Concrete Mathematics

A FOUNDATION FOR COMPUTER SCIENCE

GRAHAM * KNUTH * PATASHNIK

Contents


1. Recurrent Problems

1.1 The Tower of Hanoi 1.2 Lines in the Plane 1.3 The Josephus Problem Exercises

2. Sums

2.1 Notation 2.2 Sums and Recurrences 2.3 Manipulation of Sums 2.4 Multiple Sums 2.5 General Methods 2.6 Finite and Infinite Calculus 2.7 Infinite Sums Exercises

3. Integer functions

3.1 Floors and Ceilings 3.2 Floor/Ceiling Applications 3.3 Floor/Ceiling Recurrences 3.4 'mod': The Binary Operation 3.5 Floor/Ceiling Sums Exercises

4. Number Theory

4.1 Divisibility 4.2 Primes 4.3 Prime Examples 4.4 Factorial Factors 4.5 Relative Primality 4.6 'mod': The Congruence Relation 4.7 Independent Residues 4.8 Additional Applications 4.9 Phi and Mu Exercises

5. Binomial Coefficients

5.1 Basic Identities 5.2 Basic Practice 5.3 Tricks of the Trade 5.4 Generating Functions 5.5 Hypergeometric Functions 5.6 Hypergeometric Transformations 5.7 Partial Hypergeometric Sums 5.8 Mechanical Summation Exercises

6. Special Numbers

6.1 Stirling Numbers 6.2 Eulerian Numbers 6.3 Harmonic Numbers 6.4 Harmonic Summation 6.5 Bernoulli Numbers 6.6 Fibonacci Numbers 6.7 Continuants Exercises

7. Generating Functions

7.1 Domino Theory and Change 7.2 Basic Maneuvers 7.3 Solving Recurrences 7.4 Special Generating Functions 7.5 Convolutions 7.6 Exponential Generating Functions 7.7 Dirichlet Generating Functions Exercises

8. Discrete Probability

8.1 Definitions 8.2 Mean and Variance 8.3 Probability Generating Functions 8.4 Flipping Coins 8.5 Hashing Exercises

9. Asymptotics

9.1 A Hierarchy 9.2 0 Notation 9.3 0 Manipulation 9.4 Two Asymptotic Tricks 9.5 Buler's Summation Formula 9.6 Final Summations Exercises A Answers to Exercises B Bibliography C Credits for Exercises Index List of Tables

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