A002860 Number of Latin squares of order n; or labeled quasigroups.

1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800, 5524751496156892842531225600, 9982437658213039871725064756920320000, 776966836171770144107444346734230682311065600000

Offset: 1

Author

N. J. A. Sloane

Keywords

• hard
• nonn
• nice

Comments

Also the number of minimum vertex colorings in the n X n rook graph. - Eric W. Weisstein, Mar 02 2024

References

• 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.
• Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
• H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ronald Alter, Research Problems: How Many Latin Squares are There?, Amer. Math. Monthly 82 (1975), no. 6, 632-634. MR1537769.
• Stanley E. Bammel and Jerome Rothstein, The number of 9 X 9 Latin squares, Discrete Math., 11 (1975), 93-95.
• Daniel Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248. See p. 3241.
• Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.
• Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.
• John W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), 177-184.
• 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, Quad Squares, arXiv:2308.07455 [math.HO], 2023.
• Gloria S. Choi, Beyond Sudoku: New puzzle variants based on a novel mathematical mapping, J. High Sch. Sci. (2025) Vol. 9, No. 2, 70-84.
• Hai-Dang Dau and Nicolas Chopin, Waste-free Sequential Monte Carlo, arXiv:2011.02328 [stat.CO], 2020.
• Abdelrahman Desoky, Hany Ammar, Gamal Fahmy, Shaker El-Sappagh, Abdeltawab Hendawi, and Sameh H. Basha, Latin Square and Artificial Intelligence Cryptography for Blockchain and Centralized Systems, Int'l Conf. Adv. Intel. Sys. Informat., Proc. 9th Int'l Conf. (AISI 2023) pp. 444-455.
• Thangavelu Geetha, Amritanshu Prasad, and Shraddha Srivastava, Schur Algebras for the Alternating Group and Koszul Duality, arXiv:1902.02465 [math.RT], 2019.
• E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189-198. MR0179095 (31 #3346). Mentions this sequence. - _N. J. A. Sloane_, Mar 15 2014
• Yue Guan, Minjia Shi and Denis S. Krotov, The Steiner triple systems of order 21 with a transversal subdesign TD(3,6), arXiv:1905.09081 [math.CO], 2019.
• Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
• Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
• A.-A. A. Jucys, The number of distinct Latin squares as a group-theoretical constant, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 265-272. MR0419259 (54 #7283).
• Dieter Jungnickel and Vladimir D. Tonchev, Counting Steiner triple systems with classical parameters and prescribed rank, arXiv:1709.06044 [math.CO], 2017.
• Lintao Liu, Xuehu Yan, Yuliang Lu, and Huaixi Wang, 2-threshold Ideal Secret Sharing Schemes Can Be Uniquely Modeled by Latin Squares, National University of Defense Technology, Hefei, China, (2019).
• Brendan D. McKay, Alison Meynert and Wendy Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Des. 15 (2007), no. 2, 98-119.
• Brendan D. McKay and Eric Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
• Brendan D. McKay and Ian M. Wanless, On the number of Latin squares. Preprint 2004.
• Brendan D. McKay and Ian M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
• Jia-yu Shao and Wan-di Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.
• Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018.
• Douglas S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
• Douglas S. Stones and Ian M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
• Eric Weisstein's World of Mathematics, Latin Square.
• Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
• Eric Weisstein's World of Mathematics, Rook Graph.
• Mark B. Wells, The number of Latin squares of order 8, J. Combin. Theory, 3 (1967), 98-99.
• Krasimir Yordzhev, The bitwise operations in relation to obtaining Latin squares, arXiv preprint arXiv:1605.07171 [cs.OH], 2016.
• Index entries for sequences related to Latin squares and rectangles
• Index entries for sequences related to quasigroups

Crossrefs

Cf. A000315, A000479.

Cf. A003090, A040082, A057991.

Cf. A098679 (Latin cubes).

A row of the array in A249026.

Programs

Mathematica

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

Formula

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

Extensions

One more term (from the McKay-Wanless article) from Richard Bean, Feb 17 2004