A000315 - OEIS

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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)

1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	COMMENTS	`A reduced Latin square of order n is a 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` `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.`
	REFERENCES	`S. E. Bammel and J. Rothstein, The number of 9x9 Latin squares, Discrete Math., 11 (1975), 93-95.` `Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, SOME PROPERTIES OF LATIN SQUARES, http://ccom.uprrp.edu/~labemmy/Wordpress/wp-content/uploads/2010/11/4_Presentation_Some-Properties-of-Latin-Squares_March2009.pdf` `L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.` `J. Denes, A. D. Keedwell, editors, Latin Squares: new developments in the theory and applications, Elsevier, 1991, pp. 1, 388.` `R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.` `B. D. McKay and I. M. Wanless, Latin squares of order eleven. Preprint 2004. http://cs.anu.edu.au/~bdm/papers/ls11.pdf` `B. D. McKay and I. M. Wanless, On the number of Latin squares. Preprint 2004. http://cs.anu.edu.au/~bdm/papers/ls11.pdf` `C. R. Rao, S. K. Mitra and A. Matthai, editors, Formulae and Tables for Statistical Work. Statistical Publishing Society, Calcutta, India, 1966, p. 193.` `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, pp. 37, 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).` `D. S. Stones, The many formulae for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.` `D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.` `M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 240.`
	LINKS	`Table of n, a(n) for n=1..11.` `B. Cherowitzo, Comb. Structures Lecture Notes` `B. D. McKay, A. Meynert and W. Myrvold, Small latin squares, quasigroups and loops, J. Combin. Designs, vol. 15, no. 2 (2007) pp 98-119.` `B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.` `B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.` `N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).` `Eric Weisstein's World of Mathematics, Latin Square.` `Index entries for sequences related to Latin squares and rectangles` `Index entries for sequences related to quasigroups`
	FORMULA	`a(n) = A002860(n) / (n! * (n-1)!) = A000479(n) / (n-1)!.`
	CROSSREFS	`Cf. A000479, A002860, A003090, A040082, A057771, A057997.` `Sequence in context: A000573 A070019 A056075 * A080984 A071579 A060497` `Adjacent sequences: A000312 A000313 A000314 * A000316 A000317 A000318`
	KEYWORD	`nonn,hard,nice,more`
	AUTHOR	`N. J. A. Sloane.`
	EXTENSIONS	`Added June 1995: the 10th term was probably first computed by Eric Rogoyski` `a(11) (from the McKay-Wanless article) from Richard Bean (rwb(AT)eskimo.com), Feb 17 2004`
	STATUS	`approved`

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Last modified May 22 23:20 EDT 2013. Contains 225585 sequences.