دانلود کتاب Proofs that really count: the art of combinatorial proof

فهرست مطالب :

Cover......Page 1
Title page......Page 3
Series......Page 4
Dedication......Page 7
Foreword......Page 9
Contents......Page 13
1.1 Combinatorial Interpretation of Fibonacci Numbers......Page 15
1.2 Identities......Page 16
1.3 A Fun Application......Page 25
1.4 Notes......Page 26
1.5 Exercises......Page 27
2.1 Combinatorial Interpretation of Lucas Numbers......Page 31
2.2 Lucas Identities......Page 32
2.4 Gibonacci Identities......Page 37
2.6 Exercises......Page 46
3 Linear Recurrences......Page 49
3.1 Combinatorial Interpretations of Linear Recurrences......Page 50
3.2 Identities for Second-Order Recurrences......Page 52
3.3 Identities for Third-Order Recurrences......Page 54
3.4 Identities for fcth Order Recurrences......Page 57
3.5 Get Real! Arbitrary Weights and Initial Conditions......Page 58
3.7 Exercises......Page 59
4.1 Combinatorial Interpretation of Continued Fractions......Page 63
4.2 Identities......Page 66
4.3 Nonsimple Continued Fractions......Page 72
4.5 Notes......Page 73
4.6 Exercises......Page 74
5.1 Combinatorial Interpretations of Binomial Coefficients......Page 77
5.2 Elementary Identities......Page 78
5.3 More Binomial Coefficient Identities......Page 82
5.4 Multichoosing......Page 84
5.5 Odd Numbers in Pascal\'s Triangle......Page 89
5.6 Notes......Page 91
5.7 Exercises......Page 92
6.1 Parity Arguments and Inclusion-Exclusion......Page 95
6.2 Alternating Binomial Coefficient Identities......Page 98
6.4 Exercises......Page 103
7.1 Harmonic Numbers and Permutations......Page 105
7.2 Stirling Numbers of the First Kind......Page 107
7.3 Combinatorial Interpretation of Harmonic Numbers......Page 111
7.4 Recounting Harmonic Identities......Page 112
7.5 Stirling Numbers of the Second Kind......Page 117
7.7 Exercises......Page 120
8.1 Arithmetic Identities......Page 123
8.2 Algebra and Number Theory......Page 128
8.3 GCDs Revisited......Page 132
8.4 Lucas\' Theorem......Page 134
8.6 Exercises......Page 137
9.1 More Fibonacci and Lucas Identities......Page 139
9.2 Colorful Identities......Page 144
9.3 Some \"Random\" Identities and the Golden Ratio......Page 150
9.4 Fibonacci and Lucas Polynomials......Page 155
9.6 Open Problems and Vajda Data......Page 157
Some Hints and Solutions for Chapter Exercises......Page 161
Appendix of Combinatorial Theorems......Page 185
Appendix of Identities......Page 187
Bibliography......Page 201
Index......Page 205
About the Authors......Page 208