List of Tables
Sums and differences 55
Pascal’s triangle 155
Pascal’s triangle extended upward 164
Sums of products of binomial coefficients 169
The top ten binomial coefficient identities 174
General convolution identities 202
Stirling’s triangle for subsets 258
Stirling’s triangle for cycles 259
Basic Stirling number identities 264
Additional Stirling number identities 265
Stirling’s triangles in tandem 267
Euler’s triangle 268
Second-order Eulerian triangle 270
Stirling convolution formulas 272
Generating function manipulations 334
Simple sequences and their generating functions 335
Generating functions for special numbers 351
Asymptotic approximations 452
This book was composed at Stanford University using the TEX system for technical text developed by D. E. Knuth. The mathematics is set in a new typeface called AMS Euler (Version 2.1), designed by Hermann Zapf for the American Mathematical Society. The text is set in a new typeface called Concrete Roman and Italic, a special version of Knuth’s Computer Modern family with weights designed to blend with AMS Euler. The paper is 50-lb.-basis Bright White Finch Opaque, which has a neutral pH and a life expectancy of several hundred years. The offset printing and notch binding were done by the Courier Corporation of Westford, Massachusetts.
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills — the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists — the authors themselves rely heavily on it! — but for serious users of mathematics in virtually every discipline.
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 This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. Graham, Knuth, and Patashnik want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.
About the authors
RONALD L. GRAHAM is Chief Scientist at AT&T Labs Research. He is also University Professor of Mathematical Sciences at Rutgers University, and a former President of the American Mathematical Society. Dr. Graham is the author of six other mathematics books.
DONALD E. KNUTH is Professor Emeritus of The Art of Computer Programming at Stanford University. His prolific writings include three volumes on the Art of Computer Programming, and five books related to his TEX and META-FONT typesetting systems.
OREN PATASHNIK is a member of the research staff at the Center for Communications Research, La Jolla. He is also the author of BibTEX, a widely used bibliography processor.