MSC: Primary 05; 06; 13; 52;
Print ISBN: 978-1-4704-2724-5
Product Code: MBK/100
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Electronic ISBN: 978-1-4704-3477-9
Product Code: MBK/100.E
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AMS Member Price: $47.20
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The Mathematical Legacy of Richard P. Stanley
Share this pageEdited by Patricia Hersh; Thomas Lam; Pavlo Pylyavskyy; Victor Reiner
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 influenced and guided their own perspectives on
the subject. As a valuable 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.
Readership
Graduate students and researchers interested in combinatorics.
Table of Contents
The Mathematical Legacy of Richard P. Stanley
- Cover Cover11
- Title page iii4
- Contents vii8
- Preface ix10
- Richard Stanley’s Short Curriculum Vitae xi12
- Publications 116
- A survey of subdivisions and local ℎ-vectors 3954
- Stanley’s major contributions to Ehrhart theory 5368
- “Even more intriguing, if rather less plausible...” Face numbers of convex polytopes 6580
- The contributions of Stanley to the fabric of symmetric and quasisymmetric functions 8398
- 1. Introduction 8398
- 2. The beginnings 8499
- 3. Stanley symmetric functions and applications 87102
- 4. Chromatic symmetric functions and quasisymmetric functions 92107
- 5. A skew generalization of the RSK algorithm 95110
- 6. Flag symmetry of posets 97112
- 7. Conclusion 99114
- Acknowledgments 99114
- References 100115
- “Let \De be a Cohen-Macaulay complex …” 105120
- Stanley’s work on unimodality 119134
- Five stories for Richard 131146
- 1. Story One: P-partitions and shuffling cards 132147
- 2. Story Two: Chromatic polynomials 133148
- 3. Story Three: The Jack polynomials 134149
- 4. Story Four: Alternating permutations and poset polytopes 136151
- 5. Story Five: Stanley’s problems and Selberg’s integrals 136151
- References \@mkboth{\MakeUppercase{References }}\MakeUppercase{References } 138153
- Some new applications of the Stanley-Macdonald Pieri Rules 141156
- A Historical Survey of 𝑃-Partitions 169184
- Transitive factorizations of permutations and geometry 189204
- 1. Introduction 189204
- 2. Transitive factorizations of permutations 190205
- 3. Maps in orientable surfaces 191206
- 4. Maps in surfaces and Jack symmetric functions 193208
- 5. Maps, matrix integrals and virtual Euler characteristic 194209
- 6. Branched covers of the sphere and Hurwitz numbers 195210
- 7. The join-cut equation 196211
- 8. Hodge integrals, the moduli space of curves, and integrable hierarchies 198213
- References 199214
- Stanley’s Influence on Monomial Ideals 203218
- Introduction 203218
- 1. The upper bound conjecture for spheres 204219
- 2. Minimal resolutions of monomial ideals 205220
- 3. Squarefree lexsegment ideals 206221
- 4. Alexander duality 207222
- 5. Edge ideals of finite graphs 209224
- 6. Vertex cover ideals 213228
- 7. Polymatroidal ideals 214229
- 8. Limit depth of monomial ideals 215230
- References 216231
- Cohen-Macaulay Varieties, Geometric Complexes, and Combinatorics 219234
- 1. Introduction 219234
- 2. Proving the Cohen-Macaulay property for a union of varieties that resembles a simplicial complex 220235
- 3. The notion of depth, and Cohen-Macaulay rings 221236
- 4. Determinantal ideals, Grassmannians, and Schubert varieties 223238
- 5. Toric varieties, constructability, and shellability 224239
- 6. Reisner’s theorem 226241
- 7. Thompson’s theorem 227242
- 8. Hodge algebras, F-split rings, and F-regular rings 227242
- References 228243
- Plane partitions in the work of Richard Stanley and his school 231246
- 1. Introduction 231246
- 2. Percy Alexander MacMahon 234249
- 3. The revival of plane partitions and Richard Stanley 235250
- 4. Methods for the enumeration of plane partitions 238253
- 5. Trace generating functions 244259
- 6. The 10 symmetry classes of plane partitions 246261
- 7. Stanley’s proof of the formula for self-complementary plane partitions 251266
- 8. All problems solved? 254269
- References 258273
- Combinatorial representation theory of Lie algebras. Richard Stanley’s work and the way it was continued 263278
- 1. Introduction 263278
- 2. The Robinson-Schensted-Knuth correspondence 264279
- 3. 𝔰𝔩₂-representations and related posets 266281
- 4. The stable behavior of some graded multiplicities 270285
- 5. Jack, Macdonald, and Hall-Littlewood symmetric polynomials 271286
- Acknowledgement 274289
- References \@mkboth{\MakeUppercase{References }}\MakeUppercase{References } 275290
- Lessons I Learned from Richard Stanley 279294
- Richard Stanley through a crystal lens and from a random angle 287302
- From Poset Topology to 𝑞-Eulerian Polynomials to Stanley’s Chromatic Symmetric Functions 301316
- Stanley character polynomials 323338
- 1. Normalized characters of the symmetric groups 323338
- 2. Asymptotic representation theory of symmetric groups 324339
- 3. Free cumulants and Kerov polynomials 325340
- 4. Characters on rectangular Young diagrams 326341
- 5. Stanley character polynomial 328343
- 6. Stanley character polynomials in action: toy example of Kerov polynomials 329344
- 7. Stanley character polynomials in action: generic coefficients of Kerov polynomials 330345
- 8. The structure behind Stanley character formula 331346
- 9. Further perspectives 332347
- References 333348
- Some Problems Arising from Partition Poset Homology 335350
- Back Cover Back Cover1369