Combinatorial reasoning : an introduction to the art of counting

Written by two well-known scholars in the field, Combinatorial Reasoning: An Introduction to the Art of Counting presents a clear and comprehensive introduction to the concepts and methodology of beginning combinatorics. Focusing on modern techniques and applications, the book develops a variety of effective approaches to solving counting problems.

Balancing abstract ideas with specific topical coverage, the book utilizes real world examples with problems ranging from basic calculations that are designed to develop fundamental concepts to more challenging exercises that allow for a deeper exploration of complex combinatorial situations. Simple cases are treated first before moving on to general and more advanced cases. Additional features of the book include:

* Approximately 700 carefully structured problems designed for readers at multiple levels, many with hints and/or short answers
* Numerous examples that illustrate problem solving using both combinatorial reasoning and sophisticated algorithmic methods
* A novel approach to the study of recurrence sequences, which simplifies many proofs and calculations
* Concrete examples and diagrams interspersed throughout to further aid comprehension of abstract concepts
* A chapter-by-chapter review to clarify the most crucial concepts covered

Combinatorial Reasoning: An Introduction to the Art of Counting is an excellent textbook for upper-undergraduate and beginning graduate-level courses on introductory combinatorics and discrete mathematics.

Author Notes

William Webb received both a Ph.D. in mobile radio and an M.B.A. from Southampton University.

Dr. Webb is director of strategy at Motorola. He is a fellow of the IEEE, a senior member of the IEEE, and a chartered engineer. Dr. Webb holds four patents and has also authored Introduction to Wireless Local Loop, Second Edition; The Complete Wireless Communications Professional; and Understanding Cellular Radio (Artech House 2000, 1999, 1998). He is listed in ÂWhoÂs Who in AmericaÂ.

050

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

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