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A000315 Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.
(Formerly M3690 N1508)

%I M3690 N1508

%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 B. Cherowitzo, <a href="http://www-math.cudenver.edu/~wcherowi/courses/m6406/csln1.html#1">Comb. Structures Lecture Notes</a>

%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 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="http://www.combinatorics.org/Volume_2/volume2.html#N3">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 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/v17i1a1/0">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, 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 Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., Titov V.S., <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 (rwb(AT)eskimo.com), Feb 17 2004

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Last modified October 23 00:32 EDT 2017. Contains 293782 sequences.