THE MATHEMATICAL LEGACY OF RICHARD P. STANLEY Patricia Hersh . Thomas Lam . Pavlo Pylyavskyy . Victor Reiner . Editors

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1324 2143 2134 1243 1234 φ THE MATHEMATICAL LEGACY OF RICHARD P. STANLEY

https://doi.org/10.1090//mbk/100

THE MATHEMATICAL LEGACY OF RICHARD P. STANLEY Patricia Hersh . Thomas Lam . Pavlo Pylyavskyy . Victor Reiner . Editors

AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 05A15, 05B35, 05Exx, 06A07, 06A11, 13F55, 52B05, 52B20, 52B22, 52C35.

For additional information and updates on this book, visit www.ams.org/bookpages/mbk-100

The photo of Richard P. Stanley is courtesy of Thomas Lam.

Library of Congress Cataloging-in-Publication Data Names: Hersh, Patricia, 1973– editor. | Lam, Thomas, 1980– editor. | Pylyavskyy, Pavlo, 1982– editor. | Reiner, Victor, 1965– editor. Title: The mathematical legacy of Richard P. Stanley / Patricia Hersh, Thomas Lam, Pavlo Pylyavskyy, Victor Reiner, editors. Description: Providence, Rhode Island : American Mathematical Society, 2016. Identifiers: LCCN 2016004438 | ISBN 9781470427245 (alk. paper) Subjects: LCSH: Stanley, Richard P., 1944– | Mathematicians–United States–Biography. | Combinatorial analysis. | AMS: Combinatorics – Enumerative combinatorics – Exact enumera- tion problems, generating functions. msc | Combinatorics – Designs and configurations – Matroids, geometric lattices. msc | Combinatorics – Algebraic combinatorics – Algebraic combi- natorics. msc | Order, lattices, ordered algebraic structures – Ordered sets – Combinatorics of partially ordered sets. msc | Order, lattices, ordered algebraic structures – Ordered sets – Alge- braic aspects of posets. msc | Commutative algebra – Arithmetic rings and other special rings – Stanley-Reisner face rings; simplicial complexes. msc | Convex and discrete geometry – Poly- topes and polyhedra – Combinatorial properties (number of faces, shortest paths, etc.). msc | Convex and discrete geometry – Polytopes and polyhedra – Lattice polytopes (including rela- tions with commutative algebra and algebraic geometry). msc | Convex and discrete geom- etry – Polytopes and polyhedra – Shellability. msc | Convex and discrete geometry – Discrete geometry – Arrangements of points, flats, hyperplanes. msc Classification: LCC QA29.S6735 M38 2016 | DDC 511/.6–dc23 LC record available at http://lccn.loc.gov/2016004438

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Contents

Preface ix Richard Stanley’s Short Curriculum Vitae xi Publications, with commentary by the author Richard P. Stanley 1 A survey of subdivisions and local h-vectors Christos A. Athanasiadis 39 Stanley’s major contributions to Ehrhart theory Matthias Beck 53 “Even more intriguing, if rather less plausible...” Face numbers of convex polytopes Louis J. Billera 65 The contributions of Stanley to the fabric of symmetric and quasisymmetric functions Sara C. Billey and Peter R. W. McNamara 83 “Let Δ be a Cohen-Macaulay complex ...” Anders Bjorner¨ 105 Stanley’s work on unimodality Francesco Brenti 119 Five stories for Richard 131 Some new applications of the Stanley-Macdonald Pieri rules Adriano Garsia, Jim Haglund, Guoce Xin, and Mike Zabrocki 141 A historical survey on P -partitions Ira M. Gessel 169 Transitive factorizations of permutations and geometry I. P. Goulden and D. M. Jackson 189 Stanley’s influence on monomial ideals Takayuki Hibi 203 Cohen-Macaulay varieties, geometric complexes, and combinatorics Melvin Hochster 219

vii viii CONTENTS

Plane partitions in the work of Richard Stanley and his school C. Krattenthaler 231 Combinatorial representation theory of Lie algebras. Richard Stanley’s work and the way it was continued Cristian Lenart 263 Lessons I learned from Richard Stanley James Propp 279 Richard Stanley through a crystal lens and from a random angle Anne Schilling 287 From poset topology to q-Eulerian polynomials to Stanley’s chromatic symmetric functions John Shareshian and Michelle L. Wachs 301 Stanley character polynomials Piotr Sniady´ 323 Some problems arising from partition poset homology Sheila Sundaram 335 Preface

Richard Stanley has had a profound impact on combinatorics. We hope that this book gives readers an opportunity to learn some of the mathematics that he has touched. Stanley and his PhD advisor, Gian-Carlo Rota, were at the vanguard in trans- forming combinatorics from a disparate collection of tricks into an organized and mature area of modern mathematics. In particular, Stanley’s talent for discovering deep examples captured the attention of the mathematical community at large. This book is a recollection of this journey, putting work of the past half-century within the context of the current mathematical scene.

——————–

Stanley graduated with a B.S. from the California Institute of Technology in 1966. He then became a graduate student at , where he worked with Gian-Carlo-Rota (who was a professor at MIT). In 1971, Stanley graduated with a Ph.D. in combinatorics, entitled Ordered Structures and Partitions;the survey by Gessel in this volume discusses this work. Stanley has supervised 59 doctoral students (56 at MIT, 3 at Harvard) and mentored countless postdocs and visitors. Following this preface, we include Stanley’s abbreviated curriculum vitae and a list of his doctoral students. A hallmark of Stanley’s work has been importation of ideas born outside com- binatorics to crack combinatorial problems. Examples include: • The introduction of tools from commutative algebra (local cohomology, the Cohen-Macaulay property, canonical modules, Stanley-Reisner rings, affine semigroup rings, and invariant rings) in the enumerative theory of - face numbers of simplicial complexes, polytopes, spheres, - solutions to linear homogeneous Diophantine equations, - partition analysis. • Application of the hard Lefschetz Theorem in algebraic geometry, along with representation theory of sl2(C) and of finite groups, to questions of unimodality and Sperner theory of posets, • Application of the Aleksandrov-Fenchel inequalities from convexity to log- concavity questions. • Application of symmetric function theory to partition identities, permu- tation statistics, and enumeration of reduced decompositions. A wealth of combinatorics viewed through his distinctive lens appears in his books, which are paragons of clarity, organization and elegance: • Combinatorics and Commutative Algebra.

ix xPREFACE

• Enumerative Combinatorics, Volumes 1 and 2 (fondly referred to1 as “EC1” and “EC2”). • Algebraic Combinatorics. • Catalan Numbers. His books do not tell the full story, however. We hope that the surveys within this volume help to round out this picture. ——————– This book grew, in a sense, from two conferences held in Stanley’s honor. On June 22-26, 2004, a conference titled2 “A children’s party” was held at the Massachusetts Institute of Technology in honor of his 60th birthday. A special Stanley Festschrift Volume of The Electronic Journal of Combinatorics (Vol. 11, Issue 2, edited by Bruce Sagan) published forty papers in his honor. A decade later, on June 23-27, 2014, a conference titled “Stanley@70” was held, also at MIT, in honor of Stanley’s 70th birthday. For example, the articles of Billera and Bj¨orner in this volume were developed in conjunction with the preparation of their historical talks given at this conference. Rather than collecting research monographs, in preparing this volume we so- licited survey papers by researchers with a variety of perspectives on Stanley’s work. He also kindly accepted our invitation to contribute a short reflection on each of his own papers, for which we are extremely grateful. Within it one will find several examples of his favorite proof technique, which he calls “proof by wishful thinking”. We were delighted by the overwhelming positive response we received from our contributors. We thank the many invited commentators, all of whom took valuable time from their schedules to help us compile this volume. We would not have been able to produce this volume without the assistance of the American Mathematical Society, and we particularly thank Sergei Gelfand, Edward Dunne, and Christine Thivierge for their efforts in this endeavor. We also thank G¨unter Ziegler and Anders Bj¨orner for the excellent idea of inviting Stanley to write short reflections on his papers. Most of all, we thank Richard for teaching us the joy of combinatorics.

Acknowledgments. We would also like to acknowledge the support of the National Science Foundation, the Simons Foundation and the Sloan Foundation through the grants NSF DMS-1200730 and NSF DMS-1500987 for the first editor, NSF DMS-1160726, NSF DMS-1464693, and a Simons Fellowship for the second editor, NSF DMS-1148634, NSF DMS-1351590, and a Sloan Fellowship for the third editor, and NSF DMS-1148634 and NSF DMS-100193 for the fourth editor. Patricia Hersh (#32) Thomas Lam (#43) Pavlo Pylyavskyy (#48) Victor Reiner (#14)

1A standard joke in our field quotes the EC1 or EC2 exercise number discovered later to subsume one’s own favorite recently proven observation or result. Stanley’s presentation of so many ideas in this understated manner typifies his style. 2Richard P. Stanley is known to enjoy anagrams. Richard Stanley’s Short Curriculum Vitae

EDUCATION:

California Institute of Technology B.S. 1966 Harvard University Ph.D. 1971

EMPLOYMENT:

1965-1969 Research Scientist, Jet Propulsion Lab, Pasadena, CA (summers) 1968-1970 Teaching Assistant, Harvard University 1970-1971 C.L.E. Moore Instructor of Mathematics, M.I.T. 1971-1973 Miller Research Fellow, University of California, Berkeley 1973-1975 Assistant Professor of Mathematics, M.I.T. 1975-1979 Associate Professor of Mathematics, M.I.T. 1979-2000 Professor of Applied Mathematics, M.I.T. 1993-1996 Chair, Applied Mathematics Committee, M.I.T. 1999-2000 Academic Officer, Department of Mathematics, M.I.T. 2000-2010 Norman Levinson Professor of Applied Mathematics, M.I.T. 2010- Professor of Applied Mathematics, M.I.T.

VISITING POSITIONS:

1978-79 Visiting Associate Professor of Mathematics, UC San Diego March, 1981 Universit´e Louis Pasteur, Strasbourg, France April-May, 1981 Stockholms universitet, Sweden Jan.-June, 1986 Sherman Fairchild Distinguished Scholar, CalTech May-June, 1990 Universit¨at Augsburg, Germany September, 1990 Tokai University, Japan November, 1990 Kungliga Tekniska H¨ogskolan (KTH), Stockholm, Sweden Jan.-May, 1992 G¨oran Gustafsson Professor, KTH and Institut Mittag-Leffler Sept. 1996-June 1997 Chern Visiting Professor, UC Berkeley Sept. 1996-June 1997 General Member, MSRI, Berkeley, California Sept. 2000-June 2001 Harvard University Jan.-June, 2005 KTH and Institut Mittag-Leffler, Sweden

xi xii RICHARD STANLEY’S SHORT CURRICULUM VITAE

PROFESSIONAL ACTIVITIES (selected):

Committee on the William Lowell Putnam Competition, 1983-1986 (Chair, 1985- 1986)

Academy Scholar, Clay Research Academy (eight-day seminar for high school stu- dents), 2003-2005

HONORS AND AWARDS:

SIAM George P´olya Prize in Applied Combinatorics, 1975 Guggenheim Fellowship, 1983-84 Fellow, American Academy of Arts and Sciences (elected 1988) Member, National Academy of Sciences (elected 1995) AMS Leroy P. Steele Prize for Mathematical Exposition, 2001 Rolf Schock Prize in Mathematics, 2003 Senior Scholar, Clay Mathematics Institute, 2004 Aisenstadt Chair, Centre de Recherches Math´ematiques, U. Montreal, 2007 Honorary Doctor of Mathematics, University of Waterloo, 2007 Honorary Professorship, Nankai University, 2007.

INVITED TALKS (selected):

Jubilee Celebration, U. Stockholm 1978, invited hour address AMS annual meeting, Cincinnati, 1982, invited hour address Conference on Combinatorics, U. Waterloo, 1982, series of four lectures ICM, Warsaw, 1983, invited 45 minute address in algebra Philips Lecturer, Haverford College, 1983 Distinguished Visitor, Emory University, 1984 Brauer Lecturer, University of North Carolina, 1986 Milliman Lecturer, , 1990 Distinguished Visitors Lecture Series, University of Iowa, 1991 Chern Symposium, University of California at Berkeley, 1997, principal speaker Leonidas Alaoglu Memorial Lecture in Mathematics, CalTech, 1997 Michigan State University Visiting Lecture Series, 1998 Erd¨os Lecturer, Hebrew University, Jerusalem, 1999 Stelson Lecturer, Georgia Institute of Technology, 1999 Mathematical Challenges of the 21st Century, UCLA, 2000, invited address Hayden-Howard Lecturer, University of Kentucky, 2001 Dean Jacqueline B. Lewis Memorial Lectures, Rutgers University, 2002 Mathematics Workshop, New Plymouth, New Zealand, 2003, two lectures IAS/PCMI, Graduate Summer School Lecturer, 2004 Senior Scholar Lecture, Park City, Utah, July 7, 2004 Nankai University, Tianjin, China, 2004, series of six lectures Distinguished Lectures in Mathematics and Computer Science, Haifa, 2005 Frontier Lectures, Texas A & M University, 2005 Distinguished Lecturer, Arizona State University, 2006 Plenary speaker, International Congress of Mathematicians, Madrid, 2006 RICHARD STANLEY’S SHORT CURRICULUM VITAE xiii

14th Leonard C. Sulski Memorial Lecture, College of the Holy Cross, 2007 Kemeny Lecture Series, Dartmouth College, 2007 Minicourse on symmetric functions, KAIST, Daejeon, Korea, 2008 Distinguished Lecturer, Drexel University, 2009 Plenary speaker, IPM 20 - Combinatorics 2009, Tehran, 2009 Colloquium Lectures, American Mathematical Society, San Francisco, 2010 Clifford Lectures, Tulane University, 2010 McKnight-Zame Distinguished Lecture, University of Miami, 2011

DOCTORAL STUDENTS (and date of degree):

Ira Gessel 1977 Emden Gansner 1978 Bruce Sagan 1979 Paul Edelman 1980 Robert Proctor 1981 Jim Walker 1981 Dale Worley 1984 John Stembridge 1985 Lynne Butler 1986 Karen Collins 1986 Sheila Sundaram 1986 Francesco Brenti 1988 Mark Purtill 1990 Victor Reiner 1990 David Wagner 1990 Julian West 1990 Art Duval 1991 Tom Roby 1991 Einar Steingr´ımsson 1991 Bo-Yin Yang 1991 Clara Chan 1992 G´abor Hetyei 1994 Timothy Chow 1995 David Grabiner 1995 (Harvard) Tao-Kai Lam 1995 Glenn Tesler 1995 Christos Athanasiadis 1996 Satomi Okazaki 1996 Mikl´os B´ona 1997 Alexander Postnikov 1997 Lewis Wolfgang 1997 Patricia Hersh 1999 Wungkum Fong 2000 Mark Skandera 2000 Benjamin Joseph 2001 Federico Ardila 2003 Peter Clifford 2003 xiv RICHARD STANLEY’S SHORT CURRICULUM VITAE

Caroline (Carly) Klivans 2003 Peter McNamara 2003 Edward Early 2004 Sergi Elizalde 2004 Cilanne Boulet 2005 Thomas Lam 2005 Lauren Williams 2005 Bridget Tenner 2006 Fu Liu 2006 Jason Burns 2007 Pavlo Pylyavskyy 2007 Denis Chebikin 2008 Jingbin Yin 2009 Camillia (Cammie) Smith 2009 (Harvard) Karola M´esz´aros 2010 Hoda Bidkhori 2010 Greta Panova 2011 (Harvard) Steven Sam 2012 Nan Li 2013 Yan Zhang 2013 Taedong Yun 2013 Benjamin Iriarte 2015 Selected Published Titles in This Series

100 Patricia Hersh, Thomas Lam, Pavlo Pylyavskyy, and Victor Reiner, Editors, The Mathematical Legacy of Richard P. Stanley, 2016 99 Hung-Hsi Wu, Teaching School Mathematics: Algebra, 2016 98 Hung-Hsi Wu, Teaching School Mathematics: Pre-Algebra, 2016 96 Erica Flapan, Knots, Molecules, and the Universe, 2016 95 Koryo Miura, Toshikazu Kawasaki, Tomohiro Tachi, Ryuhei Uehara, Robert J. Lang, and Patsy Wang-Iverson, Editors, Origami6, 2015 94 Philip J. Davis, Unity and Disunity and Other Mathematical Essays, 2015 93 Linda Keen, Irwin Kra, and Rub´ıE.Rodr´ıguez, Editors, Lipman Bers, a Life in Mathematics, 2015 92 Olli Lehto, Lars Ahlfors — At the Summit of Mathematics, 2015 91 Jennifer Granville, Sarah Jones Nelson, AnnMarie Perl, Miroslav Pet˘r´ı˘cek, Klaus Friedrich Roth, and Michael Spencer, Art in the Life of Mathematicians, 2015 90 Richard Evan Schwartz, You Can Count on Monsters, 2015 89 Steven G. Krantz, How to Teach Mathematics, Third Edition, 2015 88 Reuben Hersh, Peter Lax, Mathematician, 2015 87 Gregory V. Bard, Sage for Undergraduates, 2015 86 Boris A. Khesin and Serge L. Tabachnikov, Editors, ARNOLD: Swimming Against the Tide, 2014 85 V. I. Arnold, Mathematical Understanding of Nature, 2014 84 Richard Evan Schwartz, Really Big Numbers, 2014 83 Reuben Hersh, Experiencing Mathematics, 2014 82 Lawrence C. Evans, An Introduction to Stochastic Differential Equations, 2013 81 Terence Tao, Compactness and Contradiction, 2013 80 Elaine McKinnon Riehm and Frances Hoffman, Turbulent Times in Mathematics, 2011 79 Hung-Hsi Wu, Understanding Numbers in Elementary School Mathematics, 2011 78 Jeffrey C. Lagarias, Editor, The Ultimate Challenge, 2010 77 Terence Tao, An Epsilon of Room, II, 2011 76 Barbara Gellai, The Intrinsic Nature of Things, 2010 75 John B. Conway, Mathematical Connections, 2010 74 Modelling in Healthcare, 2010 73 George G. Szpiro, A Mathematical Medley, 2010 72 Martin Aigner and Ehrhard Behrends, Editors, Mathematics Everywhere, 2010 71 Alexandre V. Borovik, Mathematics under the Microscope, 2010 70 Mark Saul, Hadamard’s Plane Geometry, 2010 69 Herbert Edelsbrunner and John L. Harer, Computational Topology, 2010 68 Paul Pollack, Not Always Buried Deep, 2009 67 Terence Tao, Poincar´e’s Legacies, Part II, 2009 66 Terence Tao, Poincar´e’s Legacies, Part I, 2009 65 Rick Gillman and David Housman, Models of Conflict and Cooperation, 2009 64 Jean-Marie De Koninck, Those Fascinating Numbers, 2009 63 Miodrag S. Petkovi´c, Famous Puzzles of Great Mathematicians, 2009 62 Colin Adams, Riot at the Calc Exam and Other Mathematically Bent Stories, 2009 61 O. A. Ivanov, Making Mathematics Come to Life, 2009

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/mbkseries/. Richard Stanley’s work in combinatorics revolutionized and reshaped the subject. His lectures, papers, and books inspired a generation of researchers. In this volume, these researchers explain how Stanley’s vision and insights LQtXHQFHGDQGJXLGHGWKHLURZQSHUVSHFWLYHVRQWKHVXEMHFW$VDYDOXDEOH bonus, this book contains a collection of Stanley’s short comments on each of his papers. This book may serve as an introduction to several different threads of ongoing research in combinatorics as well as giving historical perspective.

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