%I M2051 N0812 #127 Mar 02 2024 21:08:32

%S 1,2,12,576,161280,812851200,61479419904000,108776032459082956800,

%T 5524751496156892842531225600,9982437658213039871725064756920320000,

%U 776966836171770144107444346734230682311065600000

%N Number of Latin squares of order n; or labeled quasigroups.

%C Also the number of minimum vertex colorings in the n X n rook graph. - _Eric W. Weisstein_, Mar 02 2024

%D David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.

%D Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.

%D H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ronald Alter, <a href="http://www.jstor.org/stable/2319697">Research Problems: How Many Latin Squares are There?</a>, Amer. Math. Monthly 82 (1975), no. 6, 632-634. MR1537769.

%H S. E. Bammel and J. Rothstein, <a href="http://dx.doi.org/10.1016/0012-365X(75)90108-9">The number of 9 X 9 Latin squares</a>, Discrete Math., 11 (1975), 93-95.

%H D. Berend, <a href="https://doi.org/10.1016/j.disc.2018.08.005">On the number of Sudoku squares</a>, Discrete Mathematics 341.11 (2018): 3241-3248. See p. 3241.

%H Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, <a href="http://ccom.uprrp.edu/~labemmy/Wordpress/wp-content/uploads/2010/11/4_Presentation_Some-Properties-of-Latin-Squares_March2009.pdf">Some Properties of Latin Squares</a>, Laboratorio Emmy Noether, 2009.

%H Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, <a href="https://arxiv.org/abs/2201.00645">Sequences of the Stable Matching Problem</a>, arXiv:2201.00645 [math.HO], 2021.

%H J. W. Brown, <a href="http://dx.doi.org/10.1016/S0021-9800(68)80053-5">Enumeration of Latin squares with application to order 8</a>, J. Combin. Theory, 5 (1968), 177-184.

%H Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, <a href="https://arxiv.org/abs/2308.07455">Quad Squares</a>, arXiv:2308.07455 [math.HO], 2023.

%H Hai-Dang Dau and Nicolas Chopin, <a href="https://arxiv.org/abs/2011.02328">Waste-free Sequential Monte Carlo</a>, arXiv:2011.02328 [stat.CO], 2020.

%H Abdelrahman Desoky, Hany Ammar, Gamal Fahmy, Shaker El-Sappagh, Abdeltawab Hendawi, and Sameh H. Basha, <a href="https://doi.org/10.1007/978-3-031-43247-7_39">Latin Square and Artificial Intelligence Cryptography for Blockchain and Centralized Systems</a>, Int'l Conf. Adv. Intel. Sys. Informat., Proc. 9th Int'l Conf. (AISI 2023) pp. 444-455.

%H Thangavelu Geetha, Amritanshu Prasad, and Shraddha Srivastava, <a href="https://arxiv.org/abs/1902.02465">Schur Algebras for the Alternating Group and Koszul Duality</a>, arXiv:1902.02465 [math.RT], 2019.

%H E. N. Gilbert, <a href="http://www.jstor.org/stable/2027267">Latin squares which contain no repeated digrams</a>, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence. - _N. J. A. Sloane_, Mar 15 2014

%H Yue Guan, Minjia Shi and Denis S. Krotov, <a href="https://arxiv.org/abs/1905.09081">The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)</a>, arXiv:1905.09081 [math.CO], 2019.

%H Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, <a href="https://arxiv.org/abs/2109.01530">Fun with Latin Squares</a>, arXiv:2109.01530 [math.HO], 2021.

%H Yang-Hui He and Minhyong Kim, <a href="https://arxiv.org/abs/1905.02263">Learning Algebraic Structures: Preliminary Investigations</a>, arXiv:1905.02263 [cs.LG], 2019.

%H A.-A. A. Jucys, <a href="http://dx.doi.org/10.1016/0097-3165(76)90020-0">The number of distinct Latin squares as a group-theoretical constant</a>, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 265-272. MR0419259 (54 #7283).

%H Dieter Jungnickel and Vladimir D. Tonchev, <a href="https://arxiv.org/abs/1709.06044">Counting Steiner triple systems with classical parameters and prescribed rank</a>, arXiv:1709.06044 [math.CO], 2017.

%H Lintao Liu, Xuehu Yan, Yuliang Lu, and Huaixi Wang, <a href="https://eprint.iacr.org/2019/584">2-threshold Ideal Secret Sharing Schemes Can Be Uniquely Modeled by Latin Squares</a>, National University of Defense Technology, Hefei, China, (2019).

%H B. D. McKay, A. Meynert and W. Myrvold, <a href="http://users.cecs.anu.edu.au/~bdm/papers/ls_final.pdf">Small Latin Squares, Quasigroups and Loops</a>, J. Combin. Des. 15 (2007), no. 2, 98-119.

%H B. D. McKay and E. Rogoyski, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1n3">Latin squares of order ten</a>, Electron. J. Combinatorics, 2 (1995) #N3.

%H B. D. McKay and I. M. Wanless, <a href="http://users.cecs.anu.edu.au/~bdm/papers/ls11.pdf">On the number of Latin squares</a>. Preprint 2004.

%H B. D. McKay and I. M. Wanless, <a href="http://dx.doi.org/10.1007/s00026-005-0261-7">On the number of Latin squares</a>, Ann. Combinat. 9 (2005) 335-344.

%H J. Shao and W. Wei, <a href="http://dx.doi.org/10.1016/0012-365X(92)90722-R">A formula for the number of Latin squares.</a>, Discrete Mathematics 110 (1992) 293-296.

%H Minjia Shi, Li Xu, and Denis S. Krotov, <a href="https://arxiv.org/abs/1806.00009">The number of the non-full-rank Steiner triple systems</a>, arXiv:1806.00009 [math.CO], 2018.

%H D. S. Stones, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1a1">The many formulas for the number of Latin rectangles</a>, Electron. J. Combin 17 (2010), A1.

%H D. S. Stones and I. M. Wanless, <a href="http://dx.doi.org/10.1016/j.jcta.2009.03.019">Divisors of the number of Latin rectangles</a>, J. Combin. Theory Ser. A 117 (2010), 204-215.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LatinSquare.html">Latin Square</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumVertexColoring.html">Minimum Vertex Coloring</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.

%H M. B. Wells, <a href="http://dx.doi.org/10.1016/0097-3165(90)90015-O">The number of Latin squares of order 8</a>, J. Combin. Theory, 3 (1967), 98-99.

%H Krasimir Yordzhev, <a href="https://arxiv.org/abs/1605.07171">The bitwise operations in relation to obtaining Latin squares</a>, arXiv preprint arXiv:1605.07171 [cs.OH], 2016.

%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>

%H <a href="/index/Qua#quasigroups">Index entries for sequences related to quasigroups</a>

%F a(n) = n!*A000479(n) = n!*(n-1)!*A000315(n).

%t Table[Length[ResourceFunction["FindProperColorings"][GraphProduct[CompleteGraph[n], CompleteGraph[n], "Cartesian"], n]], {n, 5}]

%Y Cf. A000315, A000479.

%Y Cf. A003090, A040082, A057991.

%Y Cf. A098679 (Latin cubes).

%Y A row of the array in A249026.

%K hard,nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E One more term (from the McKay-Wanless article) from _Richard Bean_, Feb 17 2004