Introduction to Combinatorics

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ISBN: 9781439806227 | 1439806225
Cover: Hardcover
Copyright: 9/7/2010

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This concise text covers the fundamentals of pure mathematical combinatorics, including combinations, permutations, and more sophisticated counting techniques; generating functions; difference equations; generalizations; and special functions. It also presents graph-theoretic terminology and elementary graph theory as it arises in combinatorics. The authors provide brief introductions to more advanced ideas, such as Latin squares, block designs, matrices, and Polya's theory of counting. They also include exercises at both the elementary and advanced levels as well as additional exercises and solutions on a supplementary website.

List of Figures	p. xiii
Preface	p. xv
Introduction	p. 1
Some Combinatorial Examples	p. 1
Sets, Relations and Proof Techniques	p. 13
Two Principles of Enumeration	p. 15
Graphs	p. 18
Systems of Distinct Representatives	p. 20
Exercises 1A	p. 22
Exercises 1B	p. 23
Problems	p. 25
Fundamentals of Enumeration	p. 27
Permutations and Combinations	p. 27
Applications of P (n,k) and $$$	p. 29
Permutations and Combinations of Multisets	p. 32
Applications and Subtle Errors	p. 36
Algorithms	p. 39
Exercises 2A	p. 41
Exercises 2B	p. 43
Problems	p. 44
The Pigeonhole Principle and Ramsey's Theorem	p. 47
The Pigeonhole Principle	p. 47
Applications of the Pigeonhole Principle	p. 48
Ramsey's Theorem - the Graphical Case	p. 51
Ramsey Multiplicity	p. 54
Sum-Free Sets	p. 56
Bounds on Ramsey Numbers	p. 59
The General Form of Ramsey's Theorem	p. 63
Exercises 3A	p. 63
Exercises 3B	p. 64
Problems	p. 66
The Principle of Inclusion and Exclusion	p. 69
Unions of Events	p. 69
The Principle	p. 71
Combinations with Limited Repetitions	p. 75
Derangements	p. 77
Exercises 4A	p. 80
Exercises 4B	p. 81
Problems	p. 83
Generating Functions and Recurrence Relations	p. 85
Generating Functions	p. 85
Recurrence Relations	p. 89
From Generating Function to Recurrence	p. 94
Exponential Generating Functions	p. 95
Exercises 5A	p. 97
Exercises 5B	p. 99
Problems	p. 100
Catalan, Bell and Stirling Numbers	p. 103
Introduction	p. 103
Catalan Numbers	p. 104
Stirling Numbers of the Second Kind	p. 108
Bell Numbers	p. 113
Stirling Numbers of the First Kind	p. 114
Computer Algebra and Other Electronic Systems	p. 117
Exercises 6A	p. 119
Exercises 6B	p. 121
Problems	p. 122
Symmetries and the Pólya-Redfield Method	p. 123
Introduction	p. 123
Basics of Groups	p. 124
Permutations and Colorings	p. 130
An Important Counting Theorem	p. 131
Pólya and Redfield's Theorem	p. 134
Exercises 7A	p. 138
Exercises 7B	p. 139
Problems	p. 140
Introduction to Graph Theory	p. 143
Degrees	p. 143
Paths and Cycles in Graphs	p. 146
Maps and Graph Coloring	p. 149
Exercises 8A	p. 153
Exercises 8B	p. 155
Problems	p. 156
Further Graph Theory	p. 159
Euler Walks and Circuits	p. 159
Application of Euler Circuits to Mazes	p. 164
Hamilton Cycles	p. 166
Trees	p. 170
Spanning Trees	p. 174
Exercises 9A	p. 180
Exercises 9B	p. 184
Problems	p. 187
Coding Theory	p. 189
Errors; Noise	p. 189
The Venn Diagram Code	p. 190
Binary Codes; Weight; Distance	p. 192
Linear Codes	p. 195
Hamming Codes	p. 198
Codes and the Hat Problem	p. 200
Variable-Length Codes and Data Compression	p. 201
Exercises 10A	p. 203
Exercises 10B	p. 204
Problems	p. 205
Latin Squares	p. 207
Introduction	p. 207
Orthogonality	p. 211
Idempotent Latin Squares	p. 217
Partial Latin Squares and Subsquares	p. 219
Applications	p. 221
Exercises 11A	p. 225
Exercises 11B	p. 228
Problems	p. 229
Balanced Incomplete Block Designs	p. 231
Design Parameters	p. 232
Fisher's Inequality	p. 236
Symmetric Balanced Incomplete Block Designs	p. 238
New Designs from Old	p. 240
Difference Methods	p. 242
Exercises 12A	p. 245
Exercises 12B	p. 247
Problems	p. 248
Linear Algebra Methods in Combinatorics	p. 251
Recurrences Revisited	p. 251
State Graphs and the Transfer Matrix Method	p. 253
Kasteleyn's Permanent Method	p. 260
Exercises 13A	p. 265
Exercises 13B	p. 267
Problems	p. 268
Sets; Proof Techniques	p. 271
Sets and Basic Set Operations	p. 271
The Principle of Mathematical Induction	p. 279
Some Applications of Induction	p. 281
Binary Relations on Sets	p. 283
Exercises A	p. 285
Exercises B	p. 286
Matrices and Vectors	p. 291
Definitions	p. 291
Vector and Matrix Products	p. 294
Inverses	p. 296
Determinants	p. 298
Exercises A	p. 299
Exercises B	p. 301
Some Combinatorial People	p. 305
Solutions to Set A Exercises	p. 313
Hints for Problems	p. 341
Solutions to Problems	p. 347
References	p. 367
Index	p. 375
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