Concrete Mathematics: A Foundation for Computer Science, 2nd edition

Published by Addison-Wesley Professional (February 28, 1994) © 1994

  • Ronald L. Graham
  • Donald E. Knuth
  • Oren Patashnik

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ISBN-13: 9780134389981
Concrete Mathematics: A Foundation for Computer Science
Published 1994
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ISBN-13: 9780201558029
Concrete Mathematics: A Foundation for Computer Science
Published 1994


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  • Also available for purchase as an ebook from all major ebook resellers, including

Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics.

"More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems."

The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.

Major topics include:

  • Sums
  • Recurrences
  • Integer functions
  • Elementary number theory
  • Binomial coefficients
  • Generating functions
  • Discrete probability
  • Asymptotic methods
(Most chapters contain Exercises.)

1. Recurrent Problems.

The Tower of Hanoi.

Lines in the Plane.

The Josephus Problem.


2. Sums.


Sums and Recurrences.

Manipulation of Sums.

Multiple Sums.

General Methods.

Finite and Infinite Calculus.

Infinite Sums.


3. Integer Functions.

Floors and Ceilings.

Floor/Ceiling Applications.

Floor/Ceiling Recurrences.

'mod': The Binary Operation.

Floor/Ceiling Sums.


4. Number Theory.


Factorial Factors.

Relative Primality.

'mod': The Congruence Relation.

Independent Residues.

Additional Applications.

Phi and Mu.


5. Binomial Coefficients.

Basic Identities.

Basic Practice.

Tricks of the Trade.

Generating Functions.

Hypergeometric Functions.

Hypergeometric Transformations.

Partial Hypergeometric Sums.

Mechanical Summation.


6. Special Numbers.

Stirling Numbers.

Eulerian Numbers.

Harmonic Numbers.

Harmonic Summation.

Bernoulli Numbers.

Fibonacci Numbers.



7. Generating Functions.

Domino Theory and Change.

Basic Maneuvers.

Solving Recurrences.

Special Generating Functions.


Exponential Generating Functions.

Dirichlet Generating Functions.


8. Discrete Probability.


Mean and Variance.

Probability Generating Functions.

Flipping Coins.



9. Asymptotics.

A Hierarchy.

O Notation.

O Manipulation.

Two Asymptotic Tricks.

Euler's Summation Formula.

Final Summations.


A. Answers to Exercises.

B. Bibliography.

C. Credits for Exercises.


List of Tables. 0201558025T04062001

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