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A000315
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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)
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22
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1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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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
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.
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REFERENCES
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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.
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).
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 240.
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LINKS
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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.
Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,hard,nice,more
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AUTHOR
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EXTENSIONS
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Added June 1995: the 10th term was probably first computed by Eric Rogoyski
a(11) (from the McKay-Wanless article) from Richard Bean, Feb 17 2004
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STATUS
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approved
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