Principles and Techniques in Combinatorics

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Product Details

  • ISBN-13: 9789810211394
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 7/28/1992
  • Edition description: New Edition
  • Pages: 296
  • Product dimensions: 5.99 (w) x 8.44 (h) x 0.63 (d)

Table of Contents

Preface
Notation and Abbreviation
Contents
1 Permutations and Combinations 1
1.1 Two Basic Counting Principles 1
1.2 Permutations 6
1.3 Circular Permutations 12
1.4 Combinations 17
1.5 The Injection and Bijection Principles 27
1.6 Arrangements and Selections with Repetitions 32
1.7 Distribution Problems 40
2 Binomial Coefficients and Multinomial Coefficients 69
2.2 The Binomial Theorem 70
2.3 Combinatorial Identities 71
2.4 The Pascal's Triangle 76
2.5 Chu Shih-Chieh's Identity 78
2.6 Shortest Routes in a Rectangular Grid 85
2.7 Some Properties of Binomial Coefficients 93
2.8 Multinomial Coefficients and the Multinomial Theorem 96
3 The Pigeonhole Principle and Ramsey Numbers 119
3.2 The Pigeonhole Principle 119
3.3 More Examples 122
3.4 Ramsey Type Problems and Ramsey Numbers 129
3.5 Bounds for Ramsey Numbers 132
4 The Principle of Inclusion and Exclusion 145
4.2 The Principle 146
4.3 A Generalization 148
4.4 Integer Solutions and Shortest Routes 153
4.5 Surjective Mappings and Stirling Numbers of the Second Kind 158
4.6 Derangements and A Generalization 160
4.7 The Sieve of Eratosthenes and Euler [phi]-function 163
4.8 The 'Probleme des Menages' 169
5 Generating Functions 185
5.1 Ordinary Generating Functions 185
5.2 Some Modelling Problems 192
5.3 Partitions of Integers 196
5.4 Exponential Generating Functions 204
6 Recurrence Relations 225
6.2 Two Examples 228
6.3 Linear Homogeneous Recurrence Relations 234
6.4 General Linear Recurrence Relations 241
6.5 Two Applications 244
6.6 A System of Linear Recurrence Relations 251
6.7 The Method of Generating Functions 256
6.8 A Nonlinear Recurrence Relation and Catalan Numbers 259
6.9 Oscillating Permutations and an Exponential Generating Function 262
Bibliography 287
Answers 289
Index 297
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