Description
Universities Press Proofs that Really Count: The Art of Combinatorial Proof by T. Benjamin and Jennifer J. Quinn
Foreword
1 Fibonacci Identities
1.1 Combinatorial Interpretation of Fibonacci Numbers
1.2 Identities
1.3 A Fun Application
1.4 Notes
1.5 Exercises
2 Gibonacci and Lucas Identities
2.1 Combinatorial Interpretation of Lucas Numbers
2.2 Lucas Identities
2.3 Combinatorial Interpretation of Gibonacci Numbers
2.4 Gibonacci Identities
2.5 Notes
2.6 Exercises
3 Linear Recurrences
3.1 Combinatorial Interpretations of Linear Recurrences
3.2 Identities for Second-Order Recurrences
3.3 Identities for Third-Order Recurrences
3.4 Identities for kth Order Recurrences
3.5 Get Real! Arbitrary Weights and Initial Conditions
3.6 Notes
3.7 Exercises
4 Continued Fractions
4.1 Combinatorial Interpretation of Continued Fractions
4.2 Identities
4.3 NonsimpleContinued Fractions
4.4 Get Real Again!
4.5 Notes
4.6 Exercises
5 Binomial Identities
5.1 Combinatorial Interpretations of Binomial Coefficients
5.2 Elementary Identities
5.3 More Binomial Coefficient Identities
5.4 Multichoosing
5.5 Odd Numbers in Pascal’s Triangle
5.6 Notes
5.7 Exercises
6 Alternating Sign Binomial Identities
6.1 ParityArguments and Inclusion-Exclusion
6.2 Alternating Binomial Coefficient Identities
6.3 Notes
6.4 Exercises
7 Harmonic and Stirling Number Identities
7.1 HarmonicNumbers and Permutations
7.2 StirlingNumbers of the First Kind
7.3 Combinatorial Interpretation of Harmonic Numbers
7.4 Recounting Harmonic Identities
7.5 StirlingNumbers of the Second Kind
7.6 Notes
7.7 Exercises
8 Number Theory
8.1 Arithmetic Identities
8.2 Algebra and Number Theory
8.3 GCDs Revisited
8.4 Lucas’ Theorem
8.5 Notes
8.6 Exercises
9 Advanced Fibonacci & Lucas Identities
9.1 More Fibonacci and Lucas Identities
9.2 Colorful Identities
9.3 Some “Random” Identities and the Golden Ratio
9.4 Fibonacci and Lucas Polynomials
9.5 NegativeNumbers
9.6 Open Problems and Vajda Data
Some Hints and Solutions for Chapter Exercises
Appendix of Combinatorial Theorems
Appendix of Identities
Bibliography
Index