The Andrews Festschrift: Seventeen Papers on Classical Number Theory and Combinatorics

429 Pages·2001·12.78 MB·English

by  George E. Andrews (auth.)

Foata · Han (Eds.)
The Andrews Festschrift
Springer-Verlag Berlin Heidelberg GmbH
George Andrews
at Maralea
September 1998
A courtesy of Michael joswig
Dominique Foata • Guo-Niu Han (Eds.)
The Andrews Festschrift
Seventeen Papers on Classical Number
Theory and Combinatorics
With 12 Figures
Springer
Editors
Dominique Foata
Departement de mathematique
Universite Louis Pasteur
7, rue Rene-Descartes
67084 Strasbourg, France
E-mail: [email protected]
Guo-Niu Han
I.R.M.A. et C.N.R.S.
Universite Louis Pasteur
7, rue Rene-Descartes
67084 Strasbourg, France
E-mail: [email protected]
Mathematics Subject Classification (2000): OSAlS, OSAl7, OSA30, OSA40, OSEOS,
OSElO, llP8l, llP82, llP83, llY40, llY65, 33Dl5, 33D52, 33D65, 68Rl5
CIP - Cataloging-in-Publication data
The Andrews festschrift: seventeen papers on classical number theory and
combinatorics/Dominique Foata; Guo-Niu Han, eds.
p.cm.
Includes bibliographical references.
ISBN 978-3-540-41491-9
1. Number theory. 2. Combinatorial analysis. I. Andrews, George E., 1938-11. Foata,
Dominique. III. Han, Guo-Niu, 1965-
QA241 .A498 2001
512'.7-dc21 2001018371
ISBN 978-3-540-41491-9 ISBN 978-3-642-56513-7 (eBook)
DOI 10.1007/978-3-642-56513-7
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Preface
This volume contains the scientific contributions that old friends, colleagues,
admirers, former students have made to George Andrews on the occasion of
his sixtieth birthday: seventeen papers ranging from Classical Number Theory
(the theory of partitions dear to George Andrews) to Classical and Algebraic
Combinatorics. Most of the papers were read at the 42-nd session of the
Seminaire Lotharingien de Combinatoire that took place at Maratea, Basilicata,
from August 31 to September 6. 1998.
The meeting was made possible thanks to thefinancial supportofthe Regione
Basilicata and the dipartimento di Matematica, Universita degli Studi della
Basilicataand was organized by our colleagues from Potenza, Domenico Senato,
Elvira Di Nardo and Paolo Vitolo.
We are most thankful to George Andrews for his original paper in which he
describes how several great mathematicians, dead or living, have influenced his
mathematical career.
The second paper could only have been written by his good friend, Richard
Askey, who appears to be. in Doron Zeilberger's inimitable language, the
"guru" of special functions, George himself being the "q-guru"! Richard very
convincingly explains how the year that George spent at Madison in 1975-76
was decisive for them both.
There is no better homage to George and his familiarity with Ramanujan's
works than this long memoir by Bruce C. Berndt and Ken Ono on Ramanujan's
Unpublished Manuscript and the Tau Functions. Bruce has spent a great part
of his career deciphering the manuscripts of the great Indian number theorist.
In his contribution jointly written with Ken he presents Ramanujan's complete
manuscript, providing details and an extensive commentary.
Will the marvelous :-;avoir-faire of classical analysts dealing with special
functions be replaced by advanced computer algebra softwares in the future?
Part of it, for sure. For the time being we welcome those new maple packages
prepared by professional mathematicians, such as Frank Garvan, which are ofa
great help in dealing with q-series calculations.
In the subsequent paper G.-N. Han. A. Randrianarivony and J. Zeng have
imagined a new class ofq-secant and q-tangent numbers whose ordinary gener
atingfunctions havesimplecontinuedfraction expansions. Those new q-numbers
also have fascinating combinatorial properties. We most welcome this contribu
tion to Combinatorial Number Theory, for it may help us fill the gap existing
between geometric interpretations derived from the algebraofexponential series
and those derived from the study ofcontinued fractions.
The simple proofs of several results on representations of numbers by
quadratic forms obtained by l\1ike Hirschhorn show his great knowledge of that
part of Classical Number Theory that goes back to Dirichlet and Lorenz.
vi Preface
Dongsu Kim and Dennis Stanton calculate the generating function for words
with several simultaneous weights, all derived from the celebrated major index
statistic. Their new results can be immediately applied to integer partition
theory.
Theextension ofFranklin's bijection found by David P. Little provides a very
convincing explanation for the appearances ofcertain terms in the expansion of
the pentagonal number product, when finitely many terms are deleted in the
beginning.
George E. Andrews and Peter Paule revisit the works of MacMahon, partic
ularly his computational method for solving problems in connection with linear
homogeneous diophantine inequalities and equations. They show that Partition
Analysis can be also used for proving hypergeometric multisum identities that
arise in physics and algebraic combinatorics.
The generalized Kostka polynomials are q-analogues of the multiplicities
of a specific finite-dimensional irreducible representation of the linear group.
Their combinatorial properties are difficult to handle. Anatol N. Kirillov, Anne
Schilling and Mark Shimozono review several representations of those polyno
mials, such as the charge, path space, quasi-particle and bosonic representa
tion. They also describe a bijection between Littlewood-Richardson tableaux
and rigged configurations.
The Jacobi triple product identity has fascinated many generations ofmath
ematicians. In a short note Herbert S. Wilffinds a new and surprising number
theoretic content ofthat classical identity.
The paper byArturoCarpi and Aldode Lucaon Wordsand Repeated Factors
belongs to a long research tradition that consists of studying the unavoidable
regularities ofwords ofarbitrary large length. Here they characterize the sets of
factors ofa finite word that make possible the reconstruction ofthe entire word.
The Californian School, duly represented by Adriano Garsia, Mark Haiman
and Glenn Tesler has made an extensive study of the Macdonald polynomials
in the past ten years. They are kind enough to send us a very significant part
of their monumental project that contains explicit plethystic formulas for the
Macdonald q,t-Kostka coefficients.
The survey proposed by S. Ole Warnaar on supernomial coefficients, Bailey's
lemma and Rogers-Ramanujan-type identities indicates that the famous Bailey
lemma now has an A -extension that makes possible the derivation of natural
2
analogues ofthe classical A -Rogers-Ramanujan identities. The paper also gives
2
the description of unsolved problems, such as the supernomial inversion, the
An- 1 and higher level extensions, as well as several conjectures involving An- 1
basic hypergeometric series, string functions and cylindric partitions.
It is still a mystery ofmathematical tradition to see the triple and quintuple
product identities proved and reproved periodically. The purpose of D. Foata
and G.-N. Han, on the occasion ofthe discovery ofa septuple product identity
imagined and proved by Farkas and Kra, is to show that all those identities,
Preface vii
including the brand new onC'. can be derived by using purely "manipulatorics"
methods, and probably by computer algebra softwares in the future.
In the square-ice model Korepin et al. have derived a partition function that
is symmetric in two sets of variables. What is not clear is how to separate the
variables. Alain Lascoux, with his efficient method ofdivided differences obtains
the desired symmetric function as a product of two rectangular matrices, each
of them involving only one set ofvariables.
In the past five years Christian Krattenthaler has obtained several precise
evaluations ofvarious countings. in particular in tiling. Most ofthose countings
were first expressed in terms of determinants. The greatest difficulty was to
calculatethem. In his paperChristianunveils the cluesofhisfabulous calculating
techniques.
All in all, it has been a rewarding experience to have Lotharingiameet Penn
sylvaniainthe person ofGeorge Andrewsontheshoresoftheold Mediterranean.
Each paper in this volume has an electronic version that can be read at the
web site of the Scminaire Lotharingien de Combinatoire at Vienna under the
URL
http://www.rnat.univie.ac.at/-slc/
Accordingly, each paper may already have been referred to as Scm. Lothar.
Comb. 42 (1999), B42x. with x ranging from a to q.
Dominique Foata, Guo-Niu Han
Strasbourg, October 2000.
Table of Contents
George E. Andrews
Some Debts I Owe 1
0 0 • • • • • •
Richard Askey
The Work of George Andrews: A Madison Perspective 17
0 • • • • • • • • • • •
Bruce C. Berndt, h:en Ono
Ramanujan's Unpublished Manuscript on the Partition and Tau Functions
with Proofs and Commentary. .......................................... 39
Frank Garvan
A q-product Tutorial for a q-series Maple Package.... III
0 0 •••••• 0 •• 0 •••••• 0 • •
G.-No Han, A. Randrianarivony, J. Zeng
Un autre q-analogue des nombres d'Euler 139
Michael D. Hirschhorn
Three Classical Rewlts on Representations ofaNumber 159
0 •••••••• 0 • •
Dongsu Kim, Dennis Stanton
Simultaneous maj Statistics 167
0 ••••••••• 0 0 ••••••••• 0 •
David P. Little
An Extension ofFranklin's Bijection 179
0 ••••••••• 0 • •
George E. Andrews, Peter Paule
MacMahon's Partition Analysis IV: Hypergeometric Multisums 189
Anatol N. Kirillov, Anne Schilling, Mark Shimozono
Various Representations of the Generalized Kostka Polynomials 209
0 • •
Herbert S. Wilf
The Number-Theoretic Content ofthe Jacobi Triple Product Identity 227
Arturo Carpi, Aldo de Luca
Words and Repeated Factors 231
0 ••••••••• 0 •••••••••• 0 ••••• 0 •••• 0 •
Adriano M. Garsia, Mark Haiman, Glenn Tesler
Explicit Plethystic Formulas for Macdonald q,t-Kostka Coefficients ..... 253
0 •• 0 •
S. Ole Warnaar
Supernomial Coefficients. Bailey's Lemma and Rogers-Ramanujan-type
Identities. A Survey ofResults and Open Problems 299
00 • •
x TableofContents
Dominique Foata, Guo-Niu Han
The Triple, Quintuple and Septuple Product Identities Revisited. ............ 323
Alain Lascoux
Square-Ice Enumeration 335
Christian Krattenthaler
Advanced Determinant Calculus. ........................................ 349
Some Debts lOwe
George E. Andrewsl
1 Department of Mathematics, The PennsylvaniaState University, University Park
PA 16802 USA
andrews~math.psu.edu
Abstract. The primary objects ofthis paper are:
(1) to acknowledge my debts to a number of important mathematicians who
have passed away, and
(2) to describe some of the beginnings ofseveral themes in my research.
1 Introduction
It is impossible for me to express adequately my gratitude to all who were
involved with the 42nd Seminaire Lotharingien held in Maratea, Italy during
September 1~5, 1998. I am especially indebted to Dominique Foata who really
made the trip possible for me and managed so many aspects of the conference
so well. Also Iextend special thanks to Peter Paule who originally suggested the
role I would play. The surprise of being asked to participate in a conference to
commemorate your 60th birthday is quite overwhelming. It is impossible to feel
worthy of such an event. However, I have always told others that such events
were important for the cohesiveness of the mathematical community, and so it
would hardly be consistent for me to back out when this event was proposed.
After some thought, it occurred to me that, I might use my participation to
honor some of those who are no longer with us and who influenced me and
many others. I will follow that. same general outline in this paper.
After a belated tribute to Marco Schiitzenberger, I will describe the begin
nings ofthe paths that eventually led to: (1) Ramanujan's Lost Notebook, (2)
Bailey Chains, (3) Determinant Evaluations, and (4) Partition Analysis. I will
conclude with a look at one ofmy current paths which I havecalled the Liouville
Mystery.
2 Marco Schiitzenberger
Schiitzenberger was a grand master of mathematics and a powerful and vivid
presence. I had the good fortune to meet him at a number of European confer
ences. mostly at Oberwolfach. He had great style and great insight.
IPartially supported by National Science Foundation Grant DMS-9206993 and
by the University ofStrasbourg.
D. Foata et al. (eds.), The Andrews Festschrift
© Springer-Verlag Berlin Heidelberg 2001