zlib.pub introduction-to-combinatorial-analysis

An Introduction
to Combinatorial Analysis
John Riordan
DOVER PUBLICATIONS, INC.
Mineola, New York

Bibliographical Note
This Dover edition, first published in 2002, is an unabridged republication
of the work first published by John Wiley & Sons, Inc., New York, in 1958.
The errata list was added in a later printing.
Library of Congress Cataloging-in-Publication Data
Riordan, John, 1903–
Introduction to combinatorial analysis / John Riordan.—Dover ed.
p. cm.
Originally published: New York : John Wiley, 1958.
Includes bibliographical references and index.
eISBN 13: 978-0-486-15440-4
1. Combinatorial analysis. I. Title: Combinatorial analysis. II. Title.
QA164 .R53 2002
511’.6—dc21
2002073535
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

To E. T. Bell

Preface
COMBINATORIAL ANALYSIS, THOUGH A WELL-RECOGNIZED PART of
mathematics, seems to have a poorly defined range and position. Leibniz,
in his “ars combinatoria”, was the originator of the subject, apparently in
the sense of Netto (Lehrbuch der Combinatorik, Leipzig, 1901) as the
consideration of the placing, ordering, or choice of a number of things
together. This sense appears also in the title, Choice and Chance (W. A.
Whitworth, fifth edition, London, 1901), of one of the few books in English
on the subject. This superb title also suggests the close relation of the
subject to the theory of probability. P. A. MacMahon, in the most ambitious
treatise on the subject (Combinatory Analysis, London, vol. I, 1915, vol. II,
1916), says merely that it occupies the ground between algebra and the
higher arithmetic, meaning by the latter, as he later explains, what is now
called the Theory of Numbers.
A current American dictionary (Funk and Wagnalls New Standard, 1943)
defines “combinatoric”—a convenient single word which appears now and
then in the present text—as “a department of mathematics treating of the
formation, enumeration, and properties of partitions, variations,
combinations, and permutations of a finite number of elements under
various conditions”.
The term “combinatorial analysis” itself, seems best explained by the
following quotation from Augustus DeMorgan (Differential and Integral
Calculus, London, 1842, p. 335): “the combinatorial analysis mainly
consists in the analysis of complicated developments by means of a priori
consideration and collection of the different combinations of terms which
can enter the coefficients”.
No one of these statements is satisfactory in providing a safe and sure guide
to what is and what is not combinatorial. The authors of the three textbooks
could be properly vague because their texts showed what they meant. The
dictionary, in describing the contents of such texts, allows no room for new
applications of combinatorial technique (such as appear in the last half of
Chapter 6 of the present text in the enumeration of trees, networks, and
linear graphs). DeMorgan’s statement is admirable but half-hearted; in

present language, it recognizes that coefficients of generating functions
may be determined by solution of combinatorial problems, but ignores the
reverse possibility that combinatorial problems may be solved by
determining coefficients of generating functions.
Since the subject seems to have new growing ends, and definition is apt to
be restrictive, this lack of conceptual precision may be all for the best. So
far as the present book is concerned, anything enumerative is
combinatorial; that is, the main emphasis throughout is on finding the
number of ways there are of doing some well-defined operation. This
includes all the traditional topics mentioned in the dictionary definition
quoted above; therefore this book is suited to the purpose of presenting an
introduction to the subject. It is sufficiently vague to include new material,
like that mentioned above, and thus it is suited to the purpose of presenting
this introduction in an up-to-date form.
The modern developments of the subject are closely associated with the use
of generating functions. As appears even in the first chapter, these must be
taken in a form more general than the power series given them by P. S.
Laplace, their inventor. Moreover, for their combinatorial uses, they are to
be regarded, following E. T. Bell, as tools in the theory of an algebra of
sequences, so that despite all appearances they belong to algebra and not to
analysis. They serve to compress a great deal of development and allow the
presentation of a mass of results in a uniform manner, giving the book more
scope than would have been possible otherwise. By their means, that
central combinatorial tool, the method of inclusion and exclusion, may be
shown to be related to the use of factorial moments (which should be
attractive to the statistician). Finally, the presentation in this form fits
perfectly with the presentation of probability given by William Feller in
this series.
As to the contents, the following remarks may be useful. Chapter 1 is a
rapid survey of that part of the theory of permutations and combinations
which finds a place in books on elementary algebra, with, however, an
emphasis on the relation of these results to generating functions which both
illuminates and enlarges them. This leads to the extended treatment of
generating functions in Chapter 2, where an important result is the