Combinatorics: A Guided Tour

(MAA Textbooks)

David R. Mazur

Mathematical Association of America | 2009 | 410 páginas | rar - pdf |2,2 Mb

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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.

The text emphasizes the brands of thinking that are characteristic of combinatorics: bijective and combinatorial proofs, recursive analysis, and counting problem classification. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses.

What makes this text a guided tour are the approximately 350 reading questions spread throughout its eight chapters. These questions provide checkpoints for learning and prepare the reader for the end-of-section exercises of which there are over 470. Most sections conclude with Travel Notes that add color to the material of the section via anecdotes, open problems, suggestions for further reading, and biographical information about mathematicians involved in the discoveries.

Contents
Ch. 1 . Principles of Combinatorics
ch. 2. Distributions and Combinatorial Proofs
ch. 3. Algebraic Tools
ch. 4. Famous Number Families
ch. 5. Counting Under Equivalence
ch. 6. Combinatorics on Graphs
ch. 7. Designs and Codes
ch. 8. Partially Ordered Sets

Combinatorics: Ancient & Modern

Robin Wilson, John J. Watkins e Ronald Graham

Oxford University Press | 2013 | 392 páginas | rar - pdf | 6,9 Mb


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Who first presented Pascal's triangle? (It was not Pascal.)
Who first presented Hamiltonian graphs? (It was not Hamilton.)
Who first presented Steiner triple systems? (It was not Steiner.) 
The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: 1) it constitutes the first book-length survey of the history of combinatorics; and 2) it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. 
Individual chapters have been contributed by sixteen experts. The book opens with an introduction by Donald E. Knuth to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler's contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections by the distinguished combinatorialist, Peter J. Cameron. 
This book is not expected to be read from cover to cover, although it can be. Rather, it aims to serve as a valuable resource to a variety of audiences. Combinatorialists with little or no knowledge about the development of their subject will find the historical treatment stimulating. A historian of mathematics will view its assorted surveys as an encouragement for further research in combinatorics. The more general reader will discover an introduction to a fascinating and too little known subject that continues to stimulate and inspire the work of scholars today.

CONTENTS
Part I Introduction
Two thousand years of combinatorics 3
Donald E. Knuth
Part II Ancient Combinatorics
1. Indian combinatorics 41
Takanori Kusuba and Kim Plofker
2. China 65
Andrea Bréard
3. Islamic combinatorics 83
Ahmed Djebbar
4. Jewish combinatorics 109
Victor J. Katz
5. Renaissance combinatorics 123
Eberhard Knobloch
6. The origins of modern combinatorics 147
Eberhard Knobloch
7. The arithmetical triangle 167
A. W. F. Edwards
Part III Modern Combinatorics
8. Early graph theory 183
Robin Wilson
9. Partitions 205
George E. Andrews
10. Block designs 231
Norman Biggs and Robin Wilson
11. Latin squares 251
Lars Døvling Andersen
12. Enumeration (18th–20th centuries) 285
E. Keith Lloyd
13. Combinatorial set theory 309
Ian Anderson
14. Modern graph theory 331
Lowell Beineke and Robin Wilson
Part IV Aftermath
A personal view of combinatorics 355
Peter J. Cameron
Notes on contributors 367
Picture sources and acknowledgements 371
Index 377

Combinatorial Reasoning: An Introduction to the Art of Counting

Duane DeTemple e William Webb

Wiley | 2014 | 484 páginas | rar - pdf | 3,1 Mb

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With a focus on beginning combinatorics, this book features complete exposition of the discussed topics and over 700 exercises that illustrate the presented concepts and methodology. The book covers fundamental topics in enumeration, all of which are explored in great depth. The authors' approach is very reader-friendly and avoids the "scholarly tone" found in many other related books. Abstract ideas are grounded in familiar concrete settings, and there are far more diagrams within the book than are found in most other related texts. In addition, simple cases are treated first before general and advanced cases, and the notations are kept simple and helpful. To help readers learn how combinatorics interacts with other subjects as a tool or as an application, the book provides brief summaries of basic concepts from probability, power series, and group theory. Each chapter begins with an introduction to the topics covered within the chapter, followed by sections that are each accompanied by a large number of exercises that range from basic calculations to more challenging problems. The exercises are carefully structured to range from the routine to the more advanced. The beginning level problems are designed to confirm and develop students' understanding of the fundamental concepts and calculations, where a calculation is most often based on combinatorial reasoning rather than algebraic manipulation. More advanced exercises allow students to explore and more deeply understand the topics in the chapter and apply the techniques to more complex combinatorial situations. Each chapter also contains a brief summary of the most important concepts covered. Finally, each chapter ends with a few additional problems that further synthesize the ideas that were presented in the chapter. Chapter coverage includes: Initial EnCOUNTers with Combinatorial Reasoning; Arrangements, Selections, and Distributions; Binomial Series and Generating Functions; Alternating Sums, Inclusion-Exclusion, and Rook Polynomials; Recurrence Relations; Special Numbers; Linear Spaces and Recurrences Sequences; and Counting Symmetric Arrangements

CONTENTS
PREFACE ix
PART I THE BASICS OF ENUMERATIVE COMBINATORICS
1 Initial EnCOUNTers with Combinatorial Reasoning 3
1.1 Introduction, 3
1.2 The Pigeonhole Principle, 3
1.3 Tiling Chessboards with Dominoes, 13
1.4 Figurate Numbers, 18
1.5 Counting Tilings of Rectangles, 24
1.6 Addition and Multiplication Principles, 33
1.7 Summary and Additional Problems, 46
References, 50
2 Selections, Arrangements, and Distributions 51
2.1 Introduction, 51
2.2 Permutations and Combinations, 52
2.3 Combinatorial Models, 64
2.4 Permutations and Combinations with Repetitions, 77
2.5 Distributions to Distinct Recipients, 86
2.6 Circular Permutations and Derangements, 100
2.7 Summary and Additional Problems, 109
Reference, 112
3 Binomial Series and Generating Functions 113
3.1 Introduction, 113
3.2 The Binomial and Multinomial Theorems, 114
3.3 Newton’s Binomial Series, 122
3.4 Ordinary Generating Functions, 131
3.5 Exponential Generating Functions, 147
3.6 Summary and Additional Problems, 163
References, 166
4 Alternating Sums, Inclusion-Exclusion Principle, Rook Polynomials, and Fibonacci Nim 167
4.1 Introduction, 167
4.2 Evaluating Alternating Sums with the
DIE Method, 168
4.3 The Principle of Inclusion–Exclusion (PIE), 179
4.4 Rook Polynomials, 191
4.5 (Optional) Zeckendorf Representations and Fibonacci Nim, 202
4.6 Summary and Additional Problems, 207
References, 210
5 Recurrence Relations 211
5.1 Introduction, 211
5.2 The Fibonacci Recurrence Relation, 212
5.3 Second-Order Recurrence Relations, 222
5.4 Higher-Order Linear Homogeneous Recurrence Relations, 233
5.5 Nonhomogeneous Recurrence Relations, 247
5.6 Recurrence Relations and Generating Functions, 257
5.7 Summary and Additional Problems, 268 
References, 273
6 Special Numbers 275
6.1 Introduction, 275
6.2 Stirling Numbers, 275
6.3 Harmonic Numbers, 296
6.4 Bernoulli Numbers, 306
6.5 Eulerian Numbers, 315
6.6 Partition Numbers, 323
6.7 Catalan Numbers, 335
6.8 Summary and Additional Problems, 345
References, 352
PART II TWO ADDITIONAL TOPICS IN ENUMERATION
7 Linear Spaces and Recurrence Sequences 355
7.1 Introduction, 355
7.2 Vector Spaces of Sequences, 356
7.3 Nonhomogeneous Recurrences and Systems
of Recurrences, 367
7.4 Identities for Recurrence Sequences, 378
7.5 Summary and Additional Problems, 390
8 Counting with Symmetries 393
8.1 Introduction, 393
8.2 Algebraic Discoveries, 394
8.3 Burnside’s Lemma, 407
8.4 The Cycle Index and P´olya’s Method of Enumeration, 417
8.5 Summary and Additional Problems, 430
References, 432
PART III NOTATIONS INDEX, APPENDICES, AND SOLUTIONS TO SELECTED ODD PROBLEMS
Index of Notations 435
Appendix A Mathematical Induction 439
A.1 Principle of Mathematical Induction, 439
A.2 Principle of Strong Induction, 441
A.3 Well Ordering Principle, 442
Appendix B Searching the Online Encyclopedia of Integer Sequences (OEIS) 443
B.1 Searching a Sequence, 443
B.2 Searching an Array, 444
B.3 Other Searches, 444
B.4 Beginnings of OEIS, 444
Appendix C Generalized Vandermonde Determinants 445
Hints, Short Answers, and Complete Solutions to Selected Odd Problems 449
INDEX 467

Mathematics of Choice: Or, How to Count Without Counting

(New Mathematical Library)

Ivan Morton Niven

Mathematical Association of America   | 1975 | 202 páginas | rar - PDF - 5,1 Mb

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This text is an engaging, even addictive, introduction to basic combinatorics. Written in a fun and inviting manner, reader interest is amplified by the author’s infectious enthusiasm. This is an excellent introduce to combinations and permutations. First published in 1975, before computers and calculators were assumed to be at hand, the exercises in this book can all be done by hand on paper. Students finishing high school or in their first year of college will find this work an excellent adjunct to textbooks and lectures.

The work is arranged in a logical progression beginning with the definitions and motivations for factorials, combinations, and permutations. From there the reader moves to binomial coefficients, power sets, and Fibonacci numbers. The effect of repetitions on combinations makes a natural prelude in Chapter Four to the Inclusion-Exclusion Principle and the groundwork for basic probability. From partitions of integers the author moves into a brief and basic, yet cogent and enlightening, explanation of generating functions and some applications for them. The book also includes the Pigeonhole Principle, induction, recursion, and allied topics.

Combinatorics: A Problem Oriented Approach

(Classroom Resource Materials)
Daniel A. Marcus

The Mathematical Association of America | 1999 | 152 páginas | rar-PDF | 451 kb

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Descrição: 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.


Contents
Part I. Basics 1
Section A: Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Section B: Combinations . . . . . . . . . . . . . . . . . . . . . . . . 13
Section C: Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 31
Section D: Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Part II. Special Counting Methods 61
Section E: Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . 63
Section F: Recurrence Relations . . . . . . . . . . . . . . . . . . . . 73
Section G: Generating Functions . . . . . . . . . . . . . . . . . . . . 87
Section H: The P´olya-Redfield Method . . . . . . . . . . . . . . . . . 97
List of Standard Problems . . . . . . . . . . . . . . . . . . . . . . . . . 119
Dependence of Problems . . . . . . . . . . . . . . . . . . . . . . . . . 123
Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . 127
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Combinatorics of Permutations

Miklos Bona

 2.ª edição 

CRC | 2012  | 473 páginas | PDF  | 3 Mb

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A Unified Account of Permutations in Modern Combinatorics
A 2006 CHOICE Outstanding Academic Title, the first edition of this bestseller was lauded for its detailed yet engaging treatment of permutations. Providing more than enough material for a one-semester course, Combinatorics of Permutations, Second Edition continues to clearly show the usefulness of this subject for both students and researchers.
Expanded Chapters
Much of the book has been significantly revised and extended. This edition includes a new section on alternating permutations and new material on multivariate applications of the exponential formula. It also discusses several important results in pattern avoidance as well as the concept of asymptotically normal distributions.
New Chapter
An entirely new chapter focuses on three sorting algorithms from molecular biology. This emerging area of combinatorics is known for its easily stated and extremely difficult problems, which sometimes can be solved using deep techniques from seemingly remote branches of mathematics.
Additional Exercises and Problems
All chapters in the second edition have more exercises and problems. Exercises are marked according to level of difficulty and many of the problems encompass results from the last eight years.
A Unified Account of Permutations in Modern CombinatoricsA 2006 CHOICE Outstanding Academic Title, the first edition of this bestseller was lauded for its detailed yet engaging treatment of permutations. Providing more than enough material for a one-semester course, Combinatorics of Permutations, Second Edition continues to clearly show the usefulness of this subject for both students and researchers.Expanded ChaptersMuch of the book has been significantly revised and extended. This edition includes a new section on alternating permutations and new material on multivariate applications of the exponential formula. It also discusses several important results in pattern avoidance as well as the concept of asymptotically normal distributions.New ChapterAn entirely new chapter focuses on three sorting algorithms from molecular biology. This emerging area of combinatorics is known for its easily stated and extremely difficult problems, which sometimes can be solved using deep techniques from seemingly remote branches of mathematics. Additional Exercises and ProblemsAll chapters in the second edition have more exercises and problems. Exercises are marked according to level of difficulty and many of the problems encompass results from the last eight years.

Combinatorics

N.I A. Vilenkin

Academic Press Inc | 1971 | 311 páginas | DJVU

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Professor Naum Ya. Vilenkin is not only a distinguished mathematician but also a gifted popularizer of significant mathematics. His "Stories about Sets" (Academic Press, 1968) ranged from a discussion of infinities to a discussion of the dimension of a manifold. His present book on combinatorics is a leisurely tour which takes the reader from very simple combinatorial problems to recurrence relations and generating functions and requires no more than a high school background in mathematics. The book includes a collection of hundreds (439) of problems with solutions.

Combinatorial Mathematics for Recreation

N. Vilenkin

MIR Publishers | 1972 | 207 páginas | Djvu | 3,76 Mb

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Course in Combinatorics

2.ª Edição
J. H. van Lint, R. M. Wilson

Cambridge University Press | 2001 | 550 páginas | pdf | 2,4 Mb

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Combinatorics, a subject dealing with ways of arranging and distributing objects, involves ideas from geometry, algebra, and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become an essential tool in many scientific fields. In this second edition the authors have made the text as comprehensive as possible, dealing in a unified manner with such topics as graph theory, extremal problems, designs, colorings, and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. It is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level, and working mathematicians and scientists will also find it a valuable introduction and reference.

Cardano, the gambling scholar

Oystein Ore

Princeton University Press | 1953 | 249 páginas | djvu | 4,4 Mb

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In order not to miss many of. Cardano, the gambling scholar by ystein Ore | LibraryThing Results from Google Books. Gerolamo Cardano, Physician Extraordinaire : World Research. He has been referred to as the Gambling Scholar.. Gerolamo Cardano – Wikipedia, the free encyclopedia His book about games of chance, Liber de ludo aleae (“Book on Games of Chance”), written in 1526,. It was through gambling that Cardano financed his studies in medicine at the university of Pavia.. The Gambling Scholar, Girolamo Cardano – Bookmaker News In 1564, an Italian by the name of Girolamo Cardano wrote a book about various games then occupying the attention of many of his contemporaries, including dice-based.

102 Combinatorial Problems

Titu Andreescu, Zuming Feng

Birkhäuser Boston | 2002 | 128 páginas |  pdf |

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Descrição: 102 Combinatorial Problems consists of carefully selected problems that have been used in the training and testing of the USA International Mathematical Olympiad (IMO) team. The text provides in-depth enrichment in the important areas of combinatorics by systematically reorganizing and enhancing problem-solving tactics and strategies. The book gradually builds combinatorial skills and techniques and not only broadens the student's view of mathematics, but is also excellent for training teachers.

Análise Combinatória e Probabilidade

Augusto César de Oliveira Morgado, João Bosco Pitombeira de Carvalho, Paulo Cezar Pinto Carvalho, Pedro Fernandez

Colecção Professor de Matemática, n.º 2
Sociedade Brasileira de Matemática | 179 páginas

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Descrição: A Análise Combinatória costuma causar perplexidade a alunos e professores. De um lado, tem-se a variedade de problemas interessantes, de simples enunciados, que se enquadram no seu âmbito. Do outro lado, o grande desafio àimaginação que a soluçâo que a solução desses problemas representa, sendo aparentemente cada um deles um caso em si, não enquadrável numa teoria geral. Essa idéia aparente, contudo, não écorreta. Há princípios gerais que permitem submeter muitos desses problemas a técnicas organizadas de resolução. Expor alguns desses princípios e ensinar, mediante diversos exemplos, como aplicá-los, é uma das finalidades desse livro.

A Path to Combinatorics for Undergraduates: Counting Strategies

Titu Andreescu, Zuming Feng

Birkhäuser Boston | 2003 | 228 páginas|

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Referência em: MathEduc

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

George Pólya: Collected Papers vol. 4: Probobility; Conbinarorics; Teaching and Learning in Mathematics

George Pólya, Gian-Carlo Rota

The MIT Press | 1984 | 676 páginas | Djvu | 11,1 MB

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Contém 18 artigos sobre o ensino da Matemática de George Pólya, além de artigos teóricos sobre ensino e probabilidades

Volume IV presents 20 papers on probability, 17 on combinatorics, and 18 on the teaching and learning of mathematics. Pólya has made a number of fundamental contributions to the first two fields, including perhaps the first use of the term "central limit theorem," but his major influence on mathematics has clearly been his approach to pedagogy. Many of the papers throughout these volumes have a strongly pedagogical flavor, but the papers in the third section of this volume focus squarely on the real business of how to do mathematics—how to formulate a problem and then create a solution.

Proofs that Really Count: The Art of Combinatorial Proof

(Dolciani Mathematical Expositions)
Arthur T. Benjamin, Jennifer Quinn

The Mathematical Association of America | 2003 | 208 páginas | Djvu | 4,9 MB

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Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.

Counting

M. Koh, Tay Eng Guan, Eng Guan Tay

World Scientific Publishing Company | 2002 | 124 páginas | RAR - PDF | 2,40 Mb

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Descrição:

This book is a useful, attractive introduction to basic counting techniques for upper secondary and junior college students, as well as teachers. Younger students and lay people who appreciate mathematics, not to mention avid puzzle solvers, will also find the book interesting. The various problems and applications here are good for building up proficiency in counting. They are also useful for honing basic skills and techniques in general problem solving. Many of the problems avoid routine and the diligent reader will often discover more than one way of solving a particular problem, which is indeed an important awareness in problem solving. The book thus helps to give students an early start to learning problem-solving heuristics and thinking skills.

Índice

Contents

Preface v

1. The Addition Principle 1

2. The Multiplication Principle 9

3. Subsets and Arrangements 17

4. Applications 25

5. The Bijection Principle 35

6. Distribution of Balls into Boxes 47

7. More Applications of (BP) 53

8. Distribution of Distinct Objects into Distinct Boxes 63

9. Other Variations of the Distribution Problem 67

10. The Binomial Expansion 73

11. Some Useful Identities 77

12. Pascal's Triangle 85

x Counting

13. Miscellaneous Problems 95

Books Recommended for Further Reading 105

Answers to Exercises 107

Index 111