%I M3690 N1508 #97 Aug 21 2023 12:44:39
%S 1,1,1,4,56,9408,16942080,535281401856,377597570964258816,
%T 7580721483160132811489280,5363937773277371298119673540771840
%N Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.
%C A reduced Latin square of order n is an n X n matrix where each row and column is a permutation of 1..n and the first row and column are 1..n in increasing order. - _Michael Somos_, Mar 12 2011
%C The Stones-Wanless (2010) paper shows among other things that a(n) is 0 mod n if n is composite and 1 mod n if n is prime.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.
%D J. Denes, A. D. Keedwell, editors, Latin Squares: new developments in the theory and applications, Elsevier, 1991, pp. 1, 388.
%D R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.
%D C. R. Rao, S. K. Mitra and A. Matthai, editors, Formulae and Tables for Statistical Work. Statistical Publishing Society, Calcutta, India, 1966, p. 193.
%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, pp. 37, 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).
%D M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 240.
%H S. E. Bammel and J. Rothstein, <a href="http://dx.doi.org/10.1016/0012-365X(75)90108-9">The number of 9x9 Latin squares</a>, Discrete Math., 11 (1975), 93-95.
%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 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 B. Cherowitzo, <a href="http://math.ucdenver.edu/~wcherowi/courses/m6406/csln1.html#1">Latin Squares</a>, Comb. Structures Lecture Notes.
%H Gheorghe Coserea, <a href="/A000315/a000315.txt">Solutions for n=5</a>.
%H Gheorghe Coserea, <a href="/A000315/a000315_1.txt">Solutions for n=6</a>.
%H Gheorghe Coserea, <a href="/A000315/a000315.mzn.txt">MiniZinc model for generating solutions</a>.
%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 Brian Hopkins, <a href="http://ecajournal.haifa.ac.il/Volume2021/ECA2021_S1H1.pdf">Euler's Enumerations</a>, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.
%H B. D. McKay, A. Meynert and W. Myrvold, <a href="http://dx.doi.org/10.1002/jcd.20105">Small latin squares, quasigroups and loops</a>, J. Combin. Designs, vol. 15, no. 2 (2007) pp. 98-119.
%H B. D. McKay and E. Rogoyski, <a href="https://doi.org/10.37236/1222">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 Young-Sik Moon, Jong-Yoon Yoon, Jong-Seon No, and Sang-Hyo Kim, <a href="https://arxiv.org/abs/1810.05400">Interference Alignment Schemes Using Latin Square for Kx3 MIMO X Channel</a>, arXiv:1810.05400 [cs.IT], 2018.
%H Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, <a href="https://arxiv.org/abs/2103.11018">Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares</a>, arXiv:2103.11018 [cs.DM], 2021. Mentions this sequence.
%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 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%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 E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_9.pdf">Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares</a>, Information technologies and mathematical modeling of systems (2016), pp. 154-157. (in Russian)
%H E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, <a href="http://ceur-ws.org/Vol-1787/486-490-paper-84.pdf">Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares</a>. CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatinSquare.html">Latin Square</a>.
%H M. B. Wells, <a href="/A000170/a000170.pdf">Elements of Combinatorial Computing</a>, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
%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) = A002860(n) / (n! * (n-1)!) = A000479(n) / (n-1)!.
%Y Cf. A000479, A002860, A003090, A040082, A057771, A057997.
%K nonn,hard,nice,more
%O 1,4
%A _N. J. A. Sloane_
%E Added June 1995: the 10th term was probably first computed by Eric Rogoyski
%E a(11) (from the McKay-Wanless article) from _Richard Bean_, Feb 17 2004