@article{DiaconisShahshahani1981,
 author={Diaconis, Persi and Shahshahani, Mehrdad},
 title={Generating a random permutation with random transpositions},
 journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und verwandte Gebiete},
 volume={57}, pages={159--179}, year={1981}, doi={10.1007/BF00535487}}

@article{Thorp1973, author={Thorp, Edward O.},title={Nonrandom shuffling with applications to the game of {Faro}},journal={Journal of the American Statistical Association},volume={68},number={344},pages={842--847},year={1973},doi={10.1080/01621459.1973.10481434},url={https://www.edwardothorp.com/wp-content/uploads/2016/11/NonrandomShufflingWithApplicationsToTheGameOfFaro.pdf},note={The independent-pair shuffle is defined in Section 3.1, page 844}}
@article{Morris2009,author={Morris, Ben},title={Improved mixing time bounds for the {Thorp} shuffle and {$L$}-reversal chain},journal={Annals of Probability},volume={37},number={2},pages={453--477},year={2009},doi={10.1214/08-AOP409},url={https://arxiv.org/abs/0802.0339v1},note={Author version: arXiv:0802.0339v1, 4 February 2008; Sections 1.1 and 5, pages 2 and 10--12}}
@article{Morris2013,author={Morris, Ben},title={Improved mixing time bounds for the {Thorp} shuffle},journal={Combinatorics, Probability and Computing},volume={22},number={1},pages={118--132},year={2013},doi={10.1017/S0963548312000478},url={https://arxiv.org/abs/0912.2759v1},note={Author version: arXiv:0912.2759v1, 14 December 2009; Section 4, Lemma 7 and Theorem 8, pages 3--4}}
@book{Sagan2001,author={Sagan, Bruce E.},title={The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions},edition={2nd},series={Graduate Texts in Mathematics},volume={203},publisher={Springer},year={2001},doi={10.1007/978-1-4757-6804-6}}
@book{Stanley1999,author={Stanley, Richard P.},title={Enumerative Combinatorics, Volume 2},publisher={Cambridge University Press},year={1999}}
@article{BorceaBrandenLiggett2009,author={Borcea, Julius and Br{\"a}nd{\'e}n, Petter and Liggett, Thomas M.},title={Negative dependence and the geometry of polynomials},journal={Journal of the American Mathematical Society},volume={22},pages={521--567},year={2009},doi={10.1090/S0894-0347-08-00618-8},url={https://arxiv.org/abs/0707.2340v2},note={Theorems 4.9 and 4.20 refer to arXiv:0707.2340v2, 27 July 2008, pages 24 and 29--30}}
@article{VershikOkounkov2005,author={Vershik, Anatoly M. and Okounkov, Andrei Yu.},title={A new approach to the representation theory of the symmetric groups. {II}},year={2005},journal={Journal of Mathematical Sciences},volume={131},pages={5471--5494},doi={10.1007/s10958-005-0421-7},url={https://arxiv.org/abs/math/0503040v3},note={Equation (2.1), Proposition 5.3 and Theorem 5.8 refer to arXiv:math/0503040v3, 20 April 2005}}
@article{LarsenShalev2008,author={Larsen, Michael and Shalev, Aner},title={Characters of symmetric groups: sharp bounds and applications},journal={Inventiones Mathematicae},volume={174},pages={645--687},year={2008},doi={10.1007/s00222-008-0145-7}}
@article{KellerLifshitzSheinfeld2024,author={Keller, Nathan and Lifshitz, Noam and Sheinfeld, Ohad},title={Improved covering results for conjugacy classes of symmetric groups via hypercontractivity},journal={Forum of Mathematics, Sigma},volume={12},pages={e85},year={2024},doi={10.1017/fms.2024.95}}

@article{Morris2008,
 author={Morris, Ben}, title={The mixing time of the {Thorp} shuffle},
 journal={SIAM Journal on Computing}, volume={38}, number={2},
 pages={484--504}, year={2008}, doi={10.1137/050636231},
 url={https://arxiv.org/abs/math/0507307v1},
 note={Theorem 1 and Section 7 are cited from arXiv:math/0507307v1, 15 July 2005}}
@inproceedings{hmr2014,
 author={Hoang, Viet Tung and Morris, Ben and Rogaway, Phillip},
 title={An enciphering scheme based on a card shuffle},
 booktitle={Advances in Cryptology---CRYPTO 2012},
 series={Lecture Notes in Computer Science}, volume={7417},
 pages={1--13}, publisher={Springer}, year={2012},
 url={https://arxiv.org/abs/1208.1176v2},
 note={Section 3, Theorem 3, and Lemma 4 refer to arXiv:1208.1176v2, 21 November 2014}}
@inproceedings{mrs,
 author={Morris, Ben and Rogaway, Phillip and Stegers, Till},
 title={How to encipher messages on a small domain: deterministic encryption and the {Thorp} shuffle},
 booktitle={Advances in Cryptology---CRYPTO 2009},
 series={Lecture Notes in Computer Science}, volume={5677},
 pages={286--302}, publisher={Springer}, year={2009},
 doi={10.1007/978-3-642-03356-8_17},
 url={https://www.cs.ucdavis.edu/~rogaway/papers/thorp.pdf},
 note={Theorem 1 and Appendix A, Lemma 2, refer to the author's conference text}}
@article{MorrisRogawayStegers2018,
 author={Morris, Ben and Rogaway, Phillip and Stegers, Till},
 title={Deterministic encryption with the {Thorp} shuffle},
 journal={Journal of Cryptology}, volume={31}, number={2},
 pages={521--536}, year={2018}, doi={10.1007/s00145-017-9262-z}}
@article{Benes1964,
 author={Bene{\v s}, V. E.},
 title={Optimal rearrangeable multistage connecting networks},
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 pages={1641--1656}, year={1964},
 doi={10.1002/j.1538-7305.1964.tb04103.x}}
@article{MontenegroTetali2006,
 author={Montenegro, Ravi and Tetali, Prasad},
 title={Mathematical aspects of mixing times in {Markov} chains},
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 volume={1}, number={3}, pages={237--354}, year={2006},
 doi={10.1561/0400000003},
 url={https://faculty.uml.edu/rmontenegro/research/TCS008-journal.pdf},
 note={Section 6.4, Theorems 6.14 and 6.15, journal pages 332 and 334}}
@article{liebeck-shalev2004,
 author={Liebeck, Martin W. and Shalev, Aner},
 title={Fuchsian groups, coverings of {Riemann} surfaces, subgroup growth, random quotients and random walks},
 journal={Journal of Algebra}, volume={276}, number={2},
 pages={552--601}, year={2004}, url={https://www.ma.ic.ac.uk/~mwl/fuchs.pdf},
 note={Proposition 2.5 and Theorem 2.6 are on pages 15--16 of the linked author's text}}
@article{cll2006,
 author={Carlen, Eric A. and Lieb, Elliott H. and Loss, Michael},
 title={An inequality of {Hadamard} type for permanents},
 journal={Methods and Applications of Analysis}, volume={13}, number={1},
 pages={1--18}, year={2006}, url={https://arxiv.org/abs/math/0508096v1},
 note={Theorem 1.1 and equations (1.2)--(1.4) refer to arXiv:math/0508096v1, 4 August 2005}}
@article{araki1990,
 author={Araki, Huzihiro}, title={On an inequality of {Lieb} and {Thirring}},
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 pages={167--170}, year={1990}, doi={10.1007/BF01045887}}
@article{audenaert2007,
 author={Audenaert, Koenraad M. R.},
 title={On the {Araki--Lieb--Thirring} inequality},
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 eprint={math/0701129v2}, archivePrefix={arXiv},
 url={https://arxiv.org/abs/math/0701129v2},
 note={Theorem 1 refers to arXiv:math/0701129v2, 29 March 2007}}
@article{garcia-cuerva-parcet2004,
 author={Garc{\'i}a-Cuerva, Jos{\'e} and Parcet, Javier},
 title={Vector-valued {Hausdorff--Young} inequality on compact groups},
 journal={Proceedings of the London Mathematical Society},
 volume={88}, number={3}, pages={796--816}, year={2004},
 doi={10.1112/S0024611503014527},
 url={https://arxiv.org/abs/math/0312241v2},
 note={Section 2 and Lemma 5.1(2) refer to arXiv:math/0312241v2, 11 December 2003}}
@misc{etingof,
 author={Etingof, Pavel and Golberg, Oleg and Hensel, Sebastian and Liu, Tiankai and Schwendner, Alex and Vaintrob, Dmitry and Yudovina, Elena},
 title={Introduction to representation theory},
 year={2011}, url={https://math.mit.edu/~etingof/replect.pdf},
 note={Author's lecture text dated 10 January 2011; numbering and pages refer to this version}}
@misc{orellana-zabrocki2000,
 author={Orellana, Rosa C. and Zabrocki, Mike},
 title={Some remarks on the characters of the general {Lie} superalgebra},
 year={2000}, eprint={math/0008152v1}, archivePrefix={arXiv},
 url={https://arxiv.org/abs/math/0008152v1}, note={18 August 2000}}
@misc{mason-niese2016,
 author={Mason, Sarah K. and Niese, Elizabeth},
 title={Quasisymmetric $(k,l)$-hook {Schur} functions},
 year={2016}, eprint={1606.06942v1}, archivePrefix={arXiv},
 url={https://arxiv.org/abs/1606.06942v1}, note={22 June 2016}}
@misc{orellana-zabrocki2019,
 author={Orellana, Rosa and Zabrocki, Mike},
 title={The {Hopf} structure of symmetric group characters as symmetric functions},
 year={2019}, eprint={1901.00378v2}, archivePrefix={arXiv},
 url={https://arxiv.org/abs/1901.00378v2}, note={8 June 2019}}
@misc{teyssier-thevenin2025,
 author={Teyssier, Lucas and Th{\'e}venin, Paul},
 title={Characters of symmetric groups: sharp bounds on virtual degrees and the {Witten} zeta function},
 year={2025}, eprint={2411.04347v3}, archivePrefix={arXiv},
 url={https://arxiv.org/abs/2411.04347v3}, note={24 April 2025}}
@misc{olesker-taylor-teyssier-thevenin2025,
 author={Olesker-Taylor, Sam and Teyssier, Lucas and Th{\'e}venin, Paul},
 title={Sharp character bounds and cutoff for symmetric groups},
 year={2025}, eprint={2503.12735v2}, archivePrefix={arXiv},
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@article{FortuinKasteleynGinibre1971,
 author={Fortuin, C. M. and Kasteleyn, P. W. and Ginibre, J.},
 title={Correlation inequalities on some partially ordered sets},
 journal={Communications in Mathematical Physics}, volume={22}, number={2},
 pages={89--103}, year={1971}, doi={10.1007/BF01651330},
 url={https://math.bme.hu/~balint/oktatas/perkolacio/percolation_papers/fortuin_kasteleyn_ginibre.pdf},
 note={Proposition 1 and equation (2.1), printed page 91; proof on pages 92--93}}

@inproceedings{CzumajVocking2014,
  author = {Czumaj, Artur and V{\"o}cking, Berthold},
  title = {{Thorp} Shuffling, Butterflies, and Non-{Markovian} Couplings},
  booktitle = {Automata, Languages, and Programming},
  series = {Lecture Notes in Computer Science},
  volume = {8572},
  pages = {344--355},
  publisher = {Springer},
  year = {2014},
  doi = {10.1007/978-3-662-43948-7_29}
}

@inproceedings{Czumaj2015,
  author = {Czumaj, Artur},
  title = {Random Permutations using Switching Networks},
  booktitle = {Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing},
  pages = {703--712},
  publisher = {Association for Computing Machinery},
  year = {2015},
  doi = {10.1145/2746539.2746629},
  url = {https://www.dcs.warwick.ac.uk/~czumaj/PUBLICATIONS/CONFERENCES/Czumaj-STOC-2015-703-712.pdf},
  note = {The full-permutation construction is Theorem 1.1', Section 3.2.1, page 711}
}

@inproceedings{GelmanTaShma2014,
  author = {Gelman, Efraim and Ta-Shma, Amnon},
  title = {The {Benes} Network is {$q(q-1)/2n$}-Almost {$q$}-set-wise Independent},
  booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science},
  series = {Leibniz International Proceedings in Informatics},
  volume = {29},
  pages = {327--338},
  publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  year = {2014},
  doi = {10.4230/LIPIcs.FSTTCS.2014.327},
  url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.327}
}

@article{CarlenCorderoErausquin2009,
  author = {Carlen, Eric A. and Cordero-Erausquin, Dario},
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  number = {2},
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  year = {2009},
  doi = {10.1007/s00039-009-0001-y},
  url = {https://webusers.imj-prg.fr/~dario.cordero/Docs/articles/subadd.pdf}
}

@article{Radhakrishnan1997,
  author = {Radhakrishnan, Jaikumar},
  title = {An Entropy Proof of {Bregman's} Theorem},
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}

@article{LinialLuria2014,
  author = {Linial, Nathan and Luria, Zur},
  title = {An upper bound on the number of high-dimensional permutations},
  journal = {Combinatorica},
  volume = {34},
  number = {4},
  pages = {471--486},
  year = {2014},
  doi = {10.1007/s00493-011-2842-8},
  eprint = {1106.0649v2},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1106.0649v2},
  note = {Sections 2.2 and 3.1--3.2 refer to arXiv:1106.0649v2, 12 July 2012}
}

@book{Demailly2012,
  author = {Demailly, Jean-Pierre},
  title = {Complex Analytic and Differential Geometry},
  publisher = {Institut Fourier, Universit{\'e} de Grenoble I},
  year = {2012},
  note = {Version of June 21, 2012},
  url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf}
}

@misc{OpenAIRouting2026,
  author = {{OpenAI}},
  title = {{Routing densities and representation contraction for Thorp sweeps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf}{OAI:Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026}},
  year = {2026}
}

@misc{OpenAIOptimal2026,
  author = {{OpenAI}},
  title = {{Optimal-order mixing of the Thorp shuffle}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf}{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026}},
  year = {2026}
}

@article{Hastad2006,
 author={H{\aa}stad, Johan},
 title={The square lattice shuffle},
 journal={Random Structures \& Algorithms},
 volume={29},
 number={4},
 pages={466--474},
 year={2006},
 doi={10.1002/rsa.20131},
 url={https://johanhastad.se/jhsquare.pdf},
 note={Author version dated October 20, 2005; see also the correction}
}

@article{Hastad2016,
 author={H{\aa}stad, Johan},
 title={The square lattice shuffle, correction},
 journal={Random Structures \& Algorithms},
 volume={48},
 number={1},
 pages={213},
 year={2016},
 doi={10.1002/rsa.20620},
 url={https://johanhastad.se/nntypo.pdf},
 note={Author correction dated June 17, 2015}
}