Mathematical Essays in Honor of Gian-Carlo Rota

Progress in Mathematics Volume 161

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein Mathematical Essays in honor of Gian-Carlo Rota

Bruce E. Sagan Richard P. Stanley Editors

Birkhauser Boston • Basel • Berlin Bruce E. Sagan Richard P. Stanley Department of Mathematics Department of Mathematics Michigan State University MIT East Lansing, MI 48824 Cambridge, MA 02139

Library of Congress Cataloging-in-Publication Data

Mathematical essays in honor of Gian-Carlo Rota / Bruce E. Sagan, Richard P. Stanley, editors. p. cm. -- (Progress in mathematics ; v. 161) Includes bibliographical references. ISBN-13: 978-1-4612-8656-1 e-ISBN-13: 978-1-4612-4108-9 DOl: 10.1007/978-1-4612-4108-9 1. Mathematics I. Rota, Gian-Carlo, 1932- II. Sagan, Bruce Eli. III. Stanley, Richard P., 1944- . IV. Series: Progress in mathematics (Boston, Mass.) ; vol. 161. QA7.M34445 1998 98-15395 510--dc21 CIP

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ISBN-13: 978-1-4612-8656-1

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987654321 Contents

Preface Bruce Sagan and Richard P. Stanley ...... vii

Rotafest Program xi

MacMahon's Partition Analysis: 1. The Lecture Hall Partition Theorem George E. Andrews ...... 1

The cd-Index of Zonotopes and Arrangements Louis J. Billera, Richard Ehrenborg, and Margaret Readdy 23

Letter-Place Methods and Homotopy David A. Buchsbaum ...... 41

Classification of Trivectors in 6-D Space Wendy Chan ...... 63

Parameter Augmentation for Basic Hypergeometric Series, I William Y. C. Chen and Zhi-Guo Liu ...... 111

Unities and Negation Henry Crapo and Claude Le Conte de Poly-Barbut ...... 131

The Would-Be Method of Targeted Rings Ottavio M. D'Antona ..... ' ...... 157

Lattice Walks and Primary Decomposition Persi Diaconis, David Eisenbud, and Bernd Sturmfels ...... 173

Natural Exponential Families and Umbral Calculus A. Di Bucchianico and D.E. Loeb ...... 195

Umbral Calculus in Hilbert Space A. Di Bucchianico, D.E. Loeb, and Gian-Carlo Rota ...... 213

A Strategy for Determining Polynomial Orthogonality J. M. Freeman ...... 239 vi Contents

Plethystic Formulas and Positivity for q, t-Kostka Coefficients A.M. Garsia and J. Remmel ...... 245

An Alternative Evaluation of the Andrews-Burge Determinant C. K rattenthaler ...... 263

The Number of Points in a Combinatorial Geometry with No 8-Point-Line Minors Joseph E. Bonin and Joseph P.S. Kung ...... 271

U mbral Shifts and Symmetric Functions of Schur Type Miguel A . Mendez...... 285

An Axiomization for Cubic Algebras Colin Bailey and Joseph Oliveira ...... 305

An Elementary Proof of Roichman's Rule for Irreducible Characters of Iwahori-Hecke Algebras of Type A A run Ram ...... 335

Universal Constructions in Umbral Calculus Nigel Ray ...... 343

Hyperplane Arrangements, Parking Functions and Tree Inversions Richard P. Stanley ...... 359

More Orthogonal Polynomials as Moments Mourad E. H. Ismail and Dennis Stanton ...... 377

Difference Equations via the Classical Umbral Calculus Brian D. Taylor ...... 397

An Analogy in Geometric Homology: Rigidity and Cofactors on Geometric Graphs Walter Whiteley ...... 413

The Umbral Calculus and Identities for Hypergeometric Functions with Special Arguments Jet Wimp ...... 439

Apologies to T.S. Eliot: The Rota Nerds J.S. Yang ...... 459 Preface

In April of 1996 an array of mathematicians converged on Cambridge, Massachusetts, for the Rotafest and Umbral Calculus Workshop, two con• ferences celebrating Gian-Carlo Rota's 64th birthday. It seemed appropriate when feting one of the world's great combinatorialists to have the anniversary be a power of 2 rather than the more mundane 65. The over seventy-five par• ticipants included Rota's doctoral students, coauthors, and other colleagues from more than a dozen countries. As a further testament to the breadth and depth of his influence, the lectures ranged over a wide variety of topics from invariant theory to algebraic topology. This volume is a collection of articles written in Rota's honor. Some of them were presented at the Rotafest and Umbral Workshop while others were written especially for this Festschrift. We will say a little about each paper and point out how they are connected with the mathematical contributions of Rota himself. One of Rota's earliest loves in combinatorics was the theory of partially ordered sets (posets). It is no accident that his seminal series of papers "On the Foundations of Combinatorial Theory" begins with an article on Mobius inversion [Z. Wahrscheinlichkeitstheorie 2 (1964), 340-368]' which is most naturally done in the poset setting. In later work Rota and Metropolis [Siam J. Appl. Math. 35 (1978) 689-694] characterized the lattice of faces of the n-cube, calling the structure a cubical algebra. In the present volume, Oliveira and Bailey show that one can give a list of universal axioms for these algebras inside the variety of implication algebras. In a different direction, Crapo investigates the representation of lattices in general by unities. Crapo and Rota wrote the next Foundations installment, this one on com• binatorial geometries [Studies in Appl. Math. 49 (1970), 109-133]. These objects, also known as matroids, provide a setting which simultaneously gen• eralizes ideas from linear algebra, matching theory, and the theory of graphs. Kung and Bonin in their contribution to this Festschrift provide a bound for the number of points in a combinatorial geometry with no minor isomor• phic to the 8-point line. Whiteley is concerned with matroid applications to static rigidity and splines. Diaconis, Eisenbud, and Sturmfels develop new connections between graph theory and commutative algebra by showing how primary decomposition of an ideal describes the components of a graph arising in problems from combinatorics, statistics, and operations research. The idea of bringing techniques from algebra and other branches of math• ematics to bear on combinatorial problems is one that runs throughout Rota's work. In particular, the concept of generating function has turned out to be a Preface viii fundamental tool which was the subject of Foundations VI, coauthored with Doubilet and Stanley [in "Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability," Vol. II: Probability Theory, Uni• versity of California Press, Berkeley, CA, 1971, 267-318]. Stanley's paper in this collection deals with the generating function for distance to a cham• ber from a fixed base chamber in the extended Shi hyperplane arrangement, generalizing the well-known enumeration of trees by number of inversions. Also in a geometric vein, Billera, Ehrenborg, and Readdy prove that the flag f-vectors of zonotopes satisfy no additional affine relations over and above those for all polytopes. It follows that the cd-index, essentially the generating function for the flag f-vector, is the most efficient way of encoding the affine information. Freeman presents a method for determining when a sequence of polynomials is orthogonal from its generating function. There will be more to say about orthogonal polynomials in connection with the umbral calculus. Some of Rota's most substantial contributions have been to invariant the• ory. In particular, the ninth Foundations article, written with Doubilet and Stein [Studies in Appl. Math. 53 (1974), 185-216], gives a characteristic-free approach to this topic. Its crowning achievement is a new proof of the First Fundamental Theorem of Invariant Theory using a straightening algorithm in the letter-place algebra. In this anthology, Buchsbaum demonstrates how letter-place methods can be used to construct homotopies for resolutions of certain Weyl modules, while Chan classifies the invariants and covariants of skew symmetric tensors of step three and dimension six in a characteristic• free way. One of the important applications of invariants is to representation theory, especially of the symmetric group Sn, and to the theory of symmetric functions. Ram's paper contains an elementary proof of Roichman's formula for the irreducible characters of the type A Iwahori-Hecke algebra. Garsia and Remmel consider the Kostka-Foulkes polynomials KAp.(q, t), which are the change of basis coefficients for two symmetric function bases, and prove Macdonald's conjecture that they have nonnegative integer coefficients for the special case when A is an augmented hook and J.t is arbitrary. Partitions not only index the irreducible representations of Sn, but also have many inter• esting properties of their own. Andrews uses MacMahon's partition analysis technique to prove a theorem of Bousquet-Melou and Eriksson which gener• alizes Euler's classic result that the number of partitions of n into odd parts equals the number into distinct parts. Krattenthaler evaluates the Andrews• Burge determinant for a certain symmetry class of plane partitions using a new method involving linearly independent linear combinations of the rows or columns of the determinant. Another area in which Rota's work has had widespread influence is the umbral calculus. In Foundations III [with Mullin in "Graph Theory and its Preface ix

Applications," Academic Press, New York, NY, 1970, 168-213] and VIII [with Kahaner and Odlyzko, J. Math. Anal. Appl. 42 (1973), 685-760]' Rota put substitution techniques from the 19th century on a rigorous footing via linear operators. Previous extensions of the theory to many variables had not been done in a basis-free way. But in their contribution to this Festschrift, Loeb, Di Bucchianico, and Rota show how one can use Hilbert space to accomplish this end. Taylor is interested in applications of the classical calculus to difference equations using Bernoulli umbrae, while D' Antona seeks to provide algebraic motivation for certain symbolic substitutions using his notion of a target ring. It is implicit in Rota's article with Roman [Adv. in Math. 27(2) (1978), 95-188] and explicit in his paper with Joni [Studies in Appl. Math. 61 (1979), 93-138] that the umbral calculus is connected with a Hopf algebra structure on the algebra of polynomials and the corresponding algebra structure on the dual. Ray's article herein explores what happens to umbral calculations done in the category of coassociative coalgebras over a commutative ring rather than a field, while Mendez investigates the relationship with the Hopf alge• bra of synunetric functions. One of the main motivations for this calculus is to generalize and unify results about orthogonal polynomials and hypergeo• metric series. Four of the articles in this volume apply umbral methods in this setting: Ismail and Stanton obtain orthogonal polynomials as moments for other orthogonal polynomials, Di Bucchianico and Loeb study the relation• ship between natural exponential families of probability measures and Sheffer polynomials, Wimp addresses the problem of finding closed forms for hyper• geometric series, and Chen derives hypergeometric identities by parameter augmentation. Of course, one-sentence summaries can not do justice to the papers in this Festschrift, just as this preface only modestly surveys a small portion of the large body of Rota's important work. We encourage the reader to delve more deeply!

Bruce E. Sagan Richard P. Stanley East Lansing Cambridge August, 1997 Gian-Carlo Rota, M.LT., Cambridge, MA ROTAFEST PROGRAM

Tuesday, April 16

7:00-9:30 p.m. reception at Charles Hotel

Wednesday, April 17

9:00-12:00 morning session (Andre Joyal, chair) 9:00-9:30 Adriano Garsia, The n!-conjecture and the q, t Kostka polynomials 9:45-10:15 Stephen Grossberg, Nonlinear dynamics of neural networks 10:45-11:15 Mark Haiman, N! is all you need 11:30-12:00 Lawrence Harper, The peaks of partition numbers

2:00-5:00 afternoon session (Erwin Lutwak, chair) 2:00-2:30 Jay Goldman, Combinatorics and knot theory 2:45-3:15 Daniel Klain, Invariant valuations on convex bodies 3:45-4:15 Joseph Kung, Line sizes and the number of points in a matroid 4:30-5:00 Andrew Odlyzko, Increasing subsequences in random permutations

Thursday, April 18

9:00-12:00 morning session (Steve Tanny, chair) 9:00-9:30 Willaim Schmitt 9:45-10:15 Bruce Sagan, Beyond semimodular lattices 10:45-11: 15 Pat O'Neil: CANCELLED 11:30-12:00 David Sharp, Raleigh-Taylor instability, chaotic mixing layer and stochastic PDE's

2:00-5:00 afternoon session (Peter Doubilet, chair) 2:00-2:30 Richard Stanley, Hyperplane arrangements, inversions, and trees 2:45-3:15 Joel Stein, The future of invariant theory 3:45-4:15 Bernd Sturmfels, Lattice walks and primary decomposition 4:30-5:00 Neil White, Coxeter matroids

6:00-10:00 banquet, Hyatt Regency Hotel

Friday, April 19

9:00-12:00 morning session (Joseph Oliviera, chair) 9:00-9:30 Walter Whiteley, Two matroids from geometric homology: an analogy with digressions 9:45-10:15 Kenneth Baclawski, Politically correct ordered sets: Socially responsible combinatorics 10:45-11:15 Wendy Chan, Classification of trivectors in 6-D space 11:30-12:00 David Buchsbaum, Letter-place methods and homotopy

afternoon free

6:30 dinner at Salamander around 6:30 alternate dinner at Royal East

Saturday, April 20

9:00-12:00 morning session (Curtis Greene, chair) 9:00-9:30 Henry Crapo, Unities and negation: On the representation of lattices 9:45-10:15 Gian-Carlo Rota, Ten lessons I should have been taught 10:45-11:15 Peter Duren, Recent progress on Bergman spaces 11:30-12:00 Richard Ehrenborg, Coproducts and the cd-index

2:00-5:00 afternoon session (Michael Hawrylycz, chair) 2:00-2:30 Steven Fisk, Q-analogs of simplicial complexes 2:45-3:15 Jack Freeman Mathematical Essays in honor of Gian-Carlo Rota