although the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics.

New to the Second Edition
This second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet’s pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises.

Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Pólya’s counting theorem.

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Now with solutions to selected problems, Applied Combinatorics, Second Edition presents the tools of combinatorics from an applied point of view. This bestselling textbook offers numerous references to the literature of combinatorics and its applications that enable readers to delve more deeply into the topics.

After introducing fundamental counting rules and the tools of graph theory and relations, the authors focus on three basic problems of combinatorics: counting, existence, and optimization problems. They discuss advanced tools for dealing with the counting problem, including generating functions, recurrences, inclusion/exclusion, and Pólya theory. The text then covers combinatorial design, coding theory, and special problems in graph theory. It also illustrates the basic ideas of combinatorial optimization through a study of graphs and networks.

Customer Reviews (1)

comprehensive and lucid coverage of many algorithms
This book continues CRC's fine tradition of putting out definitive texts on a technical field. The amount of material on combinatorics is exceedingly non-trivial. I suspect few readers will actually take this book apart cover to cover. Which is a pity, because the coverage of algorithms is comprehensive and lucidly explained.

The book compares favourably to Knuth's classic 3 volume series, Art of Computer Programming, The, Volumes 1-3 Boxed Set (2nd Edition) (Vol 1-3). However, the problems for each chapter in the current book are simpler. There are few of the truly hard exercises that Knuth liberally sprinkled throughout his books.

What you may want to do with this book is perhaps only read the chapters relevant to your needs. Given on your part a reasonable pre-existing background in computing and combinatorics, then the book has a key property that its chapters are largely independent of each other. So you can get quickly to the germane sections.
... Read more

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This volume surveys the development of combinatorics since 1930 by presenting in chronological order the fundamental results of the subject proved in over five decades of original papers by: T. van Aardenne-Ehrenfest.- R.L. Brooks.- N.G. de Bruijn.- G.F. Clements.- H.H. Crapo.- R.P. Dilworth.- J. Edmonds.- P. Erdös.- L.R. Ford, Jr.- D.R. Fulkerson.- D. Gale.- L. Geissinger.- I.J. Good.- R.L. Graham.- A.W. Hales.- P. Hall.- P.R. Halmos.- R.I. Jewett.- I. Kaplansky.- P.W. Kasteleyn.- G. Katona.- D.J. Kleitman.- K. Leeb.- B. Lindström.- L. Lovász.- D. Lubell.- C. St. J.A. Nash-Williams.- G. Pólya.-R. Rado.- F.P. Ramsey.- G.-C. Rota.- B.L. Rothschild.- H.J. Ryser.- C. Schensted.- M.P. Schützenberger.- R.P. Stanley.- G. Szekeres.- W.T. Tutte.- H.E. Vaughan.- H. Whitney.

Product 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

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Developed from the authors’ introductory combinatorics course, this book focuses on a branch of mathematics which plays a crucial role in computer science. Combinatorial methods provide many analytical tools used for determining the expected performance of computer algorithms. Elementary subjects such as combinations and permutations, and mathematical tools such as generating functions and Pólya’s Theory of Counting, are covered, as are analyses of specific problems such as Ramsey Theory, matchings, and Hamiltonian and Eulerian paths.

This introduction will provide students with a solid foundation in the subject.

----

"This is a delightful little paperback which presents a day-by-day transcription of a course taught jointly by Pólya and Tarjan at Stanford University. Woods, the teaching assistant for the class, did a very good job of merging class notes into an interesting mini-textbook; he also included the exercises, homework, and tests assigned in the class (a very helpful addition for other instructors in the field). The notes are very well illustrated throughout and Woods and the Birkhäuser publishers produced a very pleasant text.

One can count on [Pólya and Tarjan] for new insights and a fresh outlook. Both instructors taught by presenting a succession of examples rather than by presenting a body of theory…[The book] is very well suited as supplementary material for any introductory class on combinatorics; as such, it is very highly recommended. Finally, for all of us who like the topic and delight in observing skilled professionals at work, this book is entertaining and, yes, instructive, reading."

—Mathematical Reviews (Review of the original hardcover edition)

"The mathematical community welcomes this book as a final contribution to honour the teacher G. Pólya."

—Zentralblatt MATH (Review of the original hardcover edition)

Product Description
The format of this book is unique in that it combines features of a traditional text with those of a problem book. The material is presented through a series of problems, about 250 in all, with connecting text; this is supplemented by a further 250 problems suitable for homework assignment. The problems are structured in order to introduce concepts in a logical order, and in a thought-provoking way. The first four sections of the book deal with basic combinatorial entities; the last four cover special counting methods. Many applications to probability are included along the way. Students from a wide range of backgrounds, mathematics, computer science or engineering will appreciate this appealing introduction. ... Read more

Customer Reviews (4)

quite good
we use it for mathematical olympiad training.students are not very good at combinatorics and the book is very good at drilling the basics into them.

it's true there are no solutions.but the solutions that are given are prototype solutions.you can get solutions to other problems from them.

the author should really come out with a version with more solutions and add more problems.that would make it much better.

problem -oriented with no solutions!
Yes, there are some solutions. I was extremely disappointed with this book. To be fair, it says " Classroom Resourse Materials". It may be a fantastic book if you are an instructor looking for problems for your students. However, I am a student. I hoped to get a book that would show me in some detail how these problems are solved. It didn't do that for even one. Not one. It did give the answer to some. It didn't even explain anything. It is what it says: a book of problems. Quite frankly, I don't think it's that hard to think of problems. The hard part is solving them. So, if you need to understand how to work problems -- this book is not for you.

problem -oriented with no solutions!
Yes, there are some solutions. I was extremely disappointed with this book. To be fair, it says " Classroom Resourse Materials". It may be a fantastic book if you are an instructor looking for problems for your students. However, I am a student. I hoped to get a book that would show me in some detail how these problems are solved. It didn't do that for even one. Not one. It did give the answer to some. It didn't even explain anything. It is what it says: a book of problems. Quite frankly, I don't think it's that hard to think of problems. The hard part is solving them. So, if you need to understand how to work problems -- this book is not for you.

A collection of excellent, yet routine problems
We all can count, but few can count really well. Combinatorics can be considered the art of sophisticated counting and it is a subject often neglected in the college curriculum. This book, with a direct, plunge aheadto the problems approach, covers all of the standard areas of thefield.
Since the amount of explanatory text is kept to a minimum, itwould not be easy to use the book as a textbook. The standard approach is afew paragraphs of text followed by a page or more of problems. Withoutsufficient additional explanation, the general student will no doubtstruggle.
However, the quality of the problems is excellent andsolutions to many are provided. Problems in combinatorics are fairlystandard issue and the author acknowledges this in a brief list of standardproblem types at the end of the book. However, it would have been animprovement if appropriate page numbers would have been associated witheach entry in the list. Another feature that adopters will find veryhelpful is a list of dependencies. The list is a simple table noting thatthe problem dependencies are such that a problem can be assigned after acertain other one has been done.
While the quality of the book isexcellent, one should think carefully before using it as a textbook. Muchbetter suited as a course supplement, it is probably best used as areference.

Published in Journal of Recreational Mathematics,reprinted with permission ... Read more

Product Description
Written by two of Gian-Carlo Rota's former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed.This book should be on the shelf of all students and researchers in combinatorics and related areas. ... Read more

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Measured geodesic laminations are a natural generalization of simple closed curves in surfaces, and they play a decisive role in various developments in two-and three-dimensional topology, geometry, and dynamical systems. This book presents a self-contained and comprehensive treatment of the rich combinatorial structure of the space of measured geodesic laminations in a fixed surface. Families of measured geodesic laminations are described by specifying a train track in the surface, and the space of measured geodesic laminations is analyzed by studying properties of train tracks in the surface. The material is developed from first principles, the techniques employed are essentially combinatorial, and only a minimal background is required on the part of the reader. Specifically, familiarity with elementary differential topology and hyperbolic geometry is assumed. The first chapter treats the basic theory of train tracks as discovered by W. P. Thurston, including recurrence, transverse recurrence, and the explicit construction of a measured geodesic lamination from a measured train track. The subsequent chapters develop certain material from R. C. Penner's thesis, including a natural equivalence relation on measured train tracks and standard models for the equivalence classes (which are used to analyze the topology and geometry of the space of measured geodesic laminations), a duality between transverse and tangential structures on a train track, and the explicit computation of the action of the mapping class group on the space of measured geodesic laminations in the surface. ... Read more

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Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study. ... Read more

Customer Reviews (1)

maybe a little long and hard to read but excellent!
Sort of the hidden underbelly of combinatorial mathematics is the relationship to
fractals and this book actually seems to admit it exists.
Some of the different types of generating function are hard to follow
and a reference to Roman's Umbral calculus or Sheffer sequences
would be useful as well, but a lot of very useful information
is here and they do mention Dr. Sloane's EIS.
Not since I got a look at the classic Riordan text have I seen
a book that packs this kind of impact on the field
of combinatorial mathematics.
This book is one for students who are serious about learning
the developing science of combinatorial analysis.
I just wish I could afford it
and not have to check it out of the library to be able to read it!
The connection of Bernoulli functions to bosons
and Euler functions to leptons isn't touched on,
but it is the reason that combinatorial analysis is important to physics.
So you can't give the book 5 stars even as good as what is presented is. ... Read more

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What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? Such questions are treated in geometric discrepancy theory. The book is an accessible and lively introduction to this area, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research. Including a wide variety of mathematical techniques (from harmonic analysis, combinatorics, algebra etc.) in action on non-trivial examples, the book is suitable for a "special topic" course for early graduates in mathematics and computer science. Besides professional mathematicians, it will be of interest to specialists in fields where a large collection of objects should be "uniformly" represented by a smaller sample (such as high-dimensional numerical integration in computational physics or financial mathematics, efficient divide-and-conquer algorithms in computer science, etc.).

From the reviews: "...The numerous illustrations are well placed and instructive. The clear and elegant exposition conveys a wealth of intuitive insights into the techniques utilized. Each section usually consists of text, historical remarks and references for the specialist, and exercises. Hints are provided for the more difficult exercises, with the exercise-hint format permitting inclusion of more results than otherwise would be possible in a book of this size..."

Allen D. Rogers, Mathematical Reviews Clippings (2001)

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Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists.

Customer Reviews (2)

great book
Great book - just be ready to get hold of some other books to fill in the details.If you're not entirely comfortable with the stuff in chapter 0 concerning modules (projectivity, injectivity), tor and ext, and homology from topology, maybe find a quick introduction to these elsewhere - while its certainly possible to learn about these things from chapter 0 of this book, it's not exactly the most painless way to do it.It was my first book on the subject, then I got Miller and Sturmfels: it would make more sense to reverse that order.Better still, get both simultaneously, and when the Stanley becomes a bit dense refer to M+S.I really loved this book, but it was damn hard at times!

This is a research monograph!
For those looking for a thorough introduction to the theory in this book, I would suggest a look at Miller and Sturmfels recent book on combinatorial commutative algebra or the book on Cohen-Macaulay rings by Bruns and Herzog. This is a great book, but I don't think it was intended as a beginners first book . Instead its style concentrates on presenting results found in research-articles ( prior to this book, there doesn't seem to have been any texts giving an overview of the theory ). It is great for looking up results, reading the needed prerequisites without going into details and finding important references. ... Read more

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This is a concise, 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 the gems of the theory. A wide spectrum of the most powerful combinatorial tools is presented, including methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A thorough discussion of recent applications to computer science illustrates the inherent usefulness of these methods. ... Read more

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Now fully updated in a third edition, this is a comprehensive textbook on combinatorial optimization. It puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete but concise proofs, also for many deep results, some of which have not appeared in print before. Recent topics are covered as well, and numerous references are provided. This third edition contains a new chapter on facility location problems, an area which has been extremely active in the past few years. Furthermore there are several new sections and further material on various topics. New exercises and updates in the bibliography were added.

Customer Reviews (2)

Kindle Version, Hard to read
I just downloaded the sample but I found myself struggling to read the symbols. Sometimes even after enlarging I couldn't read it.

Just thought I'd point this out for any kindle users like myself.

Useful yet dense!
This is the most comprehensive compilation on combinatorial optiomization I have seen so far.
Usually, Papadimitriou's book is a good place for this material - but in many cases, looking for proofs and theorems - I had to use several books:
(*) Combinatorial Optimization Algorithms and Complexity by Papadimitriou and Steiglitz.
(*) Integer and Combinatorial Optimization by Nemhauser and Wolsey
(*) Theory of linear and integer programming by Schrijver
(*) Combinatorial Optimization by Cook, Cunningham, Pulleyblank and Schrijver
(*)Combinatorial Algorithms by Kreher and Stinson

This book, on the other hand, contains so much information and so many proved theorems - it's the richest resuorce in this topic, in my humble opinion.

Using it as a graduate level textbook for an *introduction* to combinatorial optimization is kind of hard - as although it's richness, some topics are described without enough detail or examples (like the topics on network flow and bipartite graphs) - yet the authors probably assumed some previous knowledge in those topics.

I prefer using this book as a reference rather than and intoduction.

The heavy mathematical notations in this book might scare some readers, but no-fear! You quickly get used to it, and appreciate the greatness in the notations, as they make the theorems more short and to the point. On the other hand - getting back to this book for a quick review on some subject might forceyou to flip pages for a fwe minutes, just to remember the notation again.

The authors intended this book to be a graduaet level textbook or an up-to-date referencework for current research. I believe they accomplished both targets! ... Read more

Product Description
A series of important applications of combinatorics on words has emerged with the development of computerized text and string processing. The aim of this volume, the third in a trilogy, is to present a unified treatment of some of the major fields of applications. After an introduction that sets the scene and gathers together the basic facts, there follow chapters in which applications are considered in detail. The areas covered include core algorithms for text processing, natural language processing, speech processing, bioinformatics, and areas of applied mathematics such as combinatorial enumeration and fractal analysis. No special prerequisites are needed, and no familiarity with the application areas or with the material covered by the previous volumes is required. The breadth of application, combined with the inclusion of problems and algorithms and a complete bibliography will make this book ideal for graduate students and professionals in mathematics, computer science, biology and linguistics. ... Read more

Customer Reviews (1)

Lays a Strong Mathematical Foundation for work in Text Processing
I would recommend this book to students and researchers working in areas as diverse as text processing, biology and mathematics. I believe it would be a useful foundation for those working in information retrieval, information extraction, natural language processing, linguistics, bioinformatics and combinatorics.

L. Venkata Subramaniam
in IAPR Newsletter, July 2007. ... Read more

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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

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This volume is comprised of the invited lectures given at the 14th British Combinatorial Conference.The lectures survey many topical areas of current research activity in combinatorics and its applications, and also provide a valuable overview of the subject, for both mathematicians and computer scientists. ... Read more

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Combinatorics on words has arisen independently within several branches of mathematics, for instance, number theory, group theory and probability, and appears frequently in problems related to theoretical computer science. The first unified treatment of the area was given in Lothaire's Combinatorics on Words. Since its publication, the area has developed and the authors now aim to present several more topics as well as giving deeper insights into subjects that were discussed in the previous volume. An introductory chapter provides the reader with all the necessary background material. There are numerous examples, full proofs whenever possible and a notes section discussing further developments in the area. This book is both a comprehensive introduction to the subject and a valuable reference source for researchers. ... Read more