AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The Mathematics of Shuffling Cards
About this Title
Persi Diaconis, Stanford University, Stanford, CA and Jason Fulman, University of Southern California, Los Angeles, CA
Publication: AMS Non-Series Monographs
Publication Year:
2023; Volume 146
ISBNs: 978-1-4704-6303-8 (print); 978-1-4704-7290-0 (online)
DOI: https://doi.org/10.1090/mbk/146
Table of Contents
Download chapters as PDF
Front/Back Matter
Chapters
- Shuffling cards: An introduction
- Practice and history of shuffling cards
- Convergence rates for riffle shuffles
- Features
- Eigenvectors and Hopf algebras
- Shuffling and carries
- Different models for riffle shuffling
- Move to front shuffling and variations
- Shuffling and geometry
- Shuffling and algebraic topology
- Type B shuffles and shelf shuffling machines
- Descent algebras, $P$-partitions, and quasisymmetric functions
- Overhand shuffling
- âSmooshâ shuffle
- How to shuffle perfectly (randomly)
- Applications to magic tricks, traffic merging, and statistics
- Shuffling and multiple zeta values
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, DC, 1964. For sale by the Superintendent of Documents. MR 167642
- A. Aggarwal, A. Borodin, and M. Wheeler, Deformed polynuclear growth in (1+1) dimensions, arXiv:2108.06018, 2021.
- Marcelo Aguiar, Nantel Bergeron, and Kathryn Nyman, The peak algebra and the descent algebras of types $B$ and $D$, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2781â2824. MR 2052597, DOI 10.1090/S0002-9947-04-03541-X
- Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29, American Mathematical Society, Providence, RI, 2010. With forewords by Kenneth Brown and Stephen Chase and André Joyal. MR 2724388, DOI 10.1090/crmm/029
- Marcelo Aguiar and Swapneel Mahajan, Topics in hyperplane arrangements, Mathematical Surveys and Monographs, vol. 226, American Mathematical Society, Providence, RI, 2017. MR 3726871, DOI 10.1090/surv/226
- Marcelo Aguiar and Swapneel Mahajan, Coxeter groups and Hopf algebras, Fields Institute Monographs, vol. 23, American Mathematical Society, Providence, RI, 2006. With a foreword by Nantel Bergeron. MR 2225808, DOI 10.1090/fim/023
- Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math. 191 (2005), no. 2, 225â275. MR 2103213, DOI 10.1016/j.aim.2004.03.007
- Martin Aigner, A course in enumeration, Graduate Texts in Mathematics, vol. 238, Springer, Berlin, 2007. MR 2339282
- Martin Aigner and GĂŒnter M. Ziegler, Proofs from The Book, 2nd ed., Springer-Verlag, Berlin, 2001. Including illustrations by Karl H. Hofmann. MR 1801937, DOI 10.1007/978-3-662-04315-8
- David Aldous, Random walks on finite groups and rapidly mixing Markov chains, Seminar on probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 243â297. MR 770418, DOI 10.1007/BFb0068322
- David Aldous, An introduction to covering problems for random walks on graphs, J. Theoret. Probab. 2 (1989), no. 1, 87â89. MR 981766, DOI 10.1007/BF01048271
- David Aldous and Persi Diaconis, Strong uniform times and finite random walks, Adv. in Appl. Math. 8 (1987), no. 1, 69â97. MR 876954, DOI 10.1016/0196-8858(87)90006-6
- David Aldous and Persi Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 413â432. MR 1694204, DOI 10.1090/S0273-0979-99-00796-X
- Brian Allen and Ian Munro, Self-organizing binary search trees, J. Assoc. Comput. Mach. 25 (1978), no. 4, 526â535. MR 508699, DOI 10.1145/322092.322094
- Noga Alon, Minimizing the number of carries in addition, SIAM J. Discrete Math. 27 (2013), no. 1, 562â566. MR 3035466, DOI 10.1137/120890612
- Gil Alon and Gady Kozma, The probability of long cycles in interchange processes, Duke Math. J. 162 (2013), no. 9, 1567â1585. MR 3079255, DOI 10.1215/00127094-2266018
- Carmen Amarra, Luke Morgan, and Cheryl E. Praeger, Generalised shuffle groups, Israel J. Math. 244 (2021), no. 2, 807â856. MR 4344046, DOI 10.1007/s11856-021-2194-1
- Hans C. Andersen and Persi Diaconis, Hit and run as a unifying device, J. Soc. Fr. Stat. & Rev. Stat. Appl. 148 (2007), no. 4, 5â28 (English, with English and French summaries). MR 2502361
- George E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR 1634067
- NicolĂĄs Andruskiewitsch and Walter Ferrer Santos, The beginnings of the theory of Hopf algebras, Acta Appl. Math. 108 (2009), no. 1, 3â17. MR 2540954, DOI 10.1007/s10440-008-9393-1
- Roger ApĂ©ry, IrrationalitĂ© de $\zeta 2$ et $\zeta 3$, AstĂ©risque 61 (1979), 11â13 (French). Luminy Conference on Arithmetic. MR 3363457
- Hassan Aref, John R. Blake, Marko BudiĆĄiÄ et al., Frontiers of chaotic advection, Rev. Modern Phys. 89 (2017), no. 2, 025007, 66. MR 3678358, DOI 10.1103/RevModPhys.89.025007
- B. Arkin, F. Hill, S. Marks, M. Schmid, T. Walls, and G. McGraw, How we learned to cheat in online poker: a study in software security, http://bluffnakedpoker.com/PDF/developer_gambling.pdf, 1999.
- Mark Anthony Armstrong, Basic topology, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1979 original. MR 705632
- M. Arnaudon, K. Coulibaly-Pasquier, and L. Miclo, Construction of set valued dual processes on manifolds, arXiv:2012.02444, 2020.
- V. I. ArnolâČd, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295, DOI 10.1007/978-1-4757-2063-1
- Richard Arratia, A. D. Barbour, and Simon TavarĂ©, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), ZĂŒrich, 2003. MR 2032426, DOI 10.4171/000
- Richard Arratia, A. D. Barbour, and Simon TavarĂ©, Poisson process approximations for the Ewens sampling formula, Ann. Appl. Probab. 2 (1992), no. 3, 519â535. MR 1177897
- Richard Arratia and Simon TavarĂ©, The cycle structure of random permutations, Ann. Probab. 20 (1992), no. 3, 1567â1591. MR 1175278
- Sami Assaf, Persi Diaconis, and K. Soundararajan, A rule of thumb for riffle shuffling, Ann. Appl. Probab. 21 (2011), no. 3, 843â875. MR 2830606, DOI 10.1214/10-AAP701
- Sami Assaf, Persi Diaconis, and Kannan Soundararajan, Riffle shuffles with biased cuts, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), Discrete Math. Theor. Comput. Sci. Proc., AR, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012, pp. 445â456 (English, with English and French summaries). MR 2958019
- Christos A. Athanasiadis and Persi Diaconis, Functions of random walks on hyperplane arrangements, Adv. in Appl. Math. 45 (2010), no. 3, 410â437. MR 2669077, DOI 10.1016/j.aam.2010.02.001
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 242802
- Artur Avila and Giovanni Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2) 165 (2007), no. 2, 637â664. MR 2299743, DOI 10.4007/annals.2007.165.637
- Arvind Ayyer, Steven Klee, and Anne Schilling, Markov chains for promotion operators, Algebraic monoids, group embeddings, and algebraic combinatorics, Fields Inst. Commun., vol. 71, Springer, New York, 2014, pp. 285â304. MR 3308326, DOI 10.1007/978-1-4939-0938-4_{1}3
- Arvind Ayyer, Steven Klee, and Anne Schilling, Combinatorial Markov chains on linear extensions, J. Algebraic Combin. 39 (2014), no. 4, 853â881. MR 3199029, DOI 10.1007/s10801-013-0470-9
- Arvind Ayyer, Anne Schilling, Benjamin Steinberg, and Nicolas M. ThiĂ©ry, Markov chains, $\scr {R}$-trivial monoids and representation theory, Internat. J. Algebra Comput. 25 (2015), no. 1-2, 169â231. MR 3325881, DOI 10.1142/S0218196715400081
- Arvind Ayyer, Anne Schilling, Benjamin Steinberg, and Nicolas M. ThiĂ©ry, Directed nonabelian sandpile models on trees, Comm. Math. Phys. 335 (2015), no. 3, 1065â1098. MR 3320305, DOI 10.1007/s00220-015-2343-7
- Arvind Ayyer, Anne Schilling, and Nicolas M. ThiĂ©ry, Spectral gap for random-to-random shuffling on linear extensions, Exp. Math. 26 (2017), no. 1, 22â30. MR 3599002, DOI 10.1080/10586458.2015.1107868
- Axel Bacher, Olivier Bodini, Hsien-Kuei Hwang, and Tsung-Hsi Tsai, Generating random permutations by coin tossing: classical algorithms, new analysis, and modern implementation, ACM Trans. Algorithms 13 (2017), no. 2, Art. 24, 43. MR 3659246, DOI 10.1145/3009909
- M. Baker, The Buena Vista Shuffle Club, privately published, 2019.
- M. Baker and N. Bowler, Matroids over hyperfields, AIP Conference Proceedings 1978, 340010, 2018.
- MĂĄrton BalĂĄzs and DĂĄvid ZoltĂĄn SzabĂł, Comparing dealing methods with repeating cards, ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 615â630. MR 3296483
- Keith Ball and Tanguy Rivoal, IrrationalitĂ© dâune infinitĂ© de valeurs de la fonction zĂȘta aux entiers impairs, Invent. Math. 146 (2001), no. 1, 193â207 (French). MR 1859021, DOI 10.1007/s002220100168
- R. B. Bapat, Discrete multivariate distributions and generalized log-concavity, SankhyÄ Ser. A 50 (1988), no. 1, 98â110. MR 1056244
- Dave Bayer and Persi Diaconis, Trailing the dovetail shuffle to its lair, Ann. Appl. Probab. 2 (1992), no. 2, 294â313. MR 1161056
- Matthias Beck and Alan Stapledon, On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series, Math. Z. 264 (2010), no. 1, 195â207. MR 2564938, DOI 10.1007/s00209-008-0458-7
- A. Beiler, Recreations in the theory of numbers, Dover, New York, 1964.
- D. Ben, Zarrow: a lifetime of magic, Meier Yedid Magic, 2008.
- Edward A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combinatorial Theory Ser. A 15 (1973), 91â111. MR 375433, DOI 10.1016/0097-3165(73)90038-1
- Georgia Benkart, Persi Diaconis, Martin W. Liebeck, and Pham Huu Tiep, Tensor product Markov chains, J. Algebra 561 (2020), 17â83. MR 4135538, DOI 10.1016/j.jalgebra.2019.10.038
- NathanaĂ«l Berestycki and Rick Durrett, Limiting behavior for the distance of a random walk, Electron. J. Probab. 13 (2008), no. 14, 374â395. MR 2386737, DOI 10.1214/EJP.v13-490
- P. Berger, On the distribution of hand patterns in bridge: man-dealt versus computer dealt, Canadian Jour. Statistics 1 (1973), 261â266.
- Nantel Bergeron, A hyperoctahedral analogue of the free Lie algebra, J. Combin. Theory Ser. A 58 (1991), no. 2, 256â278. MR 1129117, DOI 10.1016/0097-3165(91)90061-K
- François Bergeron and Nantel Bergeron, Orthogonal idempotents in the descent algebra of $B_n$ and applications, J. Pure Appl. Algebra 79 (1992), no. 2, 109â129. MR 1163285, DOI 10.1016/0022-4049(92)90153-7
- F. Bergeron, N. Bergeron, R. B. Howlett, and D. E. Taylor, A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin. 1 (1992), no. 1, 23â44. MR 1162640, DOI 10.1023/A:1022481230120
- N. Bergeron, On Hochschild homology and Vassiliev invariants, DIMACS Conference, 1994, 1â10.
- Megan Bernstein and Evita Nestoridi, Cutoff for random to random card shuffle, Ann. Probab. 47 (2019), no. 5, 3303â3320. MR 4021252, DOI 10.1214/19-AOP1340
- Rabi N. Bhattacharya and Edward C. Waymire, Stochastic processes with applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR 1054645
- Thomas Patrick Bidigare, Hyperplane arrangement face algebras and their associated Markov chains, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)âUniversity of Michigan. MR 2695489
- Pat Bidigare, Phil Hanlon, and Dan Rockmore, A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements, Duke Math. J. 99 (1999), no. 1, 135â174. MR 1700744, DOI 10.1215/S0012-7094-99-09906-4
- Louis J. Billera and Niandong Liu, Noncommutative enumeration in graded posets, J. Algebraic Combin. 12 (2000), no. 1, 7â24. MR 1791443, DOI 10.1023/A:1008703300280
- Patrick Billingsley, Probability and measure, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2012. Anniversary edition [of MR1324786]; With a foreword by Steve Lalley and a brief biography of Billingsley by Steve Koppes. MR 2893652
- Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974. MR 375281
- Anders Björner, Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries, Building bridges, Bolyai Soc. Math. Stud., vol. 19, Springer, Berlin, 2008, pp. 165â203. MR 2484640, DOI 10.1007/978-3-540-85221-6_{5}
- Anders Björner, Note: Random-to-front shuffles on trees, Electron. Commun. Probab. 14 (2009), 36â41. MR 2481664, DOI 10.1214/ECP.v14-1445
- Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and GĂŒnter M. Ziegler, Oriented matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1993. MR 1226888
- C. Blake and I. Gent, The winnability of Klondike solitaire and many other patience games, arXiv:1906.12314, 2019.
- Dieter Blessenohl, Christophe Hohlweg, and Manfred Schocker, A symmetry of the descent algebra of a finite Coxeter group, Adv. Math. 193 (2005), no. 2, 416â437. MR 2137290, DOI 10.1016/j.aim.2004.05.007
- Thomas Bliem and Stavros Kousidis, The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics, J. Algebraic Combin. 37 (2013), no. 2, 361â380. MR 3011347, DOI 10.1007/s10801-012-0373-1
- MiklĂłs BĂłna, A walk through combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. An introduction to enumeration and graph theory; Fourth edition [of MR1936456]; With a foreword by Richard Stanley. MR 3560666
- MiklĂłs BĂłna, Combinatorics of permutations, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012. With a foreword by Richard Stanley. MR 2919720, DOI 10.1201/b12210
- Lennart Bondesson, A simple generalization of PoincarĂ©âs shuffling theorem, Probability and mathematical statistics, Uppsala Univ., Uppsala, 1983, pp. 11â15. MR 727122
- E. Borel and A. Cheron, Theorie mathematique du bridge, 2nd. ed., Gauthier Villars, Paris, 1955.
- Alexei Borodin, Persi Diaconis, and Jason Fulman, On adding a list of numbers (and other one-dependent determinantal processes), Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 639â670. MR 2721041, DOI 10.1090/S0273-0979-2010-01306-9
- Alexei Borodin, Andrei Okounkov, and Grigori Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), no. 3, 481â515. MR 1758751, DOI 10.1090/S0894-0347-00-00337-4
- Alexei Borodin and Grigori Olshanski, Representations of the infinite symmetric group, Cambridge Studies in Advanced Mathematics, vol. 160, Cambridge University Press, Cambridge, 2017. MR 3618143, DOI 10.1017/CBO9781316798577
- Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr LisonÄk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), no. 3, 907â941. MR 1709772, DOI 10.1090/S0002-9947-00-02616-7
- Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr LisonÄk, Combinatorial aspects of multiple zeta values, Electron. J. Combin. 5 (1998), Research Paper 38, 12. MR 1637378, DOI 10.37236/1376
- Mireille Bousquet-MĂ©lou, The expected number of inversions after $n$ adjacent transpositions, Discrete Math. Theor. Comput. Sci. 12 (2010), no. 2, 65â88. MR 2676666
- Douglas Bowman and David M. Bradley, Multiple polylogarithms: a brief survey, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 71â92. MR 1874522, DOI 10.1090/conm/291/04893
- Douglas Bowman and David M. Bradley, The algebra and combinatorics of shuffles and multiple zeta values, J. Combin. Theory Ser. A 97 (2002), no. 1, 43â61. MR 1879045, DOI 10.1006/jcta.2001.3194
- Douglas Bowman, David M. Bradley, and Ji Hoon Ryoo, Some multi-set inclusions associated with shuffle convolutions and multiple zeta values, European J. Combin. 24 (2003), no. 1, 121â127. MR 1957970, DOI 10.1016/S0195-6698(02)00117-8
- Edvin Bredrup and Li-Chun Zhang, Imperfectly shuffled decks in bridge, J. Appl. Statist. 25 (1998), no. 2, 173â179. MR 1649897, DOI 10.1080/02664769823160
- Yann Brenier, Topics on hydrodynamics and volume preserving maps, Handbook of mathematical fluid dynamics, Vol. II, North-Holland, Amsterdam, 2003, pp. 55â86. MR 1983589, DOI 10.1016/S1874-5792(03)80004-6
- Francesco Brenti and Volkmar Welker, The Veronese construction for formal power series and graded algebras, Adv. in Appl. Math. 42 (2009), no. 4, 545â556. MR 2511015, DOI 10.1016/j.aam.2009.01.001
- Graham Brightwell and Peter Winkler, Counting linear extensions, Order 8 (1991), no. 3, 225â242. MR 1154926, DOI 10.1007/BF00383444
- John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin types A, B and D, J. Combin. Theory Ser. A 148 (2017), 116â144. MR 3603317, DOI 10.1016/j.jcta.2016.12.003
- Francis Brown, Iterated integrals in quantum field theory, Geometric and topological methods for quantum field theory, Cambridge Univ. Press, Cambridge, 2013, pp. 188â240. MR 3098088
- Kenneth S. Brown, Semigroups, rings, and Markov chains, J. Theoret. Probab. 13 (2000), no. 3, 871â938. MR 1785534, DOI 10.1023/A:1007822931408
- Kenneth S. Brown, Semigroup and ring theoretical methods in probability, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 3â26. MR 2057147
- Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123, DOI 10.1007/978-1-4612-1019-1
- Kenneth S. Brown and Persi Diaconis, Random walks and hyperplane arrangements, Ann. Probab. 26 (1998), no. 4, 1813â1854. MR 1675083, DOI 10.1214/aop/1022855884
- AlekseÄ I. Bufetov, A central limit theorem for extremal characters of the infinite symmetric group, Funktsional. Anal. i Prilozhen. 46 (2012), no. 2, 3â16 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 46 (2012), no. 2, 83â93. MR 2977896, DOI 10.1007/s10688-012-0014-4
- Joe Buhler, David Eisenbud, Ron Graham, and Colin Wright, Juggling drops and descents, Amer. Math. Monthly 101 (1994), no. 6, 507â519. MR 1274973, DOI 10.2307/2975316
- Daniel Bump, Lie groups, 2nd ed., Graduate Texts in Mathematics, vol. 225, Springer, New York, 2013. MR 3136522, DOI 10.1007/978-1-4614-8024-2
- Steve Butler, Fan Chung, Jay Cummings, and Ron Graham, Edge flipping in the complete graph, Adv. in Appl. Math. 69 (2015), 46â64. MR 3365280, DOI 10.1016/j.aam.2015.06.002
- Steve Butler, Persi Diaconis, and Ron Graham, The mathematics of the flip and horseshoe shuffles, Amer. Math. Monthly 123 (2016), no. 6, 542â556. MR 3510970, DOI 10.4169/amer.math.monthly.123.6.542
- Florian Cajori, A history of mathematical notations, Dover Publications, Inc., New York, 1993. Including Vol. I. Notations in elementary mathematics; Vol. II. Notations mainly in higher mathematics; Reprint of the 1928 and 1929 originals. MR 3363427
- L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 32 (1958/59), 247â260. MR 104845, DOI 10.2307/3029225
- L. Carlitz, D. C. Kurtz, R. Scoville, and O. P. Stackelberg, Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 47â54. MR 309856, DOI 10.1007/BF00536689
- Henri Cartan, Differential forms, Houghton Mifflin Co., Boston, MA, 1970. Translated from the French. MR 267477
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- Roger Carter, Semisimple conjugacy classes and classes in the Weyl group, J. Algebra 260 (2003), no. 1, 99â110. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973577, DOI 10.1016/S0021-8693(02)00628-2
- Paola Cellini, Cyclic Eulerian elements, European J. Combin. 19 (1998), no. 5, 545â552. MR 1637728, DOI 10.1006/eujc.1998.0218
- Paola Cellini, A general commutative descent algebra, J. Algebra 175 (1995), no. 3, 990â1014. MR 1341755, DOI 10.1006/jabr.1995.1223
- Paola Cellini, A general commutative descent algebra. II. The case $C_n$, J. Algebra 175 (1995), no. 3, 1015â1026. MR 1341756, DOI 10.1006/jabr.1995.1224
- M. L. Cetlin, Finite automata and the simulation of the simplest forms of behavior, Uspehi Mat. Nauk 18 (1963), no. 4(112), 3â28 (Russian). MR 159734
- Sourav Chatterjee and Persi Diaconis, A central limit theorem for a new statistic on permutations, Indian J. Pure Appl. Math. 48 (2017), no. 4, 561â573. MR 3741694, DOI 10.1007/s13226-017-0246-3
- S. Chatterjee, P. Diaconis, and G. Kim, Some probability theory for the Luce-Plackett model, Stanford University technical report, 2020.
- Sourav Chatterjee, Persi Diaconis, and Elizabeth Meckes, Exchangeable pairs and Poisson approximation, Probab. Surv. 2 (2005), 64â106. MR 2121796, DOI 10.1214/154957805100000096
- Guan-Yu Chen and Laurent Saloff-Coste, The cutoff phenomenon for randomized riffle shuffles, Random Structures Algorithms 32 (2008), no. 3, 346â374. MR 2405231, DOI 10.1002/rsa.20195
- Louis H. Y. Chen, Larry Goldstein, and Qi-Man Shao, Normal approximation by Steinâs method, Probability and its Applications (New York), Springer, Heidelberg, 2011. MR 2732624, DOI 10.1007/978-3-642-15007-4
- Louis H. Y. Chen and Qi-Man Shao, Normal approximation under local dependence, Ann. Probab. 32 (2004), no. 3A, 1985â2028. MR 2073183, DOI 10.1214/009117904000000450
- Bobbie Chern, Persi Diaconis, Daniel M. Kane, and Robert C. Rhoades, Central limit theorems for some set partition statistics, Adv. in Appl. Math. 70 (2015), 92â105. MR 3388867, DOI 10.1016/j.aam.2015.06.008
- Fan Chung and Ron Graham, Edge flipping in graphs, Adv. in Appl. Math. 48 (2012), no. 1, 37â63. MR 2845506, DOI 10.1016/j.aam.2011.06.002
- F. R. K. Chung, D. J. Hajela, and P. D. Seymour, Self-organizing sequential search and Hilbertâs inequalities, J. Comput. System Sci. 36 (1988), no. 2, 148â157. 17th Annual ACM Symposium on the Theory of Computing (Providence, RI, 1985). MR 950430, DOI 10.1016/0022-0000(88)90025-6
- Mihai Ciucu, No-feedback card guessing for dovetail shuffles, Ann. Appl. Probab. 8 (1998), no. 4, 1251â1269. MR 1661184, DOI 10.1214/aoap/1028903379
- A. Cohen, A. Harmse, K. Morrison, and S. Wright, Perfect shuffles and affine groups, unpublished manuscript, 2005.
- G. L. Cola, How much shuffling is enough? A journey, Masterâs thesis, ETH Zurich, 2021.
- J. Colson, A short account of negativo-affirmative arithmetik, Philos. Trans. R. Soc. 34 (1726), 161â173.
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 460128
- Mark A. Conger, Shuffling decks with repeated card values, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)âUniversity of Michigan. MR 2709929
- Mark A. Conger and Jason Howald, A better way to deal the cards, Amer. Math. Monthly 117 (2010), no. 8, 686â700. MR 2732245, DOI 10.4169/000298910X515758
- Mark Conger and D. Viswanath, Riffle shuffles of decks with repeated cards, Ann. Probab. 34 (2006), no. 2, 804â819. MR 2223959, DOI 10.1214/009117905000000675
- M. Conger and D. Viswanath, Shuffling cards for blackjack, bridge, and other grames, arXiv:0606031, 2006.
- Joshua Cooper, Bill Kay, and Anton Swifton, Grahamâs tree reconstruction conjecture and a Waring-type problem on partitions, J. Comb. 9 (2018), no. 3, 469â488. MR 3809644, DOI 10.4310/JOC.2018.v9.n3.a3
- Ivan Corwin, Commentary on âLongest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theoremâ by David Aldous and Persi Diaconis, Bull. Amer. Math. Soc. (N.S.) 55 (2018), no. 3, 363â374. MR 3803162, DOI 10.1090/bull/1623
- E. Culbertson, Contract bridge red book on play, John Winston and Co., 1934.
- Ann Cutler and Rudolph McShane, The Trachtenberg speed system of basic mathematics, Doubleday & Co., Inc., Garden City, NY, 1960. MR 117137
- Percy Deift, Universality for mathematical and physical systems, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., ZĂŒrich, 2007, pp. 125â152. MR 2334189, DOI 10.4171/022-1/7
- V. Delecroix, Interval exchange transformations, Lecture notes, 2016.
- JesĂșs A. De Loera, Jörg Rambau, and Francisco Santos, Triangulations, Algorithms and Computation in Mathematics, vol. 25, Springer-Verlag, Berlin, 2010. Structures for algorithms and applications. MR 2743368, DOI 10.1007/978-3-642-12971-1
- Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. MR 2571413, DOI 10.1007/978-3-642-03311-7
- Y. Demirci, U. Islak, and A. Ozdemir, Edge and vertex flippings in regular and bipartite graphs, arXiv:2201.03315, 2022.
- Graham Denham, Eigenvectors for a random walk on a hyperplane arrangement, Adv. in Appl. Math. 48 (2012), no. 2, 312â324. MR 2873879, DOI 10.1016/j.aam.2010.09.009
- Luc Devroye, Nonuniform random variate generation, Springer-Verlag, New York, 1986. MR 836973, DOI 10.1007/978-1-4613-8643-8
- W. Dexter (ed.), Magic circle magic: a tribute to the memory of George Davenport and Lewis Davenport instituted by the Council of the Magic Circle, H. Clarke Publishing, 257â261, 1963.
- Persi Diaconis, From shuffling cards to walking around the building: an introduction to modern Markov chain theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 187â204. MR 1648031
- Persi Diaconis, Finite Fourier methods: access to tools, Probabilistic combinatorics and its applications (San Francisco, CA, 1991) Proc. Sympos. Appl. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1991, pp. 171â194. MR 1141927, DOI 10.1090/psapm/044/1141927
- Persi Diaconis, Mathematical developments from the analysis of riffle shuffling, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 73â97. MR 1994961, DOI 10.1142/9789812564481_{0}005
- Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture NotesâMonograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. MR 964069
- Persi Diaconis, A generalization of spectral analysis with application to ranked data, Ann. Statist. 17 (1989), no. 3, 949â979. MR 1015133, DOI 10.1214/aos/1176347251
- Persi Diaconis and Stewart N. Ethier, Gamblerâs ruin and the ICM, Statist. Sci. 37 (2022), no. 3, 289â305. MR 4444368, DOI 10.1214/21-sts826
- Persi Diaconis, Steven N. Evans, and Ron Graham, Unseparated pairs and fixed points in random permutations, Adv. in Appl. Math. 61 (2014), 102â124. MR 3267067, DOI 10.1016/j.aam.2014.05.006
- Persi Diaconis and James Allen Fill, Strong stationary times via a new form of duality, Ann. Probab. 18 (1990), no. 4, 1483â1522. MR 1071805
- Persi Diaconis, James Allen Fill, and Jim Pitman, Analysis of top to random shuffles, Combin. Probab. Comput. 1 (1992), no. 2, 135â155. MR 1179244, DOI 10.1017/S0963548300000158
- Persi Diaconis and Peter J. Forrester, Hurwitz and the origins of random matrix theory in mathematics, Random Matrices Theory Appl. 6 (2017), no. 1, 1730001, 26. MR 3612265, DOI 10.1142/S2010326317300017
- Persi Diaconis and David Freedman, Iterated random functions, SIAM Rev. 41 (1999), no. 1, 45â76. MR 1669737, DOI 10.1137/S0036144598338446
- Persi Diaconis and Jason Fulman, Carries, shuffling, and an amazing matrix, Amer. Math. Monthly 116 (2009), no. 9, 788â803. MR 2572087, DOI 10.4169/000298909X474864
- Persi Diaconis and Jason Fulman, Carries, shuffling, and symmetric functions, Adv. in Appl. Math. 43 (2009), no. 2, 176â196. MR 2531920, DOI 10.1016/j.aam.2009.02.002
- Persi Diaconis and Jason Fulman, Foulkes characters, Eulerian idempotents, and an amazing matrix, J. Algebraic Combin. 36 (2012), no. 3, 425â440. MR 2969071, DOI 10.1007/s10801-012-0343-7
- Persi Diaconis and Jason Fulman, Combinatorics of balanced carries, Adv. in Appl. Math. 59 (2014), 8â25. MR 3232807, DOI 10.1016/j.aam.2014.05.005
- Persi Diaconis, Jason Fulman, and Susan Holmes, Analysis of casino shelf shuffling machines, Ann. Appl. Probab. 23 (2013), no. 4, 1692â1720. MR 3098446, DOI 10.1214/12-aap884
- Persi Diaconis and Ronald Graham, The analysis of sequential experiments with feedback to subjects, Ann. Statist. 9 (1981), no. 1, 3â23. MR 600529
- Persi Diaconis and R. L. Graham, Spearmanâs footrule as a measure of disarray, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 262â268. MR 652736
- P. Diaconis and R. Graham, The solution to Elmsleyâs problem, Math. Horizons 14 (2007), 22â27.
- Persi Diaconis and Ron Graham, Magical mathematics, Princeton University Press, Princeton, NJ, 2012. The mathematical ideas that animate great magic tricks; With a foreword by Martin Gardner. MR 2858033
- Persi Diaconis and Ron Graham, The magic of Charles Sanders Peirce, The mathematics of various entertaining subjects. Vol. 3, Princeton Univ. Press, Princeton, NJ, 2019, pp. 161â203. MR 3931810
- Persi Diaconis, Ron Graham, Xiaoyu He, and Sam Spiro, Card guessing with partial feedback, Combin. Probab. Comput. 31 (2022), no. 1, 1â20. MR 4356453, DOI 10.1017/s0963548321000134
- Persi Diaconis, R. L. Graham, and William M. Kantor, The mathematics of perfect shuffles, Adv. in Appl. Math. 4 (1983), no. 2, 175â196. MR 700845, DOI 10.1016/0196-8858(83)90009-X
- Persi Diaconis, R. L. Graham, and J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures Algorithms 1 (1990), no. 1, 51â72. MR 1068491, DOI 10.1002/rsa.3240010105
- Persi Diaconis, Ron Graham, and Sam Spiro, Guessing about guessing: practical strategies for card guessing with feedback, Amer. Math. Monthly 129 (2022), no. 7, 607â622. MR 4457734, DOI 10.1080/00029890.2022.2069986
- Persi Diaconis, Kelsey Houston-Edwards, and Laurent Saloff-Coste, Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity, ALEA Lat. Am. J. Probab. Math. Stat. 17 (2020), no. 2, 901â991. MR 4182157, DOI 10.30757/alea.v17-35
- Persi Diaconis, Kelsey Houston-Edwards, and Laurent Saloff-Coste, Gamblerâs ruin estimates on finite inner uniform domains, Ann. Appl. Probab. 31 (2021), no. 2, 865â895. MR 4254498, DOI 10.1214/20-aap1607
- Persi Diaconis, Michael McGrath, and Jim Pitman, Riffle shuffles, cycles, and descents, Combinatorica 15 (1995), no. 1, 11â29. MR 1325269, DOI 10.1007/BF01294457
- Persi Diaconis and Soumik Pal, Shuffling cards by spatial motion, Stochastic Process. Appl. 152 (2022), 149â176. MR 4450463, DOI 10.1016/j.spa.2022.06.023
- Persi Diaconis, C. Y. Amy Pang, and Arun Ram, Hopf algebras and Markov chains: two examples and a theory, J. Algebraic Combin. 39 (2014), no. 3, 527â585. MR 3183482, DOI 10.1007/s10801-013-0456-7
- P. Diaconis, P. Rusmevichientong, B. van Roy, and X. Yan, Solitaire: man versus machine, Advances in Neural Information Processing Systems 17 (2004).
- Persi Diaconis and Laurent Saloff-Coste, Convolution powers of complex functions on $\Bbb Z$, Math. Nachr. 287 (2014), no. 10, 1106â1130. MR 3231528, DOI 10.1002/mana.201200163
- Persi Diaconis and Laurent Saloff-Coste, Comparison techniques for random walk on finite groups, Ann. Probab. 21 (1993), no. 4, 2131â2156. MR 1245303
- Persi Diaconis and Mehrdad Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 2, 159â179. MR 626813, DOI 10.1007/BF00535487
- Persi Diaconis and Mehrdad Shahshahani, Time to reach stationarity in the Bernoulli-Laplace diffusion model, SIAM J. Math. Anal. 18 (1987), no. 1, 208â218. MR 871832, DOI 10.1137/0518016
- P. Diaconis and M. Shahshahani, The subgroup algorithm for generating uniform random variables, Prob. in Eng. and Info. Sci. 1 (1987), 15â32.
- Persi Diaconis, Xuancheng Shao, and Kannan Soundararajan, Carries, group theory, and additive combinatorics, Amer. Math. Monthly 121 (2014), no. 8, 674â688. MR 3318477, DOI 10.4169/amer.math.monthly.121.08.674
- Persi Diaconis and Brian Skyrms, Ten great ideas about chance, Princeton University Press, Princeton, NJ, 2018. MR 3702017
- Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist. 26 (1998), no. 1, 363â397. MR 1608156, DOI 10.1214/aos/1030563990
- Persi Diaconis and Guanyang Wang, Bayesian goodness of fit tests: a conversation for David Mumford, Ann. Math. Sci. Appl. 3 (2018), no. 1, 287â308. MR 3781267, DOI 10.4310/AMSA.2018.v3.n1.a9
- A. B. Dieker and F. V. Saliola, Spectral analysis of random-to-random Markov chains, Adv. Math. 323 (2018), 427â485. MR 3725883, DOI 10.1016/j.aim.2017.10.034
- Robert P. Dobrow and James Allen Fill, The move-to-front rule for self-organizing lists with Markov dependent requests, Discrete probability and algorithms (Minneapolis, MN, 1993) IMA Vol. Math. Appl., vol. 72, Springer, New York, 1995, pp. 57â80. MR 1380521, DOI 10.1007/978-1-4612-0801-3_{5}
- Robert P. Dobrow and James Allen Fill, On the Markov chain for the move-to-root rule for binary search trees, Ann. Appl. Probab. 5 (1995), no. 1, 1â19. MR 1325037
- Robert P. Dobrow and James Allen Fill, Rates of convergence for the move-to-root Markov chain for binary search trees, Ann. Appl. Probab. 5 (1995), no. 1, 20â36. MR 1325038
- J. R. Dorfman, An introduction to chaos in nonequilibrium statistical mechanics, Cambridge Lecture Notes in Physics, vol. 14, Cambridge University Press, Cambridge, 1999. MR 1733454, DOI 10.1017/CBO9780511628870
- Jean-Jil Duchamps, Jim Pitman, and Wenpin Tang, Renewal sequences and record chains related to multiple zeta sums, Trans. Amer. Math. Soc. 371 (2019), no. 8, 5731â5755. MR 3937308, DOI 10.1090/tran/7516
- J. R. Duck, Rusduck âstay-stackâ system, Cardiste 1 (1957), 12â12.
- H. M. Edwards, Riemannâs zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 466039
- Richard Ehrenborg, Michael Levin, and Margaret A. Readdy, A probabilistic approach to the descent statistic, J. Combin. Theory Ser. A 98 (2002), no. 1, 150â162. MR 1897930, DOI 10.1006/jcta.2001.3233
- David Eisenbud, Alyson Reeves, and Burt Totaro, Initial ideals, Veronese subrings, and rates of algebras, Adv. Math. 109 (1994), no. 2, 168â187. MR 1304751, DOI 10.1006/aima.1994.1085
- P. Erdös, Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged) 5 (1932), 194â198.
- Niklas Eriksen, Expected number of inversions after a sequence of random adjacent transpositionsâan exact expression, Discrete Math. 298 (2005), no. 1-3, 155â168. MR 2163446, DOI 10.1016/j.disc.2004.09.015
- Henrik Eriksson, Kimmo Eriksson, and Jonas Sjöstrand, Expected number of inversions after a sequence of random adjacent transpositions, Formal power series and algebraic combinatorics (Moscow, 2000) Springer, Berlin, 2000, pp. 677â685. MR 1798262
- Pierre Eymard and Bernard Roynette, Marches alĂ©atoires sur le dual de $SU(2)$, Analyse harmonique sur les groupes de Lie (SĂ©m., Nancy-Strasbourg, 1973â1975) Lecture Notes in Math., Vol. 497, Springer, Berlin-New York, 1975, pp. 108â152 (French). MR 423457
- F. Fares, Quelques constructions dâalgebres et de coalgebres, UniversitĂ© du Quebec Ă Montreal Thesis, 1999.
- William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. 2nd ed. MR 88081
- William Feller, An introduction to probability theory and its applications. Vol. II, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 270403
- Valentin FĂ©ray and Victor Reiner, $P$-partitions revisited, J. Commut. Algebra 4 (2012), no. 1, 101â152. MR 2913529, DOI 10.1216/JCA-2012-4-1-101
- James Allen Fill, An exact formula for the move-to-front rule for self-organizing lists, J. Theoret. Probab. 9 (1996), no. 1, 113â160. MR 1371073, DOI 10.1007/BF02213737
- James Allen Fill and Lars Holst, On the distribution of search cost for the move-to-front rule, Random Structures Algorithms 8 (1996), no. 3, 179â186. MR 1603279, DOI 10.1002/(SICI)1098-2418(199605)8:3<179::AID-RSA2>3.0.CO;2-V
- Philippe Flajolet and Robert Sedgewick, Analytic combinatorics, Cambridge University Press, Cambridge, 2009. MR 2483235, DOI 10.1017/CBO9780511801655
- R. Fleischer, Elmsleyâs problem revsisted, Proceedings of 20th Japan Conference on Discrete and Computational Geometry, Graphs, and Games, 2017.
- S. Forte, Casino game protection, SLF Publishing, 2004.
- H. O. Foulkes, Eulerian numbers, Newcombâs problem and representations of symmetric groups, Discrete Math. 30 (1980), no. 1, 3â49. MR 561763, DOI 10.1016/0012-365X(80)90061-8
- James Franklin, The science of conjecture, Johns Hopkins University Press, Baltimore, MD, 2001. Evidence and probability before Pascal. MR 1893301
- Jason Fulman, The combinatorics of biased riffle shuffles, Combinatorica 18 (1998), no. 2, 173â184. MR 1656538, DOI 10.1007/PL00009814
- Jason Fulman, Semisimple orbits of Lie algebras and card-shuffling measures on Coxeter groups, J. Algebra 224 (2000), no. 1, 151â165. MR 1736699, DOI 10.1006/jabr.1999.8157
- Jason Fulman, Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting, J. Algebra 231 (2000), no. 2, 614â639. MR 1778162, DOI 10.1006/jabr.2000.8339
- Jason Fulman, Descent algebras, hyperplane arrangements, and shuffling cards, Proc. Amer. Math. Soc. 129 (2001), no. 4, 965â973. MR 1625753, DOI 10.1090/S0002-9939-00-05055-3
- Jason Fulman, Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems, J. Algebra 243 (2001), no. 1, 96â122. MR 1851655, DOI 10.1006/jabr.2001.8814
- Jason Fulman, Applications of symmetric functions to cycle and increasing subsequence structure after shuffles, J. Algebraic Combin. 16 (2002), no. 2, 165â194. MR 1943587, DOI 10.1023/A:1021177012548
- Jason Fulman, Steinâs method and non-reversible Markov chains, Steinâs method: expository lectures and applications, IMS Lecture Notes Monogr. Ser., vol. 46, Inst. Math. Statist., Beachwood, OH, 2004, pp. 69â77. MR 2118603
- Jason Fulman, A card shuffling analysis of deformations of the Plancherel measure of the symmetric group, Electron. J. Combin. 11 (2004), no. 1, Research Paper 21, 15. MR 2035315, DOI 10.37236/1774
- Jason Fulman, Card shuffling and the decomposition of tensor products, Pacific J. Math. 217 (2004), no. 2, 247â262. MR 2109933, DOI 10.2140/pjm.2004.217.247
- Jason Fulman, Separation cutoffs for random walk on irreducible representations, Ann. Comb. 14 (2010), no. 3, 319â337. MR 2737322, DOI 10.1007/s00026-010-0062-5
- Jason Fulman, Convergence rates of random walk on irreducible representations of finite groups, J. Theoret. Probab. 21 (2008), no. 1, 193â211. MR 2384478, DOI 10.1007/s10959-007-0102-1
- Jason Fulman, Steinâs method and Plancherel measure of the symmetric group, Trans. Amer. Math. Soc. 357 (2005), no. 2, 555â570. MR 2095623, DOI 10.1090/S0002-9947-04-03499-3
- Jason Fulman, Steinâs method and random character ratios, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3687â3730. MR 2386242, DOI 10.1090/S0002-9947-08-04371-7
- Jason Fulman, Descent identities, Hessenberg varieties, and the Weil conjectures, J. Combin. Theory Ser. A 87 (1999), no. 2, 390â397. MR 1704269, DOI 10.1006/jcta.1999.2964
- Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51â85. MR 1864086, DOI 10.1090/S0273-0979-01-00920-X
- Jason Fulman, Gene B. Kim, Sangchul Lee, and T. Kyle Petersen, On the joint distribution of descents and signs of permutations, Electron. J. Combin. 28 (2021), no. 3, Paper No. 3.37, 30. MR 4301305, DOI 10.37236/10222
- Jason Fulman and T. Kyle Petersen, Card shuffling and $P$-partitions, Discrete Math. 344 (2021), no. 8, Paper No. 112448, 18. MR 4257964, DOI 10.1016/j.disc.2021.112448
- K. Fulves, Charles Jordanâs best card tricks: with 265 illustrations, Dover, 1992.
- J. Gallian, Contemporary abstract algebra, 9th ed., Brooks/Cole, 2015.
- Terry Gannon, The cyclic structure of unimodal permutations, Discrete Math. 237 (2001), no. 1-3, 149â161. MR 1835657, DOI 10.1016/S0012-365X(00)00368-X
- Martin Gardner, Mathematical circus, MAA Spectrum, Mathematical Association of America, Washington, DC, 1992. More puzzles, games, paradoxes, and other mathematical entertainments from Scientific American; Revised reprint of the 1981 edition; With a preface by Donald Knuth. MR 1172317
- Adriano M. Garsia, Combinatorics of the free Lie algebra and the symmetric group, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 309â382. MR 1039352
- A. M. Garsia and J. Remmel, Shuffles of permutations and the Kronecker product, Graphs Combin. 1 (1985), no. 3, 217â263. MR 951014, DOI 10.1007/BF02582950
- A. M. Garsia and C. Reutenauer, A decomposition of Solomonâs descent algebra, Adv. Math. 77 (1989), no. 2, 189â262. MR 1020585, DOI 10.1016/0001-8708(89)90020-0
- A. M. Garsia and N. Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2003), no. 2, 217â263. MR 2019274, DOI 10.1016/S0097-3165(03)00042-6
- A. M. Garsia and N. Wallach, $r$-Qsym is free over Sym, J. Combin. Theory Ser. A 114 (2007), no. 4, 704â732. MR 2319171, DOI 10.1016/j.jcta.2006.08.009
- Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, and Jean-Yves Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), no. 2, 218â348. MR 1327096, DOI 10.1006/aima.1995.1032
- Murray Gerstenhaber and S. D. Schack, A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra 48 (1987), no. 3, 229â247. MR 917209, DOI 10.1016/0022-4049(87)90112-5
- Ira M. Gessel, A historical survey of $P$-partitions, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 169â188. MR 3617222, DOI 10.1090//mbk/100/10
- Ira M. Gessel and Christophe Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993), no. 2, 189â215. MR 1245159, DOI 10.1016/0097-3165(93)90095-P
- Ira M. Gessel and Yan Zhuang, Shuffle-compatible permutation statistics, Adv. Math. 332 (2018), 85â141. MR 3810249, DOI 10.1016/j.aim.2018.05.003
- Ira M. Gessel and Yan Zhuang, Plethystic formulas for permutation enumeration, Adv. Math. 375 (2020), 107370, 55. MR 4136604, DOI 10.1016/j.aim.2020.107370
- A. Gibbs and F. Su, On choosing and bounding probability metrics, International Statistical Review 70 (2002), 419â435.
- Peter Giblin, Graphs, surfaces and homology, 3rd ed., Cambridge University Press, Cambridge, 2010. MR 2722281, DOI 10.1017/CBO9780511779534
- J. Gil and J. Fresan, Multiple zeta values: from numbers to motives, Clay Mathematics Proceedings.
- E. Gilbert, Theory of shuffling, Technical report, Bell Laboratories, 1955.
- Steven B. Gillispie and Michael D. Perlman, The size distribution for Markov equivalence classes of acyclic digraph models, Artificial Intelligence 141 (2002), no. 1-2, 137â155. MR 1935281, DOI 10.1016/S0004-3702(02)00264-3
- P. Golle, Dealing cards in poker games, International conference on information technology, 2005, 506â511.
- Louis Gordon, Successive sampling in large finite populations, Ann. Statist. 11 (1983), no. 2, 702â706. MR 696081
- I. Gormley and T. Murphy, A latent space model for rank data, Proceedings of the 24th ICML, 90â102, 2006.
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- Andrew Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253â276. MR 1483922
- Andrew Granville and Jennifer Granville, Prime suspects, Princeton University Press, Princeton, NJ, 2019. The anatomy of integers and permutations; Illustrated by Robert J. Lewis. MR 3966460
- Curtis Greene, Posets of shuffles, J. Combin. Theory Ser. A 47 (1988), no. 2, 191â206. MR 930953, DOI 10.1016/0097-3165(88)90018-0
- David Griffeath, A maximal coupling for Markov chains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 95â106. MR 370771, DOI 10.1007/BF00539434
- Darij Grinberg, Shuffle-compatible permutation statistics II: the exterior peak set, Electron. J. Combin. 25 (2018), no. 4, Paper No. 4.17, 61. MR 3874283, DOI 10.37236/7946
- D. Grinberg and V. Reiner, Hopf algebras in combinatorics, arXiv:1409.8356, 2014.
- C. Grinstead and L. Snell, Introduction to probability, American Mathematical Society, 1997.
- M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269â504. MR 3274562, DOI 10.1007/s00222-014-0505-4
- Marshall Hall Jr., The theory of groups, Chelsea Publishing Co., New York, 1976. Reprinting of the 1968 edition. MR 414669
- Ben Hambly and Terry Lyons, Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math. (2) 171 (2010), no. 1, 109â167. MR 2630037, DOI 10.4007/annals.2010.171.109
- L. Hamilton, When random isnât random enough: lessons from an onlike poker exploit, http://www.lauradhamilton.com/random-lessons-online-poker-exploit, 2014.
- Phil Hanlon, The action of $S_n$ on the components of the Hodge decomposition of Hochschild homology, Michigan Math. J. 37 (1990), no. 1, 105â124. MR 1042517, DOI 10.1307/mmj/1029004069
- Phil Hanlon, Order and disorder in algebraic combinatorics, Math. Intelligencer 14 (1992), no. 2, 20â25. MR 1160702, DOI 10.1007/BF03025209
- Phil Hanlon and Patricia Hersh, A Hodge decomposition for the complex of injective words, Pacific J. Math. 214 (2004), no. 1, 109â125. MR 2039128, DOI 10.2140/pjm.2004.214.109
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- D. R. Heath-Brown, Artinâs conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27â38. MR 830627, DOI 10.1093/qmath/37.1.27
- W. J. Hendricks, The stationary distribution of an interesting Markov chain, J. Appl. Probability 9 (1972), 231â233. MR 292178, DOI 10.2307/3212655
- R. J. Henery, Permutation probabilities as models for horse races, J. Roy. Statist. Soc. Ser. B 43 (1981), no. 1, 86â91. MR 610382
- Patricia Hersh, Two generalizations of posets of shuffles, J. Combin. Theory Ser. A 97 (2002), no. 1, 1â26. MR 1879043, DOI 10.1006/jcta.2001.3187
- Silvia Heubach and Toufik Mansour, Combinatorics of compositions and words, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2010. MR 2531482
- Michael E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275â290. MR 1141796
- Michael E. Hoffman, Algebraic aspects of multiple zeta values, Zeta functions, topology and quantum physics, Dev. Math., vol. 14, Springer, New York, 2005, pp. 51â73. MR 2179272, DOI 10.1007/0-387-24981-8_{4}
- Michael E. Hoffman, Quasi-shuffle products, J. Algebraic Combin. 11 (2000), no. 1, 49â68. MR 1747062, DOI 10.1023/A:1008791603281
- John M. Holte, Carries, combinatorics, and an amazing matrix, Amer. Math. Monthly 104 (1997), no. 2, 138â149. MR 1437415, DOI 10.2307/2974981
- John M. Holte, Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3837â3873. MR 1389778, DOI 10.1090/S0002-9947-97-01794-7
- Christopher Hooley, On Artinâs conjecture, J. Reine Angew. Math. 225 (1967), 209â220. MR 207630, DOI 10.1515/crll.1967.225.209
- Jonathan Huang and Carlos Guestrin, Uncovering the riffled independence structure of ranked data, Electron. J. Stat. 6 (2012), 199â230. MR 2879677, DOI 10.1214/12-EJS670
- John Hamal Hubbard and Barbara Burke Hubbard, Vector calculus, linear algebra, and differential forms, Prentice Hall, Inc., Upper Saddle River, NJ, 1999. A unified approach. MR 1657732
- J. Huh, Combinatorial applications of the Hodge-Riemann relations, Proc. Int. Cong. of Math. 2018, Vol. 3, 3079â3098.
- Axel Hultman, Permutation statistics of products of random permutations, Adv. in Appl. Math. 54 (2014), 1â10. MR 3199772, DOI 10.1016/j.aam.2013.10.003
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, Adv. in Appl. Math. 112 (2020), 101960, 125. MR 4023911, DOI 10.1016/j.aam.2019.101960
- I. Martin Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. MR 2270898, DOI 10.1090/chel/359
- Daniel C. Isaksen, A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly 109 (2002), no. 9, 796â805. MR 1933702, DOI 10.2307/3072368
- Ămit IĆlak, Descent-inversion statistics in riffle shuffles, Turkish J. Math. 42 (2018), no. 2, 502â514. MR 3794483, DOI 10.3906/mat-1610-59
- Alexander R. Its, Craig A. Tracy, and Harold Widom, Random words, Toeplitz determinants, and integrable systems. I, Random matrix models and their applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 245â258. MR 1842789
- Alexander R. Its, Craig A. Tracy, and Harold Widom, Random words, Toeplitz determinants and integrable systems. II, Phys. D 152/153 (2001), 199â224. Advances in nonlinear mathematics and science. MR 1837910, DOI 10.1016/S0167-2789(01)00171-3
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, MA, 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- Svante Janson, On degenerate sums of $m$-dependent variables, J. Appl. Probab. 52 (2015), no. 4, 1146â1155. MR 3439177, DOI 10.1239/jap/1450802758
- Svante Janson, Renewal theory for $M$-dependent variables, Ann. Probab. 11 (1983), no. 3, 558â568. MR 704542
- Svante Janson, Generalized Galois numbers, inversions, lattice paths, Ferrers diagrams and limit theorems, Electron. J. Combin. 19 (2012), no. 3, Paper 34, 16. MR 2988856, DOI 10.37236/2188
- Johan Jonasson, The overhand shuffle mixes in $\Theta (n^2\log n)$ steps, Ann. Appl. Probab. 16 (2006), no. 1, 231â243. MR 2209341, DOI 10.1214/105051605000000692
- Johan Jonasson and Ben Morris, Rapid mixing of dealer shuffles and clumpy shuffles, Electron. Commun. Probab. 20 (2015), no. 20, 11. MR 3320408, DOI 10.1214/ECP.v20-3682
- Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, CA, 1968. MR 230102
- Samuel Karlin and Howard M. Taylor, A first course in stochastic processes, 2nd ed., Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 356197
- Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
- Joseph B. Keller, How many shuffles to mix a deck?, SIAM Rev. 37 (1995), no. 1, 88â89. MR 1327719, DOI 10.1137/1037006
- John G. Kemeny and J. Laurie Snell, Finite Markov chains, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 115196
- Maurice Kendall and Jean Dickinson Gibbons, Rank correlation methods, 5th ed., A Charles Griffin Title, Edward Arnold, London, 1990. MR 1079065
- Adalbert Kerber, Applied finite group actions, 2nd ed., Algorithms and Combinatorics, vol. 19, Springer-Verlag, Berlin, 1999. MR 1716962, DOI 10.1007/978-3-662-11167-3
- A. Kerber and K. Thurlings, Eulerian numbers, Foulkes characters, and Lefschetz characters of $S_n$, Semin. Lothar. 8 (1984), 31â36.
- Steven Kerckhoff, Howard Masur, and John Smillie, A rational billiard flow is uniquely ergodic in almost every direction, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 141â142. MR 799797, DOI 10.1090/S0273-0979-1985-15398-4
- S. V. Kerov, Distribution of symmetry types of high rank tensors, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), no. DifferentsialâČnaya Geometriya, Gruppy Li i Mekh. VIII, 181â186, 195 (Russian); English transl., J. Soviet Math. 41 (1988), no. 2, 995â999. MR 869584, DOI 10.1007/BF01247095
- Sergei V. Kerov and Anatol M. Vershik, The characters of the infinite symmetric group and probability properties of the Robinson-Schensted-Knuth algorithm, SIAM J. Algebraic Discrete Methods 7 (1986), no. 1, 116â124. MR 819713, DOI 10.1137/0607014
- M. Khalkhali, Very basic noncommutative geometry, arXiv:0408416, 2004.
- Donald E. Knuth, The art of computer programming. Vol. 2, Addison-Wesley, Reading, MA, 1998. Seminumerical algorithms; Third edition [of MR0286318]. MR 3077153
- Donald E. Knuth and Luis Trabb Pardo, Analysis of a simple factorization algorithm, Theoret. Comput. Sci. 3 (1976/77), no. 3, 321â348. MR 498355, DOI 10.1016/0304-3975(76)90050-5
- Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math. 396 (1989), 212â219. MR 988552, DOI 10.1515/crll.1989.396.212
- Maxim Kontsevich and Don Zagier, Periods, Mathematics unlimitedâ2001 and beyond, Springer, Berlin, 2001, pp. 771â808. MR 1852188
- M. Kresta, A. Etchells, D. Dickey, and V. Atiomo-Obeng, Advances in industrial mixing: a companion to the handbook of industrial mixing, John Wiley and Sons, 2016.
- T. Krityakierne and T. A. Thanatipanonda, No Feedback? No Worries! The art of guessing the right card, arXiv:2205.08793, 2022.
- S. Kullback, A lower bound for discrimination in term of variance, IEEE Trans. Inform. Theorey 4 (1967), 126â127.
- E. E. Kummer, Ăber die ErgĂ€nzungssĂ€tze zu den allgemeinen ReciprocitĂ€tsgesetzen, J. Reine Angew. Math. 44 (1852), 93â146 (German). MR 1578793, DOI 10.1515/crll.1852.44.93
- Hubert Lacoin, Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion, Ann. Probab. 44 (2016), no. 2, 1426â1487. MR 3474475, DOI 10.1214/15-AOP1004
- Nadia LafreniĂšre, Eigenvalues of symmetrized shuffling operators, SĂ©m. Lothar. Combin. 82B (2020), Art. 78, 12 (English, with English and French summaries). MR 4098299
- N. LafreniÚre, Valeurs propres des operateurs de melange symetrises, Ph.D. thesis, Université du Quebec à Montreal, 2019.
- Steven P. Lalley, Cycle structure of riffle shuffles, Ann. Probab. 24 (1996), no. 1, 49â73. MR 1387626, DOI 10.1214/aop/1042644707
- Steven P. Lalley, Riffle shuffles and their associated dynamical systems, J. Theoret. Probab. 12 (1999), no. 4, 903â932. MR 1729462, DOI 10.1023/A:1021636902356
- Steven P. Lalley, On the rate of mixing for $p$-shuffles, Ann. Appl. Probab. 10 (2000), no. 4, 1302â1321. MR 1810876, DOI 10.1214/aoap/1019487618
- Kenneth Lange, Applied probability, 2nd ed., Springer Texts in Statistics, Springer, New York, 2010. MR 2680838, DOI 10.1007/978-1-4419-7165-4
- Steffen L. Lauritzen, Graphical models, Oxford Statistical Science Series, vol. 17, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1419991
- Tu Quoc Thang Le and Jun Murakami, Kontsevichâs integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995), no. 2, 193â206. MR 1320252, DOI 10.1016/0166-8641(94)00054-7
- James R. Lee and Yuval Peres, Harmonic maps on amenable groups and a diffusive lower bound for random walks, Ann. Probab. 41 (2013), no. 5, 3392â3419. MR 3127886, DOI 10.1214/12-AOP779
- James R. Lee, Yuval Peres, and Charles K. Smart, A Gaussian upper bound for martingale small-ball probabilities, Ann. Probab. 44 (2016), no. 6, 4184â4197. MR 3572334, DOI 10.1214/15-AOP1073
- David A. Levin and Yuval Peres, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2017. Second edition of [ MR2466937]; With contributions by Elizabeth L. Wilmer; With a chapter on âCoupling from the pastâ by James G. Propp and David B. Wilson. MR 3726904, DOI 10.1090/mbk/107
- Daniel Levin and Mark Wildon, A combinatorial method for calculating the moments of LĂ©vy area, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6695â6709. MR 2434307, DOI 10.1090/S0002-9947-08-04526-1
- Torgny Lindvall, Lectures on the coupling method, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR 1180522
- P. Liu, Asymptotic analysis of card guessing with feedback, arXiv:1908.07718, 2019.
- Jean-Louis Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by MarĂa O. Ronco. MR 1217970, DOI 10.1007/978-3-662-21739-9
- Jean-Louis Loday and MarĂa O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), no. 2, 293â309. MR 1654173, DOI 10.1006/aima.1998.1759
- M. Lothaire, Combinatorics on words, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997. With a foreword by Roger Lyndon and a preface by Dominique Perrin; Corrected reprint of the 1983 original, with a new preface by Perrin. MR 1475463, DOI 10.1017/CBO9780511566097
- R. Duncan Luce, Individual choice behavior: A theoretical analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1959. MR 108411
- R. Duncan Luce, The choice axiom after twenty years, J. Mathematical Psychology 15 (1977), no. 3, 215â233. MR 462675, DOI 10.1016/0022-2496(77)90032-3
- Jean-Gabriel Luque and Jean-Yves Thibon, Pfaffian and Hafnian identities in shuffle algebras, Adv. in Appl. Math. 29 (2002), no. 4, 620â646. MR 1943369, DOI 10.1016/S0196-8858(02)00036-2
- G. Lusztig, A $q$-analogue of an identity of N. Wallach, Studies in Lie theory, Progr. Math., vol. 243, BirkhĂ€user Boston, Boston, MA, 2006, pp. 405â410. MR 2214256, DOI 10.1007/0-8176-4478-4_{1}5
- Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2036784, DOI 10.1093/acprof:oso/9780198506485.001.0001
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- I. Macdonald, Notes on Schubert polynomials, Laboratoire de combinatoire et dâinformatique mathematique (LACIM), Univ. du Quebec Ă Montreal, Montreal, 1991.
- R. S. MacKay, Renormalisation in area-preserving maps, Advanced Series in Nonlinear Dynamics, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1336593, DOI 10.1142/9789814354462
- S. Mahajan, Shuffles on Coxeter groups, arXiv:math/0108094, 2001.
- Shahn Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995. MR 1381692, DOI 10.1017/CBO9780511613104
- C. Malvenuto, Produits et coproduits des fonctions quasi-symetriques et de lâalgebre des descents, Laboratoire de combinatoire et dâinformatique mathematique (LACIM), Univ. du Quebec Ă Montreal, Montreal, 1994.
- Clauda Malvenuto and Christophe Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967â982. MR 1358493, DOI 10.1006/jabr.1995.1336
- Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours Spécialisés [Specialized Courses], 3. MR 1852463
- Brad Mann, How many times should you shuffle a deck of cards?, UMAP J. 15 (1994), no. 4, 303â332. MR 1439037
- John I. Marden, Analyzing and modeling rank data, Monographs on Statistics and Applied Probability, vol. 64, Chapman & Hall, London, 1995. MR 1346107
- Stuart Margolis, Franco V. Saliola, and Benjamin Steinberg, Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry, Mem. Amer. Math. Soc. 274 (2021), no. 1345, xi+135. MR 4365944, DOI 10.1090/memo/1345
- J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 222892
- John McCabe, On serial files with relocatable records, Operations Res. 13 (1965), 609â618. MR 182458, DOI 10.1287/opre.13.4.609
- Steve Medvedoff and Kent Morrison, Groups of perfect shuffles, Math. Mag. 60 (1987), no. 1, 3â14. MR 876415, DOI 10.2307/2690131
- S. Melczer, An invitation to analytic combinatorics, Springer, 2021.
- P. Meliot, Fluctuations of central measures on partitions, Proceedings of the 24th International Conference on Formal Power Series and Algrebaic Combinatorics, pp. 387â398, 2012.
- P. Meliot, A central limit theorem for the characters of the infinite symmetric group and of the infinite Hecke algera, arXiv:1105.0091, 2011.
- Alexander R. Miller, Foulkes characters for complex reflection groups, Proc. Amer. Math. Soc. 143 (2015), no. 8, 3281â3293. MR 3348771, DOI 10.1090/S0002-9939-2015-12385-4
- Alexander R. Miller, On Foulkes characters, Math. Ann. 381 (2021), no. 3-4, 1589â1614. MR 4333425, DOI 10.1007/s00208-021-02197-4
- Gregory K. Miller and Stephanie L. Fridell, A forgotten discrete distribution? Reviving the negative hypergeometric model, Amer. Statist. 61 (2007), no. 4, 347â350. MR 2411795, DOI 10.1198/000313007X245140
- S. Minch, The collected works of Alex Elmsley, Volume 1, L and L Publishing, 1991.
- Sarah Miracle and Amanda Pascoe Streib, Rapid mixing of $k$-class biased permutations, LATIN 2018: Theoretical informatics, Lecture Notes in Comput. Sci., vol. 10807, Springer, Cham, 2018, pp. 820â834. MR 3787002, DOI 10.1007/978-3-319-77404-6_{5}9
- F. Monopoli, Minimizing the frequency of carries in modular addition, arXiv:1511.02404, 2015.
- Francesco Monopoli and Imre Z. Ruzsa, Carries and the arithmetic progression structure of sets, Integers 17 (2017), Paper No. A8, 21. MR 3636629
- Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
- Ben Morris, Improved mixing time bounds for the Thorp shuffle and $L$-reversal chain, Ann. Probab. 37 (2009), no. 2, 453â477. MR 2510013, DOI 10.1214/08-AOP409
- Ben Morris, The mixing time of the Thorp shuffle, SIAM J. Comput. 38 (2008), no. 2, 484â504. MR 2411032, DOI 10.1137/050636231
- Ben Morris, Improved mixing time bounds for the Thorp shuffle, Combin. Probab. Comput. 22 (2013), no. 1, 118â132. MR 3002577, DOI 10.1017/S0963548312000478
- Ben Morris, Phillip Rogaway, and Till Stegers, How to encipher messages on a small domain: deterministic encryption and the Thorp shuffle, Advances in cryptologyâCRYPTO 2009, Lecture Notes in Comput. Sci., vol. 5677, Springer, Berlin, 2009, pp. 286â302. MR 2556963, DOI 10.1007/978-3-642-03356-8_{1}7
- Ben Morris, Phillip Rogaway, and Till Stegers, Deterministic encryption with the Thorp shuffle, J. Cryptology 31 (2018), no. 2, 521â536. MR 3771418, DOI 10.1007/s00145-017-9262-z
- S. Brent Morris, Magic tricks, card shuffling and dynamic computer memories, MAA Spectrum, Mathematical Association of America, Washington, DC, 1998. With an introduction by Martin Gardner. MR 1489235
- J. Munkres, Elements of algebraic topology, CRC Press, 1993.
- Fumihiko Nakano and Taizo Sadahiro, A generalization of carries processes and Eulerian numbers, Adv. in Appl. Math. 53 (2014), 28â43. MR 3149692, DOI 10.1016/j.aam.2013.09.005
- Fumihiko Nakano and Taizo Sadahiro, A generalization of carries process and riffle shuffles, Discrete Math. 339 (2016), no. 2, 974â991. MR 3431409, DOI 10.1016/j.disc.2015.10.025
- Evita Nestoridi, Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement, Probab. Theory Related Fields 174 (2019), no. 3-4, 929â943. MR 3980308, DOI 10.1007/s00440-018-0872-7
- Evita Nestoridi and Graham White, Shuffling large decks of cards and the Bernoulli-Laplace urn model, J. Theoret. Probab. 32 (2019), no. 1, 417â446. MR 3908920, DOI 10.1007/s10959-018-0807-3
- L. Ng, Heisenberg model, Bethe ansatz, and random walks, Harvard undergraduate senior thesis, 1996.
- Ivan Niven, Numbers: rational and irrational, New Mathematical Library, vol. 1, Random House, New York-Toronto, 1961. MR 130199
- Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative symmetric functions and an amazing matrix, Adv. in Appl. Math. 48 (2012), no. 3, 528â534. MR 2899968, DOI 10.1016/j.aam.2011.11.008
- E. Oguz, A counterexample to the shuffle compatibility conjecture, arXiv:1807.01398, 2018.
- Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
- J. M. Ottino, The kinematics of mixing: stretching, chaos, and transport, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 1001565
- Chung Yin Amy Pang, Hopf Algebras and Markov Chains, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)âStanford University. MR 4239977
- A. Pang, Markov chains from descent operators on combinatorial Hopf algebras, arXiv:1609.04312, preprint, 2016.
- A. Pang, A Hopf-algebraic lift of the down-up Markov chain on partitions to permutations, arXiv:1508.01570.
- C. Y. Amy Pang, Lumpings of algebraic Markov chains arise from subquotients, J. Theoret. Probab. 32 (2019), no. 4, 1804â1844. MR 4020688, DOI 10.1007/s10959-018-0834-0
- D. Parlett, A history of card games, Oxford University Press, 1991.
- F. Patras, Construction gĂ©omĂ©trique des idempotents eulĂ©riens. Filtration des groupes de polytopes et des groupes dâhomologie de Hochschild, Bull. Soc. Math. France 119 (1991), no. 2, 173â198 (French, with English summary). MR 1116844
- Lerna Pehlivan, On top to random shuffles, no feedback card guessing, and fixed points of permutations, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)âUniversity of Southern California. MR 2717999
- Robin Pemantle, Randomization time for the overhand shuffle, J. Theoret. Probab. 2 (1989), no. 1, 37â49. MR 981762, DOI 10.1007/BF01048267
- Robin Pemantle and Mark C. Wilson, Analytic combinatorics in several variables, Cambridge Studies in Advanced Mathematics, vol. 140, Cambridge University Press, Cambridge, 2013. MR 3088495, DOI 10.1017/CBO9781139381864
- J. Peters, The online poker cheating scandal that keeps going and going, Slate, 2013.
- T. Kyle Petersen, Descents, peaks, and P-partitions, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)âBrandeis University. MR 2708454
- T. Kyle Petersen, Enriched $P$-partitions and peak algebras, Adv. Math. 209 (2007), no. 2, 561â610. MR 2296309, DOI 10.1016/j.aim.2006.05.016
- T. Kyle Petersen, Eulerian numbers, BirkhĂ€user Advanced Texts: Basler LehrbĂŒcher. [BirkhĂ€user Advanced Texts: Basel Textbooks], BirkhĂ€user/Springer, New York, 2015. With a foreword by Richard Stanley. MR 3408615, DOI 10.1007/978-1-4939-3091-3
- R. M. Phatarfod, On the matrix occurring in a linear search problem, J. Appl. Probab. 28 (1991), no. 2, 336â346. MR 1104570, DOI 10.1017/s0021900200039723
- Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. Graduate Texts in Mathematics, 88. MR 674652
- John Pike, Eigenfunctions for Random Walks on Hyperplane Arrangements, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)âUniversity of Southern California. MR 3193075
- N. Pinter-Wollman, R. G. Wollman, R.A., S. Holmes, and D. Gordon, The effect of individual variation on the structure and function of interaction netorks in harvester ants, Journal of the Royal Society Interface 8 (2011), 1562â1573.
- Jim Pitman, Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Combin. Theory Ser. A 77 (1997), no. 2, 279â303. MR 1429082, DOI 10.1006/jcta.1997.2747
- J. W. Pitman, On coupling of Markov chains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 4, 315â322. MR 415775, DOI 10.1007/BF00532957
- J. Pitman and Z. You, Stationary 1-dependent counting processes: from runs to bivariate generating functions, arXiv:2105.08255, 2021.
- R. L. Plackett, The analysis of permutations, J. Roy. Statist. Soc. Ser. C 24 (1975), no. 2, 193â202. MR 391338, DOI 10.2307/2346567
- StĂ©phane Poirier, Cycle type and descent set in wreath products, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995), 1998, pp. 315â343. MR 1603753, DOI 10.1016/S0012-365X(97)00123-4
- StĂ©phane Poirier and Christophe Reutenauer, AlgĂšbres de Hopf de tableaux, Ann. Sci. Math. QuĂ©bec 19 (1995), no. 1, 79â90 (French, with English and French summaries). MR 1334836
- Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly 122 (2015), no. 7, 636â644. MR 3383891, DOI 10.4169/amer.math.monthly.122.7.636
- W. Poundstone, Fortuneâs formula: the untold story of the scientific betting system that beat the casinos and Wall Street, Hill and Wang Publishing, 2005.
- Maxim Rabinovich, Aaditya Ramdas, Michael I. Jordan, and Martin J. Wainwright, Function-specific mixing times and concentration away from equilibrium, Bayesian Anal. 15 (2020), no. 2, 505â532. MR 4078723, DOI 10.1214/19-BA1151
- Sarnath Ramnath and Daniel Scully, Moving card $i$ to position $j$ with perfect shuffles, Math. Mag. 69 (1996), no. 5, 361â365. MR 1438671, DOI 10.2307/2691282
- Evan Randles and Laurent Saloff-Coste, On the convolution powers of complex functions on $\Bbb {Z}$, J. Fourier Anal. Appl. 21 (2015), no. 4, 754â798. MR 3370010, DOI 10.1007/s00041-015-9386-1
- J. Reeds, unpublished manuscript, 1981.
- J. Reeds, Cracking a random number generator, Cryptologia 1 (1977), 20â26.
- Victor Reiner, Quotients of Coxeter complexes and $P$-partitions, Mem. Amer. Math. Soc. 95 (1992), no. 460, vi+134. MR 1101971, DOI 10.1090/memo/0460
- Victor Reiner, Signed permutation statistics and cycle type, European J. Combin. 14 (1993), no. 6, 569â579. MR 1248064, DOI 10.1006/eujc.1993.1059
- Victor Reiner, Franco Saliola, and Volkmar Welker, Spectra of symmetrized shuffling operators, Mem. Amer. Math. Soc. 228 (2014), no. 1072, vi+109. MR 3184410
- Vladimir Retakh and Robert Lee Wilson, Algebras associated to acyclic directed graphs, Adv. in Appl. Math. 42 (2009), no. 1, 42â59. MR 2475312, DOI 10.1016/j.aam.2008.04.002
- Christophe Reutenauer, Theorem of PoincarĂ©-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, Combinatoire Ă©numĂ©rative (Montreal, Que., 1985/Quebec, Que., 1985) Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 267â284. MR 927769, DOI 10.1007/BFb0072520
- Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799
- John Rhodes and Anne Schilling, Unified theory for finite Markov chains, Adv. Math. 347 (2019), 739â779. MR 3920838, DOI 10.1016/j.aim.2019.03.004
- Yosef Rinott and Vladimir Rotar, On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics, Ann. Appl. Probab. 7 (1997), no. 4, 1080â1105. MR 1484798, DOI 10.1214/aoap/1043862425
- Ronald Rivest, On self-organizing sequential search heuristics, Comm. ACM 19 (1976), no. 2, 63â67. MR 408303, DOI 10.1145/359997.360000
- I. Rosen, 60 minutes report: How online gamblers unmasked cheaters, 2008.
- Margit Rösler and Michael Voit, A central limit theorem for random walks on the dual of a compact Grassmannian, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 013, 18. MR 3313689, DOI 10.3842/SIGMA.2015.013
- S. Ross, A first course in probability, 9th ed., Pearson, 2014.
- Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028, DOI 10.1007/978-1-4757-6804-6
- Franco Saliola, Eigenvectors for a random walk on a left-regular band, Adv. in Appl. Math. 48 (2012), no. 2, 306â311. MR 2873878, DOI 10.1016/j.aam.2011.09.002
- Laurent Saloff-Coste, Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, 1996) Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997, pp. 301â413. MR 1490046, DOI 10.1007/BFb0092621
- Laurent Saloff-Coste, Total variation lower bounds for finite Markov chains: Wilsonâs lemma, Random walks and geometry, Walter de Gruyter, Berlin, 2004, pp. 515â532. MR 2087800
- L. Saloff-Coste and J. ZĂșñiga, Refined estimates for some basic random walks on the symmetric and alternating groups, ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 359â392. MR 2461789
- Francisco Santos, The Cayley trick and triangulations of products of simplices, Integer points in polyhedraâgeometry, number theory, algebra, optimization, Contemp. Math., vol. 374, Amer. Math. Soc., Providence, RI, 2005, pp. 151â177. MR 2134766, DOI 10.1090/conm/374/06904
- Mark Sellke, Cutoff for the asymmetric riffle shuffle, Ann. Probab. 50 (2022), no. 6, 2244â2287. MR 4499839, DOI 10.1214/22-aop1582
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 450380
- L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966), 340â357. MR 195117, DOI 10.1090/S0002-9947-1966-0195117-8
- M. Silverman, Progressive randomization of a deck of playing cards: experimental tests and statistical analysis of the riffle shuffle, Open Journal of Statistics 9 (2019), 268â298.
- Jonas Sjöstrand, Expected length of a product of random reflections, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4369â4380. MR 2957227, DOI 10.1090/S0002-9939-2012-11283-3
- Louis Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), no. 2, 255â264. MR 444756, DOI 10.1016/0021-8693(76)90182-4
- Sam Spiro, Online card games, Electron. J. Probab. 27 (2022), Paper No. 42, 15. MR 4402968, DOI 10.1214/22-ejp768
- R. Spocane, Sid II, Monge and his shuffle, Gibeciere, Vol. 10, No. 2, 2015, 165â172.
- Jonathan Derek Stadler, Schur functions, juggling, and statistics on shuffled permutations, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)âThe Ohio State University. MR 2696239
- Richard P. Stanley, Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, Providence, RI, 1972. MR 332509
- Richard P. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171â178. MR 317988, DOI 10.1016/0012-365X(73)90108-8
- R. Stanley, Eulerian partitions of a unit hypercube, Higher combinatorics, Reidel, Dordrecht/Boston, 1977, p. 49.
- Richard P. Stanley, Generalized riffle shuffles and quasisymmetric functions, Ann. Comb. 5 (2001), no. 3-4, 479â491. Dedicated to the memory of Gian-Carlo Rota (Tianjin, 1999). MR 1897637, DOI 10.1007/s00026-001-8023-7
- Richard P. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory Ser. A 31 (1981), no. 1, 56â65. MR 626441, DOI 10.1016/0097-3165(81)90053-4
- Richard P. Stanley, Alternating permutations and symmetric functions, J. Combin. Theory Ser. A 114 (2007), no. 3, 436â460. MR 2310744, DOI 10.1016/j.jcta.2006.06.008
- Richard P. Stanley, Two poset polytopes, Discrete Comput. Geom. 1 (1986), no. 1, 9â23. MR 824105, DOI 10.1007/BF02187680
- Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- Richard P. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389â496. MR 2383131, DOI 10.1090/pcms/013/08
- Dudley Stark, A. Ganesh, and Neil OâConnell, Information loss in riffle shuffling, Combin. Probab. Comput. 11 (2002), no. 1, 79â95. MR 1888184, DOI 10.1017/S0963548301004990
- Charles Stein, Approximate computation of expectations, Institute of Mathematical Statistics Lecture NotesâMonograph Series, vol. 7, Institute of Mathematical Statistics, Hayward, CA, 1986. MR 882007
- Benjamin Steinberg, Möbius functions and semigroup representation theory, J. Combin. Theory Ser. A 113 (2006), no. 5, 866â881. MR 2231092, DOI 10.1016/j.jcta.2005.08.004
- Benjamin Steinberg, Möbius functions and semigroup representation theory. II. Character formulas and multiplicities, Adv. Math. 217 (2008), no. 4, 1521â1557. MR 2382734, DOI 10.1016/j.aim.2007.12.001
- John R. Stembridge, Enriched $P$-partitions, Trans. Amer. Math. Soc. 349 (1997), no. 2, 763â788. MR 1389788, DOI 10.1090/S0002-9947-97-01804-7
- Rob Sturman, Julio M. Ottino, and Stephen Wiggins, The mathematical foundations of mixing, Cambridge Monographs on Applied and Computational Mathematics, vol. 22, Cambridge University Press, Cambridge, 2006. The linked twist map as a paradigm in applications: micro to macro, fluids to solids. MR 2265644, DOI 10.1017/CBO9780511618116
- Tze-chien Sun, A note on the unimodality of distribution functions of class $L$, Ann. Math. Statist. 38 (1967), 1296â1299. MR 216545, DOI 10.1214/aoms/1177698804
- Lucas Teyssier, Limit profile for random transpositions, Ann. Probab. 48 (2020), no. 5, 2323â2343. MR 4152644, DOI 10.1214/20-AOP1424
- Jean-Yves Thibon, The cycle enumerator of unimodal permutations, Ann. Comb. 5 (2001), no. 3-4, 493â500. Dedicated to the memory of Gian-Carlo Rota (Tianjin, 1999). MR 1897638, DOI 10.1007/s00026-001-8024-6
- Elmar Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzĂ€hlbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40â61 (German). MR 173169, DOI 10.1007/BF01114877
- Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR 1741181, DOI 10.1007/978-1-4612-1236-2
- E. Thorp, Nonrandom shuffling with applications to the game of Faro, J. Amer. Statist. Assoc. 68 (1973), 842â847.
- E. Thorp, Beat the dealer: a winning strategy for the game of twenty-one, 1962.
- E. Thorp and S. Kassouf, Beat the market: a scientific stock market system, 1967.
- Edward O. Thorp, A man for all markets, Random House, New York, 2018. From Las Vegas to Wall Street, how I beat the dealer and the market; With a foreword by Nassim Nicholas Taleb; Paperback edition of the 2017 original. MR 3791494
- Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 470099
- BĂĄlint TĂłth, Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet, Lett. Math. Phys. 28 (1993), no. 1, 75â84. MR 1224836, DOI 10.1007/BF00739568
- Craig A. Tracy and Harold Widom, On the distributions of the lengths of the longest monotone subsequences in random words, Probab. Theory Related Fields 119 (2001), no. 3, 350â380. MR 1821139, DOI 10.1007/PL00008763
- L. N. Trefethen and L. M. Trefethen, How many shuffles to randomize a deck of cards?, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), no. 2002, 2561â2568. MR 1796496, DOI 10.1098/rspa.2000.0625
- H. Turner, J. van Etten, D. Firth, and I. Kosmidis, Introduction to PlackettLuce (a Cran Package), 2018.
- J. V. Uspensky, Introduction to Mathematical Probability, McGraw Hill, NY, 1965.
- Jay-Calvin Uyemura Reyes, Random walk, semi-direct products, and card shuffling, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)âStanford University. MR 2703300
- Paul Valiant, Testing symmetric properties of distributions, SIAM J. Comput. 40 (2011), no. 6, 1927â1968. MR 2863200, DOI 10.1137/080734066
- Erik A. van Doorn, Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes, Adv. in Appl. Probab. 23 (1991), no. 4, 683â700. MR 1133722, DOI 10.2307/1427670
- Erik A. van Doorn and Philip K. Pollett, Quasi-stationary distributions for discrete-state models, European J. Oper. Res. 230 (2013), no. 1, 1â14. MR 3063313, DOI 10.1016/j.ejor.2013.01.032
- J. H. van Lint and R. M. Wilson, A course in combinatorics, 2nd ed., Cambridge University Press, Cambridge, 2001. MR 1871828, DOI 10.1017/CBO9780511987045
- Stephanie van Willigenburg, The shuffle conjecture, Bull. Amer. Math. Soc. (N.S.) 57 (2020), no. 1, 77â89. MR 4037408, DOI 10.1090/bull/1672
- Anke van Zuylen and Frans Schalekamp, The Achillesâ heel of the GSR shuffle. A note on new age solitaire, Probab. Engrg. Inform. Sci. 18 (2004), no. 3, 315â328. MR 2082470, DOI 10.1017/S0269964804183034
- A. N. Varchenko and I. M. GelâČfand, Heaviside functions of a configuration of hyperplanes, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 1â18, 96 (Russian). MR 925069
- Michelle L. Wachs, On $q$-derangement numbers, Proc. Amer. Math. Soc. 106 (1989), no. 1, 273â278. MR 937015, DOI 10.1090/S0002-9939-1989-0937015-6
- F. Wallace and M. Hamilton, Neocheating: the rising menace, Neo-tech Publishing, 1980.
- Nolan R. Wallach, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 123â151. MR 1039836, DOI 10.2969/aspm/01410123
- Di Warren and E. Seneta, Peaks and Eulerian numbers in a random sequence, J. Appl. Probab. 33 (1996), no. 1, 101â114. MR 1371957, DOI 10.2307/3215267
- Washington Post, Cracking the two biggest cheating scandals in the history of online poker, https://www.washingtonpost.com/wp-srv/investigations/poker/, 2008.
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
- A. Weiss and T. D. Rogers, The number of orientation reversing cycles in the quadratic map, Oscillations, bifurcation and chaos (Toronto, Ont., 1986) CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 1987, pp. 703â711. MR 909946
- Graham Robert White, Combinatorial Methods in Markov Chain Mixing, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)âStanford University. MR 4257207
- C. O. Williams, A card reading, The Magician Monthly 8 (1912), 67.
- P. Willmarth, The magic of Matt Schulien, Chicagoâs biggest magical entertainer, Ireland Magic Company, 1959.
- David Bruce Wilson, Mixing times of Lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab. 14 (2004), no. 1, 274â325. MR 2023023, DOI 10.1214/aoap/1075828054
- Sarah J. Witherspoon, Hochschild cohomology for algebras, Graduate Studies in Mathematics, vol. 204, American Mathematical Society, Providence, RI, [2019] ©2019. MR 3971234, DOI 10.1090/gsm/204
- Stephen Wolfram, Geometry of binomial coefficients, Amer. Math. Monthly 91 (1984), no. 9, 566â571. MR 764797, DOI 10.2307/2323743
- Makoto Yamazato, Unimodality of infinitely divisible distribution functions of class $L$, Ann. Probab. 6 (1978), no. 4, 523â531. MR 482941
- Don Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, BirkhĂ€user, Basel, 1994, pp. 497â512. MR 1341859