The mathematics of shuffling cards:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
AMS, American Mathematical Society
[2023]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 346 Seiten Illustrationen, Diagramme 26 cm |
| ISBN: | 1470463032 9781470463038 |
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Record in the Search Index
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Contents Preface Acknowledgments xi xii Chapter 1. Shuffling cards: An introduction 1.1. Riffle shuffling and total variation 1.2. The problem, its motivation, and some theorems 1.3. Outline of the book 1.4. Background for reading this book ' 1 1 3 5 8 Chapter 2. Practice and history of shuffling cards 2.1. Illustrations of some shuffles 2.2. Some early history 11 11 20 Chapter 3. Convergence rates for riffle shuffles 3.1. Basic properties of riffle shuffling 3.2. Total variation and relative entropy 3.3. Effect of cuts 3.4. Guessing strategies and games 3.5. Strong uniform times and separation distance 3.6. Coupling and riffle shuffles 3.7. Appendix: Distances between probability distributions 25 25 30 31 34 39 43 47 Chapter 4. Features 4.1. A single card 4.2. Shuffling with repeated cards 4.3. Different methods of dealing 4.4. Cycle structure, random polynomials, andLie theory 4.5. Inversions 4.6. RSK shape, shuffling, and character theory of Sx 4.7. The sign function 4.8. Other techniques 51 52 54 58 62 67 68 75 78 Chapter 5. Eigenvectors and Hopf algebras 5.1. Overview 5.2. Combinatorics of words 5.3. The main theorem 5.4. Hopf algebras and Markov chains 5.5. Restriction/induction (with shuffling and Hopf algebras) 5.6. Shuffling and Hochschild homology 5.7. Appendix 1: Uses for eigenvectors 5.8. Appendix 2: Eigenvectors in this book 81 81 83 84 86 89 91 93 97 vii
viii Chapter 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. CONTENTS 6. Shuffling and carries Introduction to carries Connection with riffle shuffles A bit of why Convergence rates of the carries chain Eigenvalues and eigenvectors of the carries chain Balanced carries Carries, number theory, and fractals Other connections Appendix: Some proofs 99 99 102 103 105 107 112 116 121 123 Chapter 7. Different models for riffle shuffling 7.1. Perfect shuffles 7.2. Thorp shuffle 7.3. Neat and clumpy shuffles: The Markov model 7.4. Dynamical systems and work of Lalley 7.5. Shuffling big decks 127 127 136 138 140 140 Chapter 8. Move-to-front shuffling and variations 8.1. Coupling and convergence rates for the move-to-frontshuffle 8.2. Formula for move-to-front shuffle 8.3. Spectral aspects of the move-to-front shuffle 8.4. Statistics of features after iterations of top-to-randomshuffle 8.5. Weighted move to front and the Plackett-Luce model 8.6. Move-to-front shuffle with Markov dependent requests 8.7. Move to root for binary search trees 8.8. Connection with Stein’s method 8.9. Random-to-random shuffle 147 148 149 151 153 155 157 159 161 163 Chapter 9. Shuffling and geometry 9.1. Hyperplane arrangements and random walks 9.2. Examples 9.3. General theory of hyperplane walks 9.4. An analog of riffle shuffling for general hyperplane arrangements 9.5. Stationary distribution, eigenfunctions, and lumped chains 9.6. Extensions 9.7. Connections and unification 9.8. Adjacent transpositions 9.9. Research problems 9.10. Appendix: Some proofs 165 166 167 171 176 179 183 191 193 193 194 Chapter
10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10. Shuffling and algebraic topology A topology teaser First steps: Homology Triangulating a product of simplices (and shuffling) The Künneth formula Translating shuffling theorems into homology A different application of shuffling to topology Final remarks 199 199 200 202 204 204 207 208
CONTENTS ix Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6. 11. Type В shuffles and shelf-shuffling machines Models of Type В shuffles Cycle structure and RSK shape Shelf shufflers Other types Research problems Appendix: Proof of Proposition 11.3.2 209 209 211 215 219 222 223 Chapter 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12. Descent algebras, P-partitions, and quasisymmetric functions Descent theory P-partitions and shuffling Quasisymmetric functions and shuffles with biased cuts Algebras of shuffles A strange inequality And then . 225 225 230 238 244 245 247 Chapter 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8. 13. Overhand shuffling Introduction to the overhand shuffle Mathematical models Some theorems Entertaining (and cheating) with overhand shuffles The over-under shuffle Coupling and the coin tossing model Braid arrangement and overhand shuffles Interval exchange maps and overhand shuffles 249 249 250 252 254 260 261 266 267 Chapter 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14. “Smoosh” shuffle Introduction Tests of mixing for smoosh shuffles A mathematical model for spatial mixing Some analysis Bells, whistles, and computer implementation A different model for spatial mixing Combining shuffles and some practical tests 271 271 272 274 277 280 281 284 How to shuffle perfectly (randomly) 287 Chapter 15. Chapter 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7. 16.8. 16.9. 16.10. 16.11. 16.12. 16. Applications to magic tricks, traffic merging, and statistics Rising sequences The Gilbreath principle Tops and bottoms are special A performable magic trick A homework problem Riffle
stacking Close stays close Reds and blacks Overhand shuffle Smooshing An application of shuffling to cars merging in traffic Statistics of permutations 295 295 297 298 298 299 300 301 301 301 302 303 304
CONTENTS X Chapter 17.1. 17.2. 17.3. 17.4. 17. Shuffling and multiple zeta values Multiple zeta values A bit of proof Chen’s integrals Periods 307 307 313 316 318 Bibliography 321 Index 341 |
| adam_txt |
Contents Preface Acknowledgments xi xii Chapter 1. Shuffling cards: An introduction 1.1. Riffle shuffling and total variation 1.2. The problem, its motivation, and some theorems 1.3. Outline of the book 1.4. Background for reading this book ' 1 1 3 5 8 Chapter 2. Practice and history of shuffling cards 2.1. Illustrations of some shuffles 2.2. Some early history 11 11 20 Chapter 3. Convergence rates for riffle shuffles 3.1. Basic properties of riffle shuffling 3.2. Total variation and relative entropy 3.3. Effect of cuts 3.4. Guessing strategies and games 3.5. Strong uniform times and separation distance 3.6. Coupling and riffle shuffles 3.7. Appendix: Distances between probability distributions 25 25 30 31 34 39 43 47 Chapter 4. Features 4.1. A single card 4.2. Shuffling with repeated cards 4.3. Different methods of dealing 4.4. Cycle structure, random polynomials, andLie theory 4.5. Inversions 4.6. RSK shape, shuffling, and character theory of Sx 4.7. The sign function 4.8. Other techniques 51 52 54 58 62 67 68 75 78 Chapter 5. Eigenvectors and Hopf algebras 5.1. Overview 5.2. Combinatorics of words 5.3. The main theorem 5.4. Hopf algebras and Markov chains 5.5. Restriction/induction (with shuffling and Hopf algebras) 5.6. Shuffling and Hochschild homology 5.7. Appendix 1: Uses for eigenvectors 5.8. Appendix 2: Eigenvectors in this book 81 81 83 84 86 89 91 93 97 vii
viii Chapter 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. CONTENTS 6. Shuffling and carries Introduction to carries Connection with riffle shuffles A bit of why Convergence rates of the carries chain Eigenvalues and eigenvectors of the carries chain Balanced carries Carries, number theory, and fractals Other connections Appendix: Some proofs 99 99 102 103 105 107 112 116 121 123 Chapter 7. Different models for riffle shuffling 7.1. Perfect shuffles 7.2. Thorp shuffle 7.3. Neat and clumpy shuffles: The Markov model 7.4. Dynamical systems and work of Lalley 7.5. Shuffling big decks 127 127 136 138 140 140 Chapter 8. Move-to-front shuffling and variations 8.1. Coupling and convergence rates for the move-to-frontshuffle 8.2. Formula for move-to-front shuffle 8.3. Spectral aspects of the move-to-front shuffle 8.4. Statistics of features after iterations of top-to-randomshuffle 8.5. Weighted move to front and the Plackett-Luce model 8.6. Move-to-front shuffle with Markov dependent requests 8.7. Move to root for binary search trees 8.8. Connection with Stein’s method 8.9. Random-to-random shuffle 147 148 149 151 153 155 157 159 161 163 Chapter 9. Shuffling and geometry 9.1. Hyperplane arrangements and random walks 9.2. Examples 9.3. General theory of hyperplane walks 9.4. An analog of riffle shuffling for general hyperplane arrangements 9.5. Stationary distribution, eigenfunctions, and lumped chains 9.6. Extensions 9.7. Connections and unification 9.8. Adjacent transpositions 9.9. Research problems 9.10. Appendix: Some proofs 165 166 167 171 176 179 183 191 193 193 194 Chapter
10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10. Shuffling and algebraic topology A topology teaser First steps: Homology Triangulating a product of simplices (and shuffling) The Künneth formula Translating shuffling theorems into homology A different application of shuffling to topology Final remarks 199 199 200 202 204 204 207 208
CONTENTS ix Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6. 11. Type В shuffles and shelf-shuffling machines Models of Type В shuffles Cycle structure and RSK shape Shelf shufflers Other types Research problems Appendix: Proof of Proposition 11.3.2 209 209 211 215 219 222 223 Chapter 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12. Descent algebras, P-partitions, and quasisymmetric functions Descent theory P-partitions and shuffling Quasisymmetric functions and shuffles with biased cuts Algebras of shuffles A strange inequality And then . 225 225 230 238 244 245 247 Chapter 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8. 13. Overhand shuffling Introduction to the overhand shuffle Mathematical models Some theorems Entertaining (and cheating) with overhand shuffles The over-under shuffle Coupling and the coin tossing model Braid arrangement and overhand shuffles Interval exchange maps and overhand shuffles 249 249 250 252 254 260 261 266 267 Chapter 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14. “Smoosh” shuffle Introduction Tests of mixing for smoosh shuffles A mathematical model for spatial mixing Some analysis Bells, whistles, and computer implementation A different model for spatial mixing Combining shuffles and some practical tests 271 271 272 274 277 280 281 284 How to shuffle perfectly (randomly) 287 Chapter 15. Chapter 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7. 16.8. 16.9. 16.10. 16.11. 16.12. 16. Applications to magic tricks, traffic merging, and statistics Rising sequences The Gilbreath principle Tops and bottoms are special A performable magic trick A homework problem Riffle
stacking Close stays close Reds and blacks Overhand shuffle Smooshing An application of shuffling to cars merging in traffic Statistics of permutations 295 295 297 298 298 299 300 301 301 301 302 303 304
CONTENTS X Chapter 17.1. 17.2. 17.3. 17.4. 17. Shuffling and multiple zeta values Multiple zeta values A bit of proof Chen’s integrals Periods 307 307 313 316 318 Bibliography 321 Index 341 |
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| index_date | 2024-07-03T22:39:52Z |
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| publisher | AMS, American Mathematical Society |
| record_format | marc |
| spelling | Diaconis, Persi 1945- Verfasser (DE-588)1044123206 aut The mathematics of shuffling cards Persi Diaconis, Jason Fulman Providence, Rhode Island AMS, American Mathematical Society [2023] © 2023 xii, 346 Seiten Illustrationen, Diagramme 26 cm txt rdacontent n rdamedia nc rdacarrier Kombinatorische Analysis (DE-588)4164746-4 gnd rswk-swf Kartenspiel (DE-588)4029798-6 gnd rswk-swf Hopf-Algebra (DE-588)4160646-2 gnd rswk-swf Mischen (DE-588)4120742-7 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd rswk-swf Mathematische Modellierung (DE-588)7651795-0 gnd rswk-swf Card shuffling / (OCoLC)fst00847027 Card games / (OCoLC)fst00847018 Distribution (Probability theory) / (OCoLC)fst00895600 Combinatorial analysis / (OCoLC)fst00868961 Hopf algebras / (OCoLC)fst00960096 Kartenspiel (DE-588)4029798-6 s Mischen (DE-588)4120742-7 s Mathematische Modellierung (DE-588)7651795-0 s Wahrscheinlichkeitsverteilung (DE-588)4121894-2 s Kombinatorische Analysis (DE-588)4164746-4 s Hopf-Algebra (DE-588)4160646-2 s DE-604 Fulman, Jason 1971- Verfasser (DE-588)1146711956 aut Erscheint auch als Online-Ausgabe 978-1-4704-7290-0 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034567159&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Diaconis, Persi 1945- Fulman, Jason 1971- The mathematics of shuffling cards Kombinatorische Analysis (DE-588)4164746-4 gnd Kartenspiel (DE-588)4029798-6 gnd Hopf-Algebra (DE-588)4160646-2 gnd Mischen (DE-588)4120742-7 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd Mathematische Modellierung (DE-588)7651795-0 gnd |
| subject_GND | (DE-588)4164746-4 (DE-588)4029798-6 (DE-588)4160646-2 (DE-588)4120742-7 (DE-588)4121894-2 (DE-588)7651795-0 |
| title | The mathematics of shuffling cards |
| title_auth | The mathematics of shuffling cards |
| title_exact_search | The mathematics of shuffling cards |
| title_exact_search_txtP | The mathematics of shuffling cards |
| title_full | The mathematics of shuffling cards Persi Diaconis, Jason Fulman |
| title_fullStr | The mathematics of shuffling cards Persi Diaconis, Jason Fulman |
| title_full_unstemmed | The mathematics of shuffling cards Persi Diaconis, Jason Fulman |
| title_short | The mathematics of shuffling cards |
| title_sort | the mathematics of shuffling cards |
| topic | Kombinatorische Analysis (DE-588)4164746-4 gnd Kartenspiel (DE-588)4029798-6 gnd Hopf-Algebra (DE-588)4160646-2 gnd Mischen (DE-588)4120742-7 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd Mathematische Modellierung (DE-588)7651795-0 gnd |
| topic_facet | Kombinatorische Analysis Kartenspiel Hopf-Algebra Mischen Wahrscheinlichkeitsverteilung Mathematische Modellierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034567159&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT diaconispersi themathematicsofshufflingcards AT fulmanjason themathematicsofshufflingcards |