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This graduate level text is distinguished both by the range of topics and the novelty of the material it treats--more than half of the material in it has previously only appeared in research papers. The first half of this book introduces the characteristic and matchings polynomials of a graph. It is instructive to consider these polynomials together because they have a number of properties in common. The matchings polynomial has links with a number of problems in combinatorial enumeration, particularly some of the current work on the combinatorics of orthogonal polynomials. This connection is discussed at some length, and is also in part the stimulus for the inclusion of chapters on orthogonal polynomials and formal power series. Many of the properties of orthogonal polynomials are derived from properties of characteristic polynomials. The second half of the book introduces the theory of polynomial spaces, which provide easy access to a number of important results in design theory, coding theory and the theory of association schemes. This book should be of interest to second year graduate text/reference in mathematics. ... Read more

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This monograph integrates unitary symmetry and combinatorics, showing in great detail how the coupling of angular momenta in quantum mechanics is related to binary trees, trivalent trees, cubic graphs, MacMahon s master theorem, and other basic combinatorial concepts. A comprehensive theory of recoupling matrices for quantum angular momentum is developed. For the general unitary group, polynomial forms in many variables called matrix Schur functions have the remarkable property of giving all irreducible representations of the general unitary group and are the basic objects of study. The structure of these irreducible polynomials and the reduction of their Kronecker product is developed in detail, as is the theory of tensor operators.

Contents: Quantum Angular Momentum; Composite Systems; Graphs and Adjacency Diagrams; Generating Functions; The D -Polynomials: Form; Operator Actions in Hilbert Space; The D-Polynomials: Structure; The General Linear and Unitary Groups; Tensor Operator Theory; Compendium A: Basic Algebraic Objects; Compendium B: Combinatorial Objects. ... Read more

Customer Reviews (1)

Agreat synthesis of Mathematical Physics, algebra, and combinatorics
This book is a masterpiece! It is a great proof that everything is related, and that
Natute, when analyzed deeply, reduces to combinatorics.One particulary attractive
tidbit was the pointing out that MacMahon's Master theorem is important in physics, and in fact a variant of it was rediscovered by nobelist Julian Schwinger of QED fame.
This book is too long to be read cover-to-cover, but it is so well-written that
if you open it at any chapter, it would still be understandable. This is the most
comprehensive treatment of the Racah symbol I am aware of. Anyone who is interested in
seeing how algebraic combinatorics can be used effectively in mathematical physics, should read at least parts of this book. ... Read more

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From the reviews: "Do you know M.Padberg's Linear Optimization and Extensions? […] Now here is the continuation of it, discussing the solutions of all its exercises and with detailed analysis of the applications mentioned. Tell your students about it. […] For those who strive for good exercises and case studies for LP this is an excellent volume." Acta Scientiarum Mathematicarum

Customer Reviews (2)

Computational and Mathematical Excellence
For nearly 30 years, Padberg has been a leader in computational integer programming and in combinatorial optimization theory.

In practice, Padberg has helped to design and implement "branch-and-cut" methods for finding exact optimal solutions to large traveling salesman problems, and this approach is a method of choice for finding approximately optimal solutions to tough industrial problems.The book provides the mathematical and computational background for understanding branch-and-cut; the established mathematical texts by Nemhauser and Wolsey and by Schrijver are less detailed and more condensed, and omit numerical issues.The treatment of modern simplex algorithms for linear programming---updating LU factorizations and using column- and constraint-generation and -purging---is excellent, and a large bibliography contains recent references.Besides industrial and Berlin-airlift scheduling problems, the book contains TSP examples of circuit-board wiring, U.S. state capitals, and Odysseus!

Three more highlights: The double description algorithm receives a complete description, and this is useful for combinatorial geometers.The discussion of integer-arithmetic and complexity theory is very readable, and these technical topics are slighted by interior-point books (besides Wright's quickie), despite their importance in integer programming and combinatorial optimization. The discussion of interior-point algorithms emphasizes projective geometry, a beautiful theory that has inspired so much of optimization theory---besides Karmarkar's interior-point algorithm,Dantzig's simplex algorithm, Fenchel duality, Davidon's conic algorithm for nonlinear optimization, etc.).

The book is not a comprehensive survey of linear programming,
and lacks a treatment of Nesterov's theory of self-concordant barrier-functions.Also, no treatment is given of pivoting algorithms besides Dantzig's (e.g., Terlaky's criss-cross method, Todd's oriented matroid algorithm).

A good reference for Linear Programming Theory
This book is certainly a very good reference for theoretical topics of linear programming. It covers the Simplex method and the Ellipsoid algorithms. It also covers the geometry of linear programming (polyhedraand polytopes, etc). It certainly covers more topics than most other linearprogramming texts.As expected, a book writen for theoretical topics iscertainly not easy to read, especially for people with no training in doingrigorous mathematical proofs. Also, not many examples or illustrations aregiven in this book, and this might be a problem for some readers. ... Read more

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A collaborative effort of the friends, students, and admirers of the mathematician Gian-Carlo Rota, focusing on a number of themes from Rota's work. Major topics include algebraic combinatorics and theoretical computer science, catlan numbers and combinatorics on words, and recurrent themes of Gian-Carlo Rota's mathematical thought. Also includes the Fubini Lectures. ... Read more

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This gradual, systematic introduction to the main concepts of combinatorics is the ideal text for advanced undergraduate and early graduate courses in this subject. Each of the book's three sections—Existence, Enumeration, and Construction—begins with a simply stated first principle, which is then developed step by step until it leads to one of the three major achievements of combinatorics: Van der Waerden's theorem on arithmetic progressions, Pólya's graph enumeration formula, and Leech's 24-dimensional lattice.

Along the way, Professor Martin J. Erickson introduces fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise new and innovative questions and observations. His carefully chosen end-of-chapter exercises demonstrate the applicability of combinatorial methods to a wide variety of problems, including many drawn from the William Lowell Putnam Mathematical Competition. Many important combinatorial methods are revisited several times in the course of the text—in exercises and examples as well as theorems and proofs. This repetition enables students to build confidence and reinforce their understanding of complex material.

Mathematicians, statisticians, and computer scientists profit greatly from a solid foundation in combinatorics. Introduction to Combinatorics builds that foundation in an orderly, methodical, and highly accessible manner. ... Read more

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For one-/two- semester, freshman courses in Discrete Mathematics.This carefully organized, very readable text covers every essential topic in discrete mathematics in a logical fashion. Placing each topic in context, it covers concepts associated with discrete mathematical systems that have applications in computer science, engineering, and mathematics. The author introduces more basic concepts at the freshman level than are found in other texts, in a simple, accessible form. Introductory material is balanced with extensive coverage of graphs, trees, recursion, algebra, theory of computing, and combinatorics. Extensive examples throughout the text reinforce concepts. ... Read more

Customer Reviews (2)

It is high on my list of potential textbooks for a course in discrete math
I occasionally teach discrete mathematics at the college level and so I am always trolling in a search for a better textbook. That is the context within which I examined this book. The coverage of the basic topics is exactly the way I prefer in both order and content. The initial chapters are:

*) Truth tables, logic and proofs
*) Set theory
*) Logic, integers and proofs
*) Functions and matrices

I would never use a book that did not cover logic, set theory, relations and functions in that order.
The remaining topics in the order they appear are:

*) Algorithms and recursion
*) Graphs, directed graphs and trees
*) Number theory
*) Counting and probability
*) Algebraic structures
*) Number theory revisited
*) Recursion revisited
*) Counting continued
*) Generating functions
*) Graphs revisited
*) Trees
*) Networks
*) Theory of computation
*) Theory of codes
*) Enumeration of colors
*) Rings, integral domains and fields
*) Group and semigroup characters
*) Applications of number theory

As can be seen from this list, there is much more material here than can be covered in a one-semester course. However, the sections are independent enough that selection can be done. For example, the "revisited" chapters could be skipped. This revisiting means that the depth of coverage is there if you have need of it. A large number of exercises are included at the end of the chapters and solutions to many of them are included.
I do not know if and when I will teach discrete math again. If that situation does arise again, I will give this book strong consideration as the choice of text.

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

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A gentle introduction to the highly sophisticated world ofdiscrete mathematics, Mathematical Problems and Proofs presentstopics ranging from elementary definitions and theorems to advancedtopics -- such as cardinal numbers, generating functions, propertiesof Fibonacci numbers, and Euclidean algorithm. This excellent primerillustrates more than 150 solutions and proofs, thoroughly explainedin clear language. The generous historical references and anecdotesinterspersed throughout the text create interesting intermissions thatwill fuel readers' eagerness to inquire further about the topics andsome of our greatest mathematicians. The author guides readers throughthe process of solving enigmatic proofs and problems, and assists themin making the transition from problem solving to theorem proving.
At once a requisite text and an enjoyable read, MathematicalProblems and Proofs is an excellent entrée to discretemathematics for advanced students interested in mathematics,engineering, and science. ... Read more

Customer Reviews (1)

A Basketful of Gems
This is a very interesting book.If you have mastered the bare essentials of set theory (through the upside down A for "for all" and the backwards E for "there exists"), AND either 1) the bare essentials of combinatorics (through Pascal's triangle), or 2) the bare essentials of number theory (through the definition of the Moebius function and the statement of the Chinese Remainder theorem), or 3) the bare essentials of geometry (through the law of cosines), AND if you are very talented in mathematics, then this book is a "MUST READ".It matters not whether you are a high school student or a professional mathematician.You will find new and fruitful insights and quite a few interesting problems in this book.For the beginner, there are several tantalizing (but somewhat oversimplified) references to advanced topics such as Paul Cohen's proof of the independence of the continuum hypotheses and Wiles' proof of Fermat's last theorem. For the professional there are footnotes with references to little known and suprising results obtained in the 20th century.But, unlike the claims in the "editorial review", this book neither prepares you to read the literaure nor is it a store house of exercises which will help you take your problem solving abilities to the next level.The "editorial review" is "off", but the book is "right on".Only the title is unfortunate.Where it will help the career mathematician is in the "beer hall" or "coffee house" or "tea time" discussions with other mathematicians.It is just chock full of beautiful little "gems" which can be shared with one's friends.This is the kind of beautiful stuff that makes mathematics truly interesting and exciting and though I have searched for a book like this for the last 35 years, this is by far the best in its class which I have found.Dr. John Aiken, Jan 5, 2003. ... Read more

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Parallelism or concurrency is one of the fundamental concepts in computer science. But in spite of its importance, theoretical methods to handle concurrency are not yet sufficiently developed. This volume presents a comprehensive study of Mazurkiewicz' trace theory from an algebraic-combinatorial point of view. This theory is recognized as an important tool for a rigorous mathematical treatment of concurrent systems. The volume covers several different research areas, and contains not only known results but also various new results published nowhere else. Chapter 1 introduces basic concepts. Chapter 2 gives a straight path to Ochmanski's characterization of recognizable trace languages and to Zielonka's theory of asynchronous automata. Chapter 3 applies the theory of traces to Petri nets. A kind of morphism between nets is introduced which generalizes the concept of synchronization. Chapter 4 provides a new bridge between the theory of string rewriting and formal power series. Chapter 5 is an introduction to a combinatorial theory of rewriting on traces which can be used as an abstract calculus for transforming concurrent processes. ... Read more

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Building from basics and demonstrating the relationships among the various branches of combinatorics, Victor Bryant presents the results in a straightforward way. Numerous examples and exercises including hints and solutions are included throughout and serve to lead the reader to some of the deeper results of the subject, many of which are usually excluded from introductory texts. ... Read more

Customer Reviews (2)

Definitely worth buying!
Excellent book! Anyone the least interested in combinatorics or graph theory should definitely take a closer look at this book. It's well written with great illustrations and examples. There are also plenty of excercises with both hints and solutions.

Beginner!
Beginner! HA! In this case beginner is defined as PhD. Mathematics professor. If you don't have a in-depth math background, do not purchase this book. ... Read more

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This book describes direct and recursive methods for the construction of combinatorial designs. It is ideally suited to the statistician through its discussion of how the designs currently used in experimental work have been obtained and through its coverage of other known and potentially useful designs. It is equally suited to the needs of the combinatorialist, with its stress on the statistical motivation for studying particular finite structures and its suggestions of open problems in the construction of new designs with useful properties for the experimentalist.Designs are discussed within a unified framework, showing the interplay between elegant structure and practical use. ... Read more

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This book integrates recent developments and related applications in $q$-series with a historical development of the field, focusing on major breakthroughs and the author's own research interests. The author develops both the important analytic topics (Bailey chains, integrals, and constant terms) and applications to additive number theory. He concludes with applications to physics and computer algebra and a section on results closely related to Ramanujan's ``Lost Notebook.'' With its wide range of applications, the book will interest researchers and students in combinatorics, additive number theory, special functions, statistical mechanics, and computer algebra. It is understandable to even a beginning graduate student in mathematics who has a background in advanced calculus and some mathematical maturity. ... Read more

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A One-Stop Source of Known Results, a Bibliography of Papers on the Subject, and Novel Research Directions

Focusing on a very active area of research in the last decade, Combinatorics of Compositions and Words provides an introduction to the methods used in the combinatorics of pattern avoidance and pattern enumeration in compositions and words. It also presents various tools and approaches that are applicable to other areas of enumerative combinatorics.

After a historical perspective on research in the area, the text introduces techniques to solve recurrence relations, including iteration and generating functions. It then focuses on enumeration of basic statistics for compositions. The text goes on to present results on pattern avoidance for subword, subsequence, and generalized patterns in compositions and then applies these results to words. The authors also cover automata, the ECO method, generating trees, and asymptotic results via random compositions and complex analysis.

Highlighting both established and new results, this book explores numerous tools for enumerating patterns in compositions and words. It includes a comprehensive bibliography and incorporates the use of the computer algebra systems Maple™ and Mathematica^®, as well as C++ to perform computations.

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Combinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Pólya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques.

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Hardbound. ... Read more

Customer Reviews (2)

A collection of papers, not a handbook at all
An handbook is usually a little fact book with just the facts
presented in as understandable a fashion as possible.
You have to look at this book from a price to benefit point of view.
Most of these research / survey papers are just not presented in a teaching mode: more of a theorem and proof mode with references.
The price tag per volume is over 200 dollars.
My bet is that libraries and the authors are the main market,
with most poor graduate students and post grads not being able to afford
the volumes at a price for both at of $400.
The material is dense and not presented in enough detail that
the result is very useful to a student.
Here we have more ego than scholarship as a result?

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

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This volume is a collection of survey papers in combinatorics that have grown out of lectures given in the workshop on Probabilistic Combinatorics at the Paul Erdös Summer Research Center in Mathematics in Budapest. The papers, reflecting the many facets of modern-day combinatorics, will be appreciated by specialists and general mathematicians alike: assuming relatively little background, each paper gives a quick introduction to an active area, enabling the reader to learn about the fundamental results and appreciate some of the latest developments. An important feature of the articles, very much in the spirit of Erdös, is the abundance of open problems. ... Read more

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The first chapter of this book provides a brief treatment ofthe basics of the subject. The other chapters deal with the variousdecompositions of non-negative matrices, Birkhoff type theorems, thestudy of the powers of non-negative matrices, applications of matrixmethods to other combinatorial problems, and applications ofcombinatorial methods to matrix problems and linear algebra problems.

The coverage of prerequisites has been kept to a minimum.Nevertheless, the book is basically self-contained (an Appendixprovides the necessary background in linear algebra, graph theory andcombinatorics). There are many exercises, all of which are accompaniedby sketched solutions. Audience: The book is suitable for a graduate course as well asbeing an excellent reference and a valuable resource formathematicians working in the area of combinatorial matrix theory. ... Read more