Book Description
This carefully organized, very readable book covers everyessential topic in discrete mathematics in a logical fashion. Placingeach topic in context, it covers concepts associated with discretemathematical systems that have applications in computer science, engineering,and mathematics. The author introduces more basic concepts at thefreshman level than are found in other books, in a simple, accessibleform. Introductory material is balanced with extensive coverage ofgraphs, trees, recursion, algebra, theory of computing, and combinatorics.Extensive examples throughout the text reinforce concepts. More combinatorics/algebraic structures than in most books. Detailed discussion of and strong emphasis on proofs. Extensive, in-depth presentation of topics. Large selection of applied and computational problems, ranging from the elementary to the more advanced. More topics in probability and more statistical interpretations than other texts. Comprehensive discussion of topics such as finite state machines, automata, and languages. Earlier introduction of matrices and relations, Boolean algebras and circuits than most texts. Includes algorithms for many constructive tasks that occur in discrete systems. ... Read more

Customer Reviews (1)

A pretty good book
While the table of contents is very good, too much time is wasted in developing preliminaries. The text should quickly move into the core of the subject, but instead spends 220 pages developing the elementary material. A good reference for teachers of discrete mathematics with emphasis in combinatorics and not in computer science. ... Read more

Book Description
The discrete mathematics and theoretical computer science communities have recently witnessed explosive growth in the area of algorithmic combinatorics on words. The next generation of research on combinatorics of partial words promises to have a substantial impact on molecular biology, nanotechnology, data communication, and DNA computing. Delving into this emerging research area, Algorithmic Combinatorics on Partial Words presents a mathematical treatment of combinatorics on partial words designed around algorithms and explores up-and-coming techniques for solving partial word problems as well as the future direction of research.

This five-part book begins with a section on basics that covers terminology, the compatibility of partial words, and combinatorial properties of words. The book then focuses on three important concepts of periodicity on partial words: period, weak period, and local period. The next part describes a linear time algorithm to test primitivity on partial words and extends the results on unbordered words to unbordered partial words while the following section introduces some important properties of pcodes, details a variety of ways of defining and analyzing pcodes, and shows that the pcode property is decidable using two different techniques. In the final part, the author solves various equations on partial words, presents binary and ternary correlations, and covers unavoidable sets of partial words.

Setting the tone for future research in this field, this book lucidly develops the central ideas and results of combinatorics on partial words. ... Read more

Book 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 (3)

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

Book Description
By concentrating on counting problems, Introduction to Combinatoricsconveys basic ideas of its subject. Topics include combinations, permutations, the inclusion-exclusion principles, partitions, Stirling's Formula, generating functions, recurrence relations, groups, group actions, and graphs. The final two chapters discuss the application of group theory to counting patterns, via Burnside's Theorem and Polya's Theorem. Slomson's approach is to begin with concrete problems, and to use them as a lead-in to general theory. Numerous exercises-most of which are provided with detailed answers-are included for the advanced student. Among the applications considered are approaches to probability problems, especially in card games. ... Read more

Book Description
Probabilistic methods have become a vital tool in the arsenal ofevery combinatorialist. The theory of random graphs is still aprime area for the use of probabilistic methods, and, over theyears, these methods have also proved of paramount importance inmany associated areas such as the design and analysis ofcomputer algorithms. In recent years, probabilisticcombinatorics has undergone revolutionary changes as the resultof the appearance of some exciting new techniques such asmartingale inequalities, discrete isoperimetric inequalities,Fourier analysis on groups, eigenvalue techniques, branchingprocesses, and rapidly mixing Markov chains. The aim of thisvolume is to review briefly the classical results in the theoryof random graphs and to present several of the important recentdevelopments in probabilistic combinatorics, together with someapplications.

The first paper contains a brief introduction to the theory ofrandom graphs. The second paper reviews explicit constructionsof random-like graphs and discusses graphs having a variety ofuseful properties. Isoperimetric inequalities, of paramountimportance in probabilistic combinatorics, are covered in thethird paper. The chromatic number of random graphs is presentedin the fourth paper, together with a beautiful inequality due toJanson and the important and powerful Stein-Chen method forPoisson approximation. The aim of the fifth paper is to presenta number of powerful new methods for proving that a Markov chainis "rapidly mixing" and to survey various related questions,while the sixth paper looks at the same topic in a verydifferent context. For the random walk on the cube, theconvergence to the stable distribution is best analyzed throughFourier analysis; the final paper examines this topic andproceeds to several more sophisticated applications. Openproblems can be found throughout each paper. ... Read more

Book Description

A Path to Combinatorics for Undergraduates is a lively introduction not only to combinatorics, but also to mathematical ingenuity, rigor, and the joy of solving puzzles. This unique approach to combinatorics is centered around unconventional, essay-type combinatorial examples, followed by a number of carefully selected, challenging problems and extensive discussions of their solutions. Topics encompass permutations and combinations, binomial coefficients and their applications, bijections, inclusions and exclusions, and generating functions. Each chapter features fully-worked problems, including many from Olympiads and other competitions, as well as a number of problems original to the authors; at the end of each chapter are further exercises to reinforce understanding, encourage creativity, and build a repertory of problem-solving techniques.

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

Book Description
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

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

Book Description
This volume contains nine invited papers that survey many areas of current research in combinatorics both on the theoretical and practical side.Several papers may be regarded as summarizing our present state of knowledge in a particular topic. ... Read more

Book Description
This book evolved from several courses in combinatorics and graph theory given at Appalachian State University and UCLA. Chapter 1 focuses on finite graph theory, including trees, planarity, coloring, matchings, and Ramsey theory. Chapter 2 studies combinatorics, including the principle of inclusion and exclusion, generating functions, recurrence relations, Pólya theory, the stable marriage problem, and several important classes of numbers. Chapter 3 presents infinite pigeonhole principles, König's lemma, and Ramsey's theorem, and discusses their connections to axiomatic set theory. The text is written in an enthusiastic and lively style. It includes results and problems that cross subdisciplines, emphasizing relationships between different areas of mathematics. In addition, recent results appear in the text, illustrating the fact that mathematics is a living discipline. The text is primarily directed toward upper-division undergraduate students, but lower-division undergraduates with a penchant for proof and graduate students seeking an introduction to these subjects will also find much of interest.
... Read more

Customer Reviews (4)

Glorious
The authors of this book have managed to teach, in a complete and thorough manner, enough material to fill a book more than twice the size of this one.Do not mistake its brevity for a Rudin-esque lack of explanation, or for a lack of substance.Explanations are provided, a good deal of material is covered, and the book remains so concise and to the point that I have no complaints whatsoever.

Very few math books lend themselves well to being read cover-to-cover, but the unassuming nature of this book makes it perfect for a leisurely and fun read, or for a classroom.

Fans of the writing style of Joseph Gallian's "Contemporary Abstract Algebra" will enjoy this book's ability to present material in a friendly way without oversimplifying.

Buy this book.

A truly elegant introduction to combinatorics
Unlike its competitors, this book states simple concepts simply.It gives an excellent selection of the most important techniques and examples, without endlessly repeated "real-world" applications.In 80 pages, it covers the most interesting topics in graph theory, including:Cayley's tree-counting theorem, vertex coloring (with proof of the 5-Color Theorem), Hall matching theorem, Ramsey numbers, and stable marriage. Another 80 pages contains the main concepts of enumeration: elementary combinations (poker hands), inclusion-exclusion, generating functions for Fibonacci and Catalan numbers, Polya counting of symmetry classes, Stirling numbers.There is final section on infinite sets and graphs.

The book covers quite as much as similar ones of twice the length.Finally, a textbook which is not afraid to be brief!

Perfect book for self teaching
I am a math student with Indiana University working out of this book for independent study credit.When my adviser and I sat down to discuss books, we sifted through 10 to 12 books, and it was clear from the start that this book was the best.And I haven't changed my mind since.The book is clear, concise, and easy to read.Excellent for anyone who is teaching themselves, which of course means it's great for a full course with actual instructors.

What an introductory book on combinatorics should be
Little did I expect of a book that has seemingly not garnered that much attention among professors and students (at least the lack of reviews in Amazon.com might serve as a confirming instance of that speculation), but come exam time, the book proved that such prejudice is outright foolish.

The authors must really love both the field and writing about it, for their overflowing exuberance readily transfers to the pages. Pictures and humor are never a scarcity here: the authors took no shame providing both; a curious but pleasurably fresh anomaly in the often dry and coldly serious world that is mathematics writing. ... Read more

Book Description
WINNER of a CHOICE Outstanding Academic Title Award for 2006!As linear orders, as elements of the symmetric group, modeled by matrices, modeled by graphspermutations are omnipresent in modern combinatorics. They are omnipresent but also multifaceted, and while several excellent books explore particular aspects of the subject, no one book has covered them all. Even the classic results are scattered in various resources.Combinatorics of Permutations offers the first comprehensive, up to date treatment of both enumerative and extremal combinatorics and looks at permutation as linear orders and as elements of the symmetric group. The author devotes two full chapters to the young but active area of pattern avoidance. He explores the quest for the Stanley-Wilf conjecture and includes the recent and spectacular Marcus-Tardos proof of this problem. He examines random permutations and Standard Young Tableaux and provides an overview of the very rich algebraic combinatorics of permutations. The final chapter takes an in-depth look at combinatorial sorting algorithms.The author's style is relaxed, entertaining, and clearly reflects his enthusiasm for the "serious fun" the subject holds. Filled with applications from a variety of fields and exercises that draw upon recent research results, this book serves equally well as a graduate-level text and a reference for combinatorics researchers. ... Read more

Customer Reviews (2)

wait for second printing
This is a nice bookwhich deserved a better
proofreading before going to press.In its
current incarnation it is marred with numerous
typos not only in the text but in the exercises
as well.It can be very frustrating when the
reader is essentially asked to guess the author's
intentions.My advice to prospective buyers is to
wait for the second printing when these errors
will hopefully be cleaned up.

Accessible Cutting-Edge Research Engagingly written
If you are impatient to go to the front-line of research
in combinatorics, this book is a great short-cut.
It is beautifully written by one of the greatest authorities
in the field, and after you finish reading it, you are
ready to do your own research. ... Read more

Book Description
Counting: The Art of Enumerative Combinatorics provides an introduction to discrete mathematics that addresses questions that begin, How many ways are there to...For example, ¿How many ways are there to order a collection of 12 ice cream cones if 8 flavors are available?of the book the reader should be able to answer such nontrivial counting questions as, ¿How many ways are there to color the faces of a cube if ¿k¿ colors are available with each face having exactly one color?¿ or ¿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? are no prerequisites, this book can be used for college courses in combinatorics at the sophomore level for either computer science or mathematics students. The first five chapters have served as the basis for a graduate course for in-service teachers. Chapter 8 introduces graph theory. ... Read more

Book 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.

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

Book Description

This book is a carefully written exposition of Coxeter groups, an area of mathematics which appears in algebra, geometry, and combinatorics. In this book, the combinatorics of Coxeter groups has mainly to do with reduced expressions, partial order of group elements, enumeration, associated graphs and combinatorial cell complexes, and connections with combinatorial representation theory. While Coxeter groups have already been exposited from algebraic and geometric perspectives, this book will be presenting the combinatorial aspects of Coxeter groups. The authors have included an exposition of Coxeter groups along with a rich variety of exercises, ranging from easy to very difficult, giving the book the unique character of serving as both a textbook and a monograph.

Book Description

Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineering. The book contains eleven chapters written by experts in their respective fields, and covers a wide spectrum of high-interest problems across these discipline domains. Among the contributing authors are Richard Karp of UC Berkeley and Robert Tarjan of Princeton; both are at the pinnacle of research scholarship in Graph Theory and Combinatorics. The chapters from the contributing authors focus on "real world" applications, all of which will be of considerable interest across the areas of Operations Research, Computer Science, Applied Mathematics, and Engineering. These problems include Internet congestion control, high-speed communication networks, multi-object auctions, resource allocation, software testing, data structures, etc. In sum, this is a book focused on major, contemporary problems, written by the top research scholars in the field, using cutting-edge mathematical and computational techniques.

Book Description
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

Book Description

Book Description

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 (3)

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.

I dont read this book yet!
I dont read this book yet! ... Read more

Book Description

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