Concrete Mathematics

A FOUNDATION FOR COMPUTER SCIENCE

GRAHAM * KNUTH * PATASHNIK

Contents

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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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