*250*
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SELECTED WORKS OF

RICHARD P. STANLEY Patricia Hersh Thomas Lam Pavlo Pylyavskyy Victor Reiner Editors

American Mathematical Society

SELECTED WORKS OF

RICHARD P. STANLEY

SELECTED WORKS OF

RICHARD P. STANLEY Patricia Hersh Thomas Lam Pavlo Pylyavskyy Victor Reiner Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Joan S. Birman (Chair)

Jane Gilman

Kenneth A. Ribet 2010 Mathematics Subject Classiﬁcation. Primary 05A15, 05B35, 05Exx, 06A07, 06A11, 13F55, 52B05, 52B20, 52B22, 52C35.

Library of Congress Cataloging-in-Publication Data Names: Stanley, Richard P., 1944– | Hersh, Patricia, 1973- editor. | Lam, Thomas, 1980- editor. | Pylyavskyy, Pavlo, 1982-editor. | Reiner, Victor, 1965- editor. Title: Selected works of Richard P. Stanley / Patricia Hersh, Thomas Lam, Pavlo Pylyavskyy, Victor Reiner, editors. Description: Providence, Rhode Island : American Mathematical Society, c2017. | Series: Collected works ; 25. Identiﬁers: LCCN 2016006716 | ISBN 9781470416829 (alk. paper) Subjects: LCSH: Stanley, Richard P., 1944- | Combinatorial analysis. | Mathematicians–United States. | AMS: Combinatorics – Enumerative combinatorics – Exact enumeration problems, generating functions. msc | Combinatorics – Designs and conﬁgurations – Matroids, geometric lattices. msc | Combinatorics – Algebraic combinatorics – Algebraic combinatorics. msc | Order, lattices, ordered algebraic structures – Ordered sets – Combinatorics of partially ordered sets. msc | Order, lattices, ordered algebraic structures – Ordered sets – Algebraic aspects of posets. msc | Commutative algebra – Arithmetic rings and other special rings – Stanley-Reisner face rings; simplicial complexes. msc | Convex and discrete geometry – Polytopes and polyhedra – Combinatorial properties (number of faces, shortest paths, etc.). msc | Convex and discrete geometry – Polytopes and polyhedra – Lattice polytopes (including relations with commutative algebra and algebraic geometry). msc | Convex and discrete geometry – Polytopes and polyhedra – Shellability. msc | Convex and discrete geometry – Discrete geometry – Arrangements of points, ﬂats, hyperplanes. msc Classiﬁcation: LCC QA29.S6735 S73 2017 | DDC 511/.6–dc23 LC record available at http:// lccn.loc.gov/2016006716

c 2017 by the American Mathematical Society. All rights reserved. Printed in the United States of America. A complete list of permissions and acknowledgments can be found after the Foreword. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

22 21 20 19 18 17

Contents Foreword Patricia Hersh, Thomas Lam, Pavlo Pylyavskyy, and Victor Reiner

vii

Permissions & Acknowledgments

ix

Richard Stanley’s Short Curriculum Vitae

xiii

Photos

xvii

The early years

1

How the Upper Bound Conjecture was proved

3

Theory and application of plane partitions: part 1

11

Theory and application of plane partitions: part 2

33

Modular elements of geometric lattices

55

Supersolvable lattices

59

Linear homogeneous Diophantine equations and magic labelings of graphs

81

Acyclic orientations of graphs

107

Combinatorial reciprocity theorems

115

The Upper Bound Conjecture and Cohen-Macaulay rings

175

Combinatorial reciprocity theorems

183

Binomial posets, M¨ obius inversion, and permutation enumeration

195

Eulerian partitions of a unit hypercube

217

Hilbert functions of graded algebras

219

The number of faces of a simplicial convex polytope

247

Diﬀerentiably ﬁnite power series

251

Weyl groups, the hard Lefschetz theorem, and the Sperner property

265

Two combinatorial applications of the Aleksandrov-Fenchel inequalities

283

Linear Diophantine equations and local cohomology

293

Some aspects of groups acting on ﬁnite posets

313

RICHARD P. STANLEY

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CONTENTS

(with Anders Bj¨orner and Adriano M. Garsia) An introduction to Cohen-Macaulay partially ordered sets

343

An introduction to combinatorial commutative algebra

375

On the number of reduced decompositions of elements of Coxeter groups

391

A baker’s dozen of conjectures concerning plane partitions

405

Unimodality and Lie superalgebras

415

Two poset polytopes

435

Generalized H-vectors, intersection cohomology of toric varieties, and related results 451

vi

Diﬀerential posets

479

Log-concave and unimodal sequences in algebra, combinatorics, and geometry

523

Some combinatorial properties of Jack symmetric functions

559

Subdivisions and local h-vectors

599

(with Sara Billey and William Jockusch) Some combinatorial properties of Schubert polynomials

647

(with Sergey Fomin) Schubert polynomials and the nilCoxeter algebra

677

Flag f -vectors and the cd-index

689

A symmetric function generalization of the chromatic polynomial of a graph

707

Irreducible symmetric group characters of rectangular shape

737

Increasing and decreasing subsequences and their variants

749

Promotion and evacuation

785

A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group

809

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Foreword This volume collects some of Richard Stanley’s most inﬂuential papers. One ﬁnds in them a recurring theme: innocent counting problems can reveal deep structure, connecting them to algebra, geometry and topology. Many basic combinatorial objects, such as graphs, partially ordered sets, generating functions, simplicial complexes, polytopes, partitions, and reduced decompositions ﬁnd themselves illuminated by the theory of symmetric functions, representation theory, commutative algebra, and algebraic geometry. We hope that the reader is as surprised, delighted, and inspired by Stanley’s revelations as we have been. Patricia Hersh Thomas Lam Pavlo Pylyavskyy Victor Reiner

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Permissions & Acknowledgments The American Mathematical Society gratefully acknowledges the kindness of the following individuals and institutions in granting permission to reprint material in this volume: Centrum Wiskunde & Informatica Richard P. Stanley, Combinatorial reciprocity theorems. Combinatorics, Part 2: Graph theory, foundations, partitions and combinatorial geometry (Proc. Adv. Study Inst., Breukelen, 1974), pp. 107-118. Math Centre Tracts, No. 56, Math. Centrum, Amsterdam, 1974. Duke University Press “Linear homogeneous Diophantine equations and magic labelings of graphs”, Duke Mathematical Journal Volume 40 (1973), pp. 607-632. Copyright 1973. Duke University Press. All rights reserved. Republished by permission of the present publisher, Duke University Press. www.dukeupress.edu Karen Edwards Photograph of Richard Stanley with Anders Bj¨ orner and Tom Roby. Elsevier Reprinted from Advances in Mathematics, Volume 14, No. 2, Richard P. Stanley, “Combinatorial reciprocity theorems”, 194-253 (1974), with kind permission from Elsevier. Reprinted from European Journal of Combinatorics, Volume 5, No. 4, Richard P. Stanley, “On the number of reduced decompositions of elements of Coxeter groups”, 359-372 (1984), with kind permission from Elsevier. Reprinted from Advances in Mathematics, Volume 111, No. 1, R. P. Stanley, “A symmetric function generalization of the chromatic polynomial of a graph”, 166-194 (1995), with kind permission from Elsevier. Reprinted from Discrete Mathematics, Volume 306, Nos. 10-11, Richard P. Stanley, “Acyclic orientations of graphs”, 905-909 (2006), with kind permission from Elsevier. Reprinted from Advances in Mathematics, Volume 77, No. 1, Richard P. Stanley, “Some combinatorial properties of Jack symmetric functions”, 76-115 (1989), with kind permission from Elsevier. RICHARD P. STANLEY

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PERMISSIONS & ACKNOWLEDGMENTS

Reprinted from Journal of Combinatorial Theory, Series A, Volume 31, No. 1, Richard P. Stanley, “Two combinatorial applications of the Aleksandrov-Fenchel inequalities”, 56-65 (1981), with kind permission from Elsevier. Reprinted from Journal of Combinatorial Theory, Series A, Volume 32, No. 2, Richard P. Stanley, “Some aspects of groups acting on ﬁnite posets”, 132-161 (1982), with kind permission from Elsevier. Reprinted from Advances in Mathematics, Volume 103, No. 2, Sergey Fomin and Richard P. Stanley, “Schubert Polynomials and the NilCoxeter Algebra”, 196207 (1994), with kind permission from Elsevier. Reprinted from Journal of Combinatorial Theory, Series A, Volume 20, No. 3, Richard P. Stanley, “Binomial posets, M¨ obius inversion, and permutation enumeration”, 336-356 (1976), with kind permission from Elsevier. Reprinted from Advances in Mathematics, Volume 35, No. 3, Richard P. Stanley, “The number of faces of a simplicial convex polytope”, 236-238 (1980), with kind permission from Elsevier. Reprinted from Advances in Mathematics, Volume 28, No. 1, Richard P. Stanley, “Hilbert functions of graded algebras”, 57-83 (1978), with kind permission from Elsevier. Reprinted from Enumeration and Design (D. M. Jackson and S. A. Vanstne, eds.), “An introduction to combinatorial commutative algebra”, 3-18 (1984), with kind permission from Elsevier. Reprinted from European Journal of Combinatorics, Volume 1, No. 2, R. P. Stanley, “Diﬀerentiably Finite Power Series”, 175-188 (1980), with kind permission from Elsevier. European Mathematical Society Richard P. Stanley, “Increasing and decreasing subsequences and their variants” in Sanz-Sol´e, M., Soria, J., Varona, J. L. and Verdr, J. (eds.), Proceedings of the International Congress of Mathematicians, Madrid 2006, European Mathematical Society, 2006, pp. 545-579. Ira M. Gessel Photograph of Richard Stanley. Curtis Greene Two group photographs, including Richard Stanley. Patricia Hersh Photograph of Richard Stanley with Louis Billera and Michelle Wachs. Thomas Lam Photographs of Richard Stanley. Mathematical Society of Japan Richard P. Stanley, “Generalized H-vectors, intersection cohomology of toric varieties, and related results”, in Commutative Algebra and Combinatorics (M. Nagata and H. Matsumura, eds.), Advanced Studies in Pure Mathematics (ASPM)

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PERMISSIONS & ACKNOWLEDGMENTS

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11, Kinokuniya, Tokyo, and North-Holland, Amsterdam/New York, 1987, pp. 187213. Tom Roby Photograph of Richard Stanley and Tom Zaslavsky. Seminaire Lotharingien de Combinatoire Richard P. Stanley, “Irreducible symmetric group characters of rectangular shape”, S. Lothar. Combin. 50 (2003/04), Art. B50d, 11 pp. (electronic). Society for Industrial and Applied Mathematics Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. c Alg. Disc. Meth., 1 (1980), 168-184. Copyright 1980 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved. Springer Science+Business Media With kind permission from Springer Science+Business Media: Annals of Combinatorics, “How the Upper Bound Conjecture Was Proved”, Richard P. Stanley, Volume 18, no. 3 (2014). Springer eBook, “An Introduction to Cohen-Macaulay Partially Ordered Sets”, Ordered sets (Banﬀ, Alta., 1981), pp. 583-615, Anders Bj¨orner, Adriano M. Garsia, and Richard P. Stanley, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 83, Reidel, Dordrecht-Boston, Mass., 1982. Higher Combinatorics (M. Aigner, ed.), Proceedings of the NATO Advanced Study Institute held in Berlin, Germany, September 1-10, 1976, “Eulerian Partitions of a Unit Hypercube”, Richard P. Stanley, Reidel, Dordrecht-Boston, Volume 31, 1977, p. 49. Discrete & Computational Geometry, “Two poset polytopes”, Richard P. Stanley, Volume 1, No. 1 (1986). Springer eBook, A Baker’s Dozen of Conjectures Concerning Plane Partitions, Richard P. Stanley, 1986. Journal of Algebraic Combinatorics, “Some Combinatorial Properties of Schubert Polynomials”, Sara Billey, Richard Stanley, and William Jockusch, Volume 2, No. 4 (1993). Inventiones Mathematicae, “Linear Diophantine equations and local cohomology”, Richard P. Stanley, Volume 68, No. 2 (1982). Mathematische Zeitschrift, “Flag f -vectors and the cd-index”, Richard P. Stanley, Volume 216, No. 1 (1994). algebra universalis, “Modular elements of geometric lattices”, Richard P. Stanley, Volume 1, No. 1 (1971). algebra universalis, “Supersolvable lattices”, R. P. Stanley, Volume 2, No. 1 (1972). John Wiley and Sons Richard P. Stanley, “Log-concave and unimodal sequences in algebras, combinatorics, and geometry”, Annals of the New York Academy of Sciences, 36 pages. 2006.

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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

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RICHARD STANLEY’S SHORT CURRICULUM VITAE

PROFESSIONAL ACTIVITIES (selected): Committee on the William Lowell Putnam Competition, 1983-1986 (Chair, 19851986) Academy Scholar, Clay Research Academy (eight-day seminar for high school students), 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, University of Washington, 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

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RICHARD STANLEY’S SHORT CURRICULUM VITAE

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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

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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

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Photos

Top row, left-to-right: David Grabiner, Sergi Elizalde, Peter McNamara, Victor Reiner, Steven Sam, Yan Zhang, Nan Li, Einar Steingrimsson. 2nd row from top, left-to-right: Bo-Yin Yang, Caroline Klivans, Federico Ardila, Mark Skandera, Glenn Tesler, Edward Early, Patricia Hersh, Camillia Smith Barnes. 3rd row from top, left-to-right: Mark Purtill, Art Duval, Timothy Chow, Tom Roby, Greta Panova, Thomas Lam, Gabor Hetyei. Bottom row, left-to-right: Francesco Brenti, Karen Collins, Richard Stanley, Ira Gessel, Emden Gansner, Robert Proctor, John Stembridge. (Photo courtesy of Curtis Greene.)

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PHOTOS

Richard P. Stanley with many of his mathematical descendants and collaborators. (Photo courtesy of Curtis Greene.)

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PHOTOS

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Richard P. Stanley at his 70th birthday conference. (Photo courtesy of Ira Gessel.)

RICHARD P. STANLEY

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PHOTOS

Richard Stanley, Thomas Roby, and Anders Bj¨ orner. (Photo courtesy of Karen Edwards.)

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PHOTOS

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Richard Stanley and Thomas Zaslavsky. (Photo courtesy of Tom Roby.)

RICHARD P. STANLEY

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PHOTOS

Louis Billera, Michelle Wachs, and Richard Stanley. (Photo courtesy of Patricia Hersh.)

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THE EARLY YEARS My ﬁrst memory of a career plan dates back to the age of seven or eight, when I was much enamored with Paul Winchell, Jerry Mahoney, Edgar Bergen, and Charlie McCarthy and wanted to become a professional ventriloquist. At that time I was living in Tahawus, New York, a small mining town in the Adirondack Mountains which subsequently was moved (in 1963), to the nearby town of Newcomb1 . For many years Tahawus was perhaps the nicest ghost town in the Northeast, but much of it is now dismantled. Fortunately I did not continue with the goal of becoming a ventriloquist since the job market these days for such talent is even weaker than for mathematicians. I became interested in astronomy and for several years wanted to be an astronomer. From the ages of nine to thirteen I lived in Lynchburg, Virginia. One of my friends knew a woman named Mrs. Cochran (who I thought of as an elderly person). Mrs. Cochran saw that I liked mathematics (though at that time I had no special interest in the subject), so she taught me the standard synthetic algorithm for ﬁnding the square root of a positive real number x. If y = ai+1 10i+1 + ai+2 10i+2 + · · · is the approximation thus far, then the next digit ai will be the largest integer such that (y + 10i ai )2 ≤ x, or 10−2i y + 2 · 10−i ai + a2i ≤ 10−2i x. This rule seemed like complete magic to me. I understood why the analogous synthetic algorithm for division (“long division”) worked, but I had not the slightest understanding of the square root algorithm. Why, for instance, the mysterious factor of 2 in the term 2 · 10−i ? I asked Mrs. Cochran about this, but she only replied that I would understand it when I was older. I have no idea whether she actually understood it herself. I did spend lots of time computing square roots and trying to impress my parents that I could determine whether a positive integer was a perfect square. About that time I became aware of a copy of the CRC Handbook of Chemistry and Physics in our house, which my father used for his work as a chemical engineer. There was a section on mathematics, most of which was completely incomprehensible to me. However, in there appeared the synthetic square root algorithm and the corresponding algorithm for the cube 1

For the history of Tahawus, see L. A. Gereau, Tahawus Memories 1941–1963, Hungry Bear Publishing, Saranac Lake, NY.

1 RICHARD P. STANLEY

1

root. (Did anyone actually use this horrible algorithm?) If the square root algorithm was mysterious, then the cube root one was utterly mind-boggling. This to me was the ultimate arcane mathematical mystery which would be forever beyond my comprehension. At the age of fourteen I moved from Lynchburg to Savannah, Georgia2 . Shortly before then I had switched my main interest from astronomy to nuclear physics. In my ﬁrst day of school in the ninth grade at Wilder Junior High School I sat next to a classmate named Irvin Asher. (He later went on to obtain a B.S. and Ph.D. in physics from M.I.T., moved to Israel, and died in 2010.) He was totally absorbed in a complicated mathematical computation. I asked him what he was doing, and he explained (though of course not in these terms) that he was working out the synthetic algorithm for ﬁnding higher roots than the cubic! He had already worked out fourth through eighth roots (say, since I don’t remember exactly) and was now working on ninth roots. (He was essentially computing the coeﬃcients of the polynomials (x + 1)n . Needless to say we were both unaware of binomial coeﬃcients and the binomial theorem.) It seemed to me that I had a transcendent genuis for a classmate — at the very least equal to Newton and Einstein. This experience on the one hand was a jolt to my ego (since I had always been the top math student in my class), and on the other stirred up an interest in mathematics. I became determined to learn as much mathematics as I possibly could. It was on this fateful day that I was bitten by the mathematical bug and became incurably infected. My ﬁrst step in my mathematical self-education was to purchase the Barnes and Noble Outline of College Algebra and read it in several weeks. I then started reading every popular or recreational book on mathematics in the Savannah Public Library and the Savannah High School library. Thus began my journey in the realm of mathematics. 2

In case anyone is curious, here are the details of where I lived as a child. I was born in Manhattan in 1944. At the time my mother was living with her parents in Larchmont, New York, while my father was overseas during World War II. When my father returned we moved to New York City, where my father worked for my mother’s father trying to set up a wire business. Around a year later my father got a job with National Lead Company in Tahawus. About seven years after that I moved to Arlington, Massachusetts, and in a little over a year to Lynchburg. Then after four years (in 1958) I moved to Savannah. After I graduated from high school in Savannah, my parents moved to New Martinsville, West Virginia; Marietta, Georgia (a suburb of Atlanta); Sparks, Nevada (a suburb of Reno); and ﬁnally Englewood, Colorado (a suburb of Denver). My father (now deceased) kept moving or being transferred to diﬀerent chemical plants.

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Ann. Comb. 18 (2014) 533-539

I Annals of Combinatorics

DOI 10.1007/s00026-014-0238-5 P ublished online July 2, 2014 © Springer Basel 2014

How the Upper Bound Conjecture Was Proved Richard R Stanley Department o f Mathematics, Massachusetts Institute o f Technology, Cambridge, MA 021394307, USA [email protected] Received September 10, 2013

Mathematics Subject Classification: 05E40

Abstract. We give a short history of how the Upper Bound Conjecture for Spheres was proved.

Keywords: Upper Bound Conjecture, magic square, face ring, Cohen-Macaulay ring

I can trace the path of the proof of the Upper Bound Conjecture (UBC) for spheres back to high school. I went to high school in Savannah, Georgia. Around the age of 13 I became interested in mathematics, but there was no one who could give me proper guidance. One person, most likely an Armstrong Junior College* professor named Stubbs, did suggest that I join the Mathematical Association of America and receive their journal The American Mathematical Monthly, so I did this. Almost all of the articles were above my head. One paper [4], however, really caught my atten tion. It was entitled “Preferential arrangements.” A preferential arrangement of an /2-element set S would nowadays be called an ordered set partition. It is essentially a linear ordering of the elements of 5, allowing ties. More precisely, it is a sequence (Z?i,...,#fc) of pairwise disjoint nonempty subsets of S whose union is S. Thus there are thirteen preferential arrangements of {1, 2,3}, given by 1-2-3,1-3-2,2-1-3, 2-3-1, 3-1-2, 3-2-1, 12-3, 13-2, 23-1, 1-23, 2-13, 3-12, 123. Let f ( n ) denote the number of preferential arrangements of an 72-element set. The paper by Gross states the result that f(n ) =D" (2 —e1)-1 |z=0, which is equivalent to the generating function (due to Cayley) 1 £ / ( « ) n!

2 -e z

( 1. 1)

This seemed to me to be a truly amazing fact. I did not understand why it was true, but just the statement was pure magic. Why was there this mysterious connection between a discrete counting problem and the exponential function of calculus? * Now Armstrong Atlantic State University.

Birkhauser

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3

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R.P. Stanley

The next step occurred when I was an undergraduate at Caltech and took a course in analytic number theory from Tom Apostol. I was fascinated by the Riemann zeta function £ (s) and browsed through the book The Zeta Function of Riemann by E. C. Titchmarsh. On page 7 appears the following result. Let g(n) be the number of ways to write the integer n as an ordered product of integers all greater than 1. For instance, g(12) = 8, corresponding to 2 • 2 • 3, 2 • 3 • 2, 3 • 2 • 2, 2 • 6, 6 • 2, 3 • 4, 4 • 3, 12. Thus g{n) is some kind of “multiplicative analogue” of f(n ). Moreover, if n is a product of k distinct primes then g(n) = f(k ). Equation (1.2.15) asserts that y 8(n) „>i «*

1 2 -£ (* )'

( 1.2)

It was immediately clear to me that this formula was some kind of analogue of Equa tion (1.1), but at that time I had no idea how to make this feeling more precise. I also did not realize that this was the kind of question that a research mathematician might work on. When I was a junior at Caltech my adviser Marshall Hall asked me in the spring what I planned to do that summer. I told him that I was thinking of attending an NSF “math camp” for undergraduates (I can’t recall exactly what it was called), a precur sor of today’s REUs, somewhere in Oklahoma. Marshall Hall said something like, “Why would you want to go all the way to Oklahoma, Richard? Wouldn’t you rather stay here and work at JPL?” JPL was the Jet Propulsion Laboratory, operated by Cal tech for NASA. JPL was responsible for missions involving unmanned extraterrestrial spacecraft. I had no idea that working at JPL was even a possibility, but Marshall Hall arranged for me to be interviewed by Edward Posner, who was originally trained in ring theory and had become the head of the group that developed error-correcting codes used by the spacecraft to send and receive information. I was quite nervous going into this interview, worrying about how to answer questions like “What do you think you can contribute to our mission?” and “How do you expect to benefit from working at JPL?” However, the first question was “What are the Sylow theorems?” After this the interview went smoothly and I got the job. It was very exciting to be at JPL when missions such as Mariner and Voyager to the moon and other planets were in operation. I continued working there every summer until one year after I completed graduate school. I was working with a very strong group of mathematicians, including (in addition to Posner) Len Baumert, Bob McEliece, Gene Rodemich, and Howard Rumsey. Consultants and/or visitors in cluded Elwyn Berlekamp, Solomon Golomb, Irwin Jacobs, Gus Solomon, Herb Tay lor, and Andrew Viterbi. I can remember that Jacobs and Viterbi had recently started a company called Linkabit. Some of my JPL colleagues thought that this was a risky mistake in judgment. A perusal of the internet will allow the reader to judge this for himself or herself. In some work on coding theory, Bob McEliece needed a formula that turned out to be equivalent to the computation of the Mobius function of the lattice of partitions of a set. This result, originally due to Marcel Paul Schutzenberger and independently Roberto W. Frucht and Gian-Carlo Rota, appeared in a famous paper [7] of Rota (my future thesis adviser). I found this paper of Rota extremely interesting, both for the beautiful combinatorics and for the way that it unified seemingly disparate topics. In

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particular, I realized that if £ denotes the £ function of a locally finite poset P and if s < t in P , then (2 — t) is the number of chains s = to < t\ < • • • < = t from s to t. The similarity to Equations (1.1) and (1.2) was apparent. I realized that the theory of incidence algebras appearing in Rota’s paper must be the key for unifying the two formulas. What was needed was a connection with generating functions. At that time I did not have the proper perspective and came up with an unsatisfying contrived explanation. I don’t remember exactly which summer I had the ideas described above, but most likely 1967, one year after becoming a graduate student at Harvard. When I returned to Harvard after that summer I mentioned my work to some professors. Again I can not remember the precise details, but someone told me that Gian-Carlo Rota was at M.I.T. In fact, he had just returned from a two-year hiatus at Rockefeller University. I do remember that when Raoul Bott at some stage heard about my working for Rota, he asked me why I wanted to go into such a Mickey Mouse subject! However, Bott changed his tune later and showed great respect for Rota-style combinatorics. At any rate, I made an appointment with Rota and had a long, fruitful discussion. He in fact had been thinking along similar lines and had the key idea of looking for an isomor phism between rings of generating functions and subalgebras of incidence algebras. Very quickly I saw how to use this idea to develop the unification of various classes of generating functions. Peter Doubilet made some further contributions, eventually culminating in the paper [3]. What does all this have to do with the Upper Bound Conjecture? Because I en joyed algebra, I began wondering while at JPL what one could say about the structure of the incidence algebra I(P) of a finite poset P (over a field K). The first question that came to mind was the classification of two-sided ideals. It wasn’t difficult to show that for finite posets, the lattice of two-sided ideals of the incidence algebra I(P) is isomorphic to the lattice of order ideals of the set of intervals of P, ordered by reverse inclusion. In particular, the number of two-sided ideals is finite. I wondered what this number was when P is an ^-element chain, in which case 7(P) is isomorphic to the algebra of n x n upper triangular matrices over K. By a simple bijection I noticed that this number is the number of 2 x n matrices whose entries are 1, 2,..., 2n (each number appearing exactly once), with each row and column increasing. (At that time I had no idea that I was looking at a special class of standard Young tableaux.) By a very laborious argument which I have thankfully forgotten, I was able to show that this number is the Catalan number (2" ). When I mentioned this work to Bob McEliece, he told me that these 2 x n matri ces reminded him of plane partitions, a subject inaugurated by P. A. MacMahon. (For the history of plane partitions, see the notes to Chapter 7 in [9].) I started looking at MacMahon’s great opus [6], which I found quite fascinating. Plane partitions are a two-dimensional analogue of ordinary (linear) partitions. Thus it is natural to con sider extensions to higher dimensions, as first suggested by MacMahon. MacMahon was unable to make any progress on the enumeration of three-dimensional (or solid) partitions, so I decided to take a look. The only work in this area was a paper of E. M. Wright [10], in which he enumerated a very special class of solid partitions. To be more specific, let N denote the infinite chain 0 < 1 < 2 < • • •. The enumer ation of d-dimensional partitions of n is equivalent to the enumeration of ^-element

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order ideals of N^+1. Wright enumerated the n-element order ideals of the poset V x N2, where V was the three-element order ideal of N 2 with one minimal and two maximal elements. This suggested the question of enumerating ^-element order ide als of P x N 2 for any finite poset P. Wright used an induction argument where the base case was V x N. Thus I first needed to enumerate ^-element order ideals of P x N . Equivalently, one needs to count order-preserving maps a : P —>N according This turned out to be a very interesting question in itself to the sum n = Y*teP and was the inspiration for my doctoral thesis. No further progress has been made on the original question of the exact enumeration of solid partitions, though there have been interesting asymptotic results, e.g., [2 ]. A couple of years after obtaining my Ph.D. in 1970^ David Smith, a mathemat ics professor at Duke University, told me about an interesting conjecture which he thought might be related to my work. This was the Anand-Dumir-Gupta conjec ture [1] on “magic squares.” These magic squares are much weaker creatures than those considered in recreational mathematics; they are n x n matrices of nonnega tive integers whose rows and columns all have the same sum. Let Hn(r) denote the number of such magic squares with row and column sums r. Part of the conjecture of Anand-Dumir-Gupta is that for fixed n, Hn(r) is a polynomial in r. For instance, it is very easy to see that H\{r) = 1 and Hz(r) — r- h i. P. A. MacMahon [6 , §407] computed in 1916 that H$(r) = (r^ 5) — (r^ ), and David Smith himself computed H$(r) around 1970. He was right on the nose about my interest in this problem. Magic squares seemed very similar to the objects (called P-partitions, where P is a finite partially ordered set) studied in my thesis. Both magic squares and P-partitions involved nonnegative integer solutions to linear equalities and inequalities. I analyzed carefully the computation of i /3 (r) by MacMahon in the hope that it could be generalized. It was based on a general technique (now called the ElliottMacMahon algorithm) developed by himself and E. B. Elliott which MacMahon called the “syzygetic method.” After quite a bit of work on trying to figure out exactly what syzygies were and how they behaved, I finally realized that the key t To be precise, I had fulfilled all the requirements for obtaining a Ph.D. in 1970 but did not apply for a degree that year, so I began a postdoc at M.I.T. in fall 1970 without officially having a Ph.D. This is because the Vietnam War was in progress. I had a student deferment from the draft. On June 23, 1970, my draft eligibility would come to an end because of reaching the age of 26. That year there was also a draft lottery. Each day d of the year was randomly assigned a different number f ( d) from 1 to 365. If you were bom on day d then you could not be drafted until the f(d)t h day of the year. June 23 corresponded to 109, so I would be eligible (assuming no student deferment) on April 19. My student deferment would expire when classes ended in May, so I decided not to graduate although in practice draft boards were reluctant to give a graduate student deferment for more than four years. Indeed, my draft board (located in Savannah) informed me that my request for a further year’s deferment was denied, and that I would be reclassified 1-A (available to be drafted) on April 19. Thus I had about a two month period of draft eligibility. (As an amusing aside, in April I had to take a draft physical. In the middle of this exam, which was at a nearby army base, while we were standing around in our underwear, someone phoned in a bomb threat. We had to wait outside for about 20 minutes while the area was checked. One of the examiners told us that this happened every time the group from Cambridge was examined, and that we were lucky that this was April and not January.) Although I passed the draft physical, I never heard from my draft board so once June 23 arrived I was safe. I could also mention that M.I.T. had some lawyers who were engaged in helping persons associated with M.I.T. avoid the draft. During my period of eligibility they told me not to worry, even if I were drafted they could claim that my being at M.I.T. was essential for the nation’s welfare, and that such claims had always been honored.

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to the polynomiality of Hn{r) was the famous Hilbert syzygy theorem from 1890*. Moreover, I could use MacMahon’s syzygetic method to prove the other parts of the Anand-Dumir-Gupta conjecture, which concerned certain properties of the polyno mial Hn(r). This work resulted in the paper [8]. In this paper appears a geomet ric interpretation of MacMahon’s algorithm which, among other things, relates the polynomials Hn(r) (and some more general polynomials) to certain triangulations of polytopes, in particular, the number /• of /-dimensional faces of such triangulations for all / (encoded by the f-vector of the triangulation). At that time I had no interest in the / ’s themselves. This situation changed when I attended a talk of Peter McMullen at Stanford, most likely in 1973. There I learned of the beautiful Upper Bound Conjecture for spheres and of McMullen’s proof for convex polytopes based on the line shellings of Bruggesser and Mani. The UBC for spheres gives an explicit upper bound /• (n, d) for the number of /-dimensional faces of a triangulation of a (d — 1)-dimensional sphere with n vertices. This bound is achieved by the cyclic polytopes, so if true is best possible. I wondered whether the / ’s appearing in my paper on the Anand-DumirGupta conjecture could have anything to do with the UBC and whether the algebraic machinery I had developed could give some information about the /• ’s. At that time this thought was just wild speculation. In retrospect it is remarkable that it turned out to be so fruitful. I did not have any special insight into why commutative algebra might be related to the UBC; I simply noticed that the UBC was about the number of /-dimensional faces of some geometric object and that the machinery used to prove the Anand-Dumir-Gupta conjecture was also related to such numbers. It was clear from the beginning that the rings I looked at in my paper on magic squares (which later turned out to be just the coordinate rings of projective toric vari eties) were not going to be general enough to prove the UBC. I needed a ring whose Hilbert function had a certain relation to the /-vector of a triangulation A of a sphere. It wasn’t long before I realized that a certain graded algebra AT[A] = K[jq , . .. , xn\/I (now called theface ring or Stanley-Reisner ring of A), where I is an ideal of the poly nomial ring K[x\ generated by certain squarefree monomials, would have the right Hilbert function. At that time I thought that this ring was too simple to be of use. In particular, the variety associated with such a ring is just a union of linear subspaces, hardly of great interest to algebraic geometers. I had been discussing my work with Ken Baclawski. He told me about the pa per [5] of Mel Hochster. In this paper Hochster shows that the rings I used in my paper on magic squares (among others) are Cohen-Macaulay rings. This was the first time I had heard of Cohen-Macaulay rings. Hochster’s paper involved a lot of in teresting polyhedral combinatorics, and I began to wonder whether Cohen-Macaulay rings might have something to do with the UBC. It wasn’t clear to me what effect the Cohen-Macaulay propery of a ring had on its Hilbert function. I asked some algebraic geometers at M.I.T. about this, including Michael Artin. Artin said something like, “Aren’t Cohen-Macaulay rings free over something? This is due to Hironaka . . . ” Though it was unclear to me exactly what Artin meant, I could see immediately that * The Hilbert syzygy theorem is a result in commutative algebra. I had taken a course in graduate school on commutative algebra that I did not find very interesting. It did not cover the Hilbert syzygy theorem. I had to learn quite a bit of commutative algebra from scratch in order to understand the work of Hilbert.

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being “free over something” could have an influence on the Hilbert function. At this point my memory becomes a little murky. I started having discussions with David Eisenbud, who became an invaluable resource. Eventually it became clear to me that Cohen-Macaulay graded algebras R = R q (BR\ ® • • •, generated by R\, where Rq — K and K is infinite, were finitely-generated free modules over a polynomial subring K[0\ , . . . , 0^], with 0* R\. (This result is essentially equivalent to the Noether normalization applied to R.) It turns out that if A triangulates a sphere and if K[A] is Cohen-Macaulay, then the freeness result yields a bound on the /-vector of A which exactly coincides with the UBC — a truly astonishing “coincidence” ! At this point I was starting to feel that my approach might actually lead somewhere. Could it be just an accident that the Cohen-Macaulay property gave precisely the result I was looking for? It remained to show that AT[A] is Cohen-Macaulay when A triangulates a sphere. At that time I had nowhere enough background in commutative and homological algebra to prove this result. However, I was able to use Hochster’s work to show that a very special class of triangulated spheres satisfied the UBC. These spheres may be described as follows: let ? b e a convex d-polytope in R d with integer vertices. Suppose that V has a triangulation A, also with integer vertices, which is primitive (the ^-dimensional faces have volume 1/d\, the minimum possible), and every face of A whose interior lies in the interior of V has dimension greater than [d/2\. Then the boundary of A (a triangulation of a (d — l)-sphere) satisfies the UBC. This result is too special to be of much interest (and in fact it can be deduced from the UBC for convex polytopes proved earlier by McMullen), but it does provide some indication that it might be possible to show that AT[A] is Cohen-Macaulay when A triangulates a sphere. Although I did not have the background to prove the needed result about AT[A], the same could not be said of Mel Hochster. He independently defined this ring, having no idea at the time of its potential applicability to combinatorics, as a kind of combinatorial analogue of the rings which he had considered in [5]. He gave to his student Gerald Reisner the problem of determining when AT[A] was Cohen-Macaulay, or more generally, computing the depth of IST[A]. In 1974 I attended the International Congress of Mathematicians in Vancouver. I was scheduled to give a ten-minute submitted (uninvited) talk for which I planned to discuss my partial result mentioned above. I attended an invited talk by Victor Klee in which he stated that the UBC for spheres was one of the main open problems related to polyhedra. Soon after that I ran into David Eisenbud at the meeting. He informed me that Reisner had found a complete characterization of Cohen-Macaulay face rings A"[A], In particular, AT[A] was Cohen-Macaulay whenever A triangulated a sphere. Thus the UBC for spheres was proved! This was the greatest “math high” of my career, which lasted throughout the meeting. A little after talking to David Eisenbud, I ran into Peter McMullen (who was also an invited speaker, though his talk was unrelated to the UBC) and could inform him of the great news. About a day later was my ten-minute talk, in which I could say that the UBC had been proved two days earlier.

Morals. 1. The shortest path may not be the best.

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2. Even if you don’t arrive at your destination, the journey can still be worthwhile. References 1. Anand, H., Dumir, V.C., Gupta, H.: A combinatorial distribution problem. Duke Math. J. 33(4), 757-769 (1966) 2. Baiakrishnan, S., Govindarajan, S., Prabhakar, N.S.: On the asymptotics of higher dimen sional partitions. J. Phys. A: Math. Theor. 45, #055001 (2012) 3. Doubilet, P., Rota, G.-C., Stanley, R.: On the foundations of combinatorial theory. VI. The idea of generating function. In: Le Cam, L.M., Neyman, J., Scott, E.L. (eds.) Pro ceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971). Vol. II: Probability Theory, pp. 267-318. Univ. California Press, Berkeley, Calif. (1972) 4. Gross, O.A.: Preferential arrangements. Amer. Math. Monthly 69, 4 -8 (1962) 5. Hochster, M.: Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. of Math. (2) 96, 318-337 (1972) 6. MacMahon, P.A.: Combinatory Analysis. Volumes 1, 2. Cambridge University Press, Cambridge (1916); reprinted by Chelsea, New York (1960), and by Dover, New York (2004) 7. Rota, G.-C.: On the foundations o f combinatorial theory I. Theory o f Mobius functions. Z. Wahrscheinlichkeitstheorie 2, 340-368 (1964) 8. Stanley, R.: Linear homogeneous Diophantine equations and magic labelings o f graphs. Duke Math. J. 40, 607-632 (1973) 9. Stanley, R.: Enumerative Combinatorics, Vol. 2. Cambridge University Press, Cambridge (1999) 10.

Wright, E.M.: The generating function o f solid partitions. Proc. Roy. Soc. Edinburgh Sect. A 67, 185-195 (1965/1967)

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Studies in Applied Mathematics. Vol. L No. 2, June 1971. Copyright © 1971 by The Massachusetts Institute of Technology

Theory and Application of Plane Partitions: Part 1 S By Richard P. Stanley

I.

Introduction 1. Definitions

II. Symmetric Functions 2. The four basic symmetric functions 3. Relations among the symmetric functions 4. An inner product

III. Schur Functions 5. 6. 7. 8. 9. 10. 11. 12. 13.

The combinatorial definition The correspondence o f Knuth Kosta’s theorem and the orthonormality o f the Schur functions Further properties o f Knuth’s correspondence The dual correspondence The classical definition o f the Schur functions The Jacobi-Trudi identity Skew plane partitions and the multiplication of Schur functions Frobenius’ formula for the characters o f the symmetric group

IV. Enumeration of Column-Strict Plane Partitions 14. 15. 16. 17.

Part restrictions Shape restrictions, hook lengths, and contents Row and column restrictions Young tableaux, ballot problems, and Schensted’s theorem

V. Enumeration of Ordinary Plane Partitions 18. Row, column, and part bounds 19. The conjugate trace and trace o f a plane partition 20. Asymptotics

VI. Conclusion 21. Open problems * The research was supported by the Air Force Office o f Scientific Research AF 44620-70-C-0079. 167

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I. Introduction 1. Definitions A partition X of a non-negative integer n can be regarded as a decreasing sequence of positive integers,

Xx > X2 > • - > Xr > 0

(1)

satisfying = n. We say that X has r parts. Because of the linear nature of the array (1), we also refer to X as a linear partition of n. Similarly a partition of n into distinct parts may be regarded as a strictly decreasing array of positive integers,

Xx > X2 > • • • > Xr > 0,

(2)

satisfying = n. Such a partition is called a strict partition of n. We denote partitions in three w ays: (i) A i— n signifies that X is a partition of n (a notation due to Philip Hall [35]); (ii) X = (X1,X2, . ..) signifies that the parts of X are Xx > X2 > . . . , (iii) X =

nu ^ »i(j+i).

for a U U ^ 1-

The non-zero entries ntj > 0 are called the parts of n. If there are Xt parts in the ith row of 7c, so that for some r,

X1 > X2 > • • • > Xr > 2r+ x = 0, then we call the partition Xx > X2 > • • • > Xr of the integer p = X1 + X2 + • • • + Xr the shape of n, denoted by X. We also say that n has r rows and p parts. Similarly if X\ is the num ber of parts in the ith column of 77;, then for some c,

X\ > X'2 > ••• > X'c > Xc+1 = 0. The partition X[ > X2 > • • • > X'c of p is the conjugate partition to X [6, Ch. 19.2], denoted X\ and we say that n has c columns.

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If the non-zero entries of n are strictly decreasing in each row, we say that n is row-strict, Column-strict is similarly defined. If n is both row-strict and columnstrict, we say that n is row and column-strict. II. Symmetric functions 2. The four basic symmetric functions The wide variety of results known about plane partitions can be unified greatly by appealing to the theory of symmetric functions. We use a method involving elementary linear algebra, due to Philip Hall [35]. Let A n denote the set of all homogeneous symmetric functions of degree n in the infinitely many indeterminates x x , x 2, . . . , with coefficients in the field Q of rational numbers. We regard elements of An merely as formal expressions. A n has the structure of a vector space over Q. We can also make the A fs into a graded algebra,

A = A0 ® A x © A2 © ..., by defining m ultiplication to be ordinary power series multiplication. We are interested in studying various bases for the vector space An. If 2 i- n, define

kx = Xx^x^2 . . . ,

(4)

where the sum m ation sign indicates that we are to form all distinct monomials in the x /s with exponents Xl9X29... (in some order). The k f s are known as the monomial symmetric functions. It is easily seen that the k f s form a basis for A n as X runs over all partitions of n. Thus A nhas dimension p(n)9the num ber of partitions of n. F or an introduction to the function p(n\ see Hardy and W right [6, Ch. 19]. If we wish to specialize certain values of x i9 we indicate this by notation such as k i l ( X i , X 2,X3) = XyX2 + XtX3 + x 2x 3 k„(x,x2, x 3, . . . ) = xn + x 2/I + x 3w + ••• = *7(1 - x n) (here kn denotes kx where X = (n, 0 , 0 , . . . ) = ( n 1)). We also use x and y to denote the vectors (xx , x 2 , . . . ) and ( j ^ , y2 >• • •)> so M x) = .. and kx(y) = X y ^y 22 ___The x f’s and y{ s are to be regarded as independent indeterminates. Any basis which can be obtained from kx via a matrix with integral coefficients and determ inant + 1 is called an integral basis. We now consider two im portant integral bases. Define

K = E

n

h* =

(5)

The h f s are the complete homogeneous symmetric functions. Also define

a„ = k hx.

(8)

Note that since ak and hx are multiplicative, 0 preserves m ultiplication and is therefore an autom orphism of the algebra A. The basic properties of 4>and 9 a r e : 3.1. P r o po s i t i o n . 4> is symmetric. 3.2. P r o po s i t i o n . 92 — 1. 3.3. P r o po s i t i o n . The s f s are eigenvectors for 6; indeed,

9sx = ( - 1 )n_rsA,

A=

if 2 i- n,

, . . . , 2,).

The key to proving these relations lies in observing that we have the generating functions

n (1

x (t) 1

n

i= 1

Z V

(9)

antn.

(10)

n= 0

i= 1

( i + X,t) =

Z n= 0

It follows from (9) that 00

00

n o - ^ U= l

r

00

00

1 = n Z My)*/1= Z Z kx(x)hx(y), i=l m= 0 n = 0 X'-n

which we abbreviate to

no- ) 1=Zk kx(x)hx(y)x$ j

Now suppose

hx(y) =

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Zfi ‘frxnKiyX

kx(x) =

ZV 'I'xMx)-,

(ii)

Theory and Application of Plane Partitions: Part 1

where i// =

171

Then from (11) we get n

i,j= 1

(i - x iyj) 1 = Z

n,v \

Z fa n 'I'J kn(yA ix Y

I

A

( 12)

Since the left-hand side of (11) is symmetric in the x /s and y f s, we also have n

i,j =1

(! - x iyj) 1 = Z kx(y)hx(x)A

(13)

Since the distinct products k^(y)hv(x) are linearly independent, we see from com paring (12) and (13) that Z ^A^Av =

(