Table Of ContentFINITE
OPERATOR
CALCULUS
Academic Press Rapid Manuscript Reproduction
FINITE
O PERA TO R
CA LCU LU S
GIAN-CARLO ROTA
WITH THE COLLABORATION OF
P. Doubilet
G Greene
D. Kahaner
A. Odlyzko, and
R. Stanley
Academic Press, Inc.
New York San Francisco London 1975
A Subsidiary of Harcoyrt Brace Jovanovich, Publishers
Copyright © 1975, by A cademic Press, Inc.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
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ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
Rota, Gian-Carlo, (date)
Finite operator calculus.
Bibliography: p.
1. Combinatorial enumeration problems. 2. Lin-
ear operators. 3. Generating functions. Com-
mutative rings. 5. Valuation theory. I. Doubilet,
P., joint author. II. Title.
QA164.8.R67 515'.72 75-30776
ISBN 0-12-596650-4
PRINTED IN THE UNITED STATES OF AMERICA
To Nelson Dunford,
with affection and gratitude
CONTENTS
Introduction IX
CHAPTER 1 THE NUMBER OF PARTITIONS OF A SET 1
CHAPTER 2 FINITE OPERATOR CALCULUS 7
1. Introduction 7
2. Basic Polynomials 8
3. The First Expansion Theorem 13
4. The Pincherle Derivative 16
5. Sheffer Polynomials 20
6. Recurrence Formulas 25
7. Umbral Composition 27
8. Cross-Sequences 34
9. Eigenfunction Expansions 38
10. Hermite Polynomials 43
11. Laguerre Polynomials 48
12. Vandermonde Convolution 55
13. Examples and Applications 58
14. Problems and History 72
15. Acknowledgments 77
16. References 77
CHAPTER 3 THE IDEA OF GENERATING FUNCTION 83
1. Introduction 83
2. Notation and Terminology 85
3. Structure of the Incidence Algebra
4. Reduced Incidence Algebra 92
5. The Large Incidence Algebra 92
6. Residual Isomorphism 111
7. Algebras of Dirichlet Type 115
8. Algebras of Full Binomial Type 122
9. Algebras of Triangular Type 127
10. References 132
CONTENTS
CHAPTER 4 THE VALUATION RING 135
1. Introduction 135
2. Notation 135
3. The Valuation Ring 136
4. The Characteristic 139
5. Applications 143
6. Open Problems 145
7. References 146
CHAPTER 5 VALUATION RING AND MÖBIUS ALGEBRA 149
1. The Möbius Algebra of a Lattice 150
2. Partially Ordered Sets 152
3. Identities in the Möbius Algebra 155
4. References 158
VIII
INTRODUCTION
One always goes back to one’s first love, say the French. The proverb is accurate
so far as mathematics goes, and the present collection of papers could be taken as
a confirmation of it.
Solving, or at least understanding, a problem in combinatorics means many
things to many people, depending on their background and purposes. To me,
understanding a problem in combinatorics means reducing it to a problem in
linear operators or at least to modules over a commutative ring.
The first chapter (written by myself alone) shows how a classical problem in
enumeration is easily solved by using linear functionals on a vector space. It serves
as a motivation for the second chapter (written by myself with some collabora
tion from Kahaner and Odlyzko), which gives a systematic development—perhaps
the first one—of some formal aspects of the calculus of finite differences, stressing
special polynomials and identities. After comparing these two chapters, the reader
may be led to suspect that the second may be used for enumeration much as
partitions are enumerated in the first. Such a suspicion will be further
strengthened by a reading of the paper by Mullin and myself, which is not in
cluded here because the theory is still incomplete. There is in fact a full corres
pondence between set-theoretic constructions and the calculus of Chapter 2,
which will be presented elsewhere.
Chapter 3 is intended to carry out a message which again awaits further develop
ment. The calculus of finite differences is the algebraization of the reduced inci
dence algebra of a Boolean algebra. Upon replacing a Boolean algebra by some
other incidence algebra, other similar “calculi” are obtained, which are more
suitable for other combinatorial problems. For example, in studying graphs the
right “calculus” is the large reduced incidence algebra of the lattices of contrac
tions. This chapter, written by myself with much collaboration from P. Doubilet
and R. Stanley, takes the first step in this direction. I have since found that in
many cases reduced incidence algebras are actually Hopf algebras. This allows one
to introduce shift-invariant operators and to extend all of the techniques of
Chapter 2, but again the full devleopment will probably take several years.
Chapter 4 was written by myself alone; its purpose is to introduce the valuation
ring of a distributive lattice and to show how to compute with it. It seems
astonishing that this simple construction should have been overlooked—probably
because people were distracted by the all but useless Boolean ring-until I pre
sented it in 1967. Much further work has been done since, particularly by L.
Geissinger.
IX
