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A309985
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Maximum determinant of an n X n Latin square.
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8
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OFFSET
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0,3
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COMMENTS
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a(n) = A301371(n) for n <= 7. a(8) < A301371(8) = 27296640, a(9) < A301371(9) = 933251220.
a(10) = 36843728625, conjectured. See Stack Exchange link. - Hugo Pfoertner, Sep 29 2019
A328030(n) <= a(n) <= A301371(n). - Hugo Pfoertner, Dec 02 2019
It is unknown, but very likely, that A301371(n) > a(n) also holds for all n > 9 - Hugo Pfoertner, Dec 12 2020
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LINKS
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Table of n, a(n) for n=0..9.
Brendan McKay, Latin squares.
Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).
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EXAMPLE
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An example of an 8 X 8 Latin square with maximum determinant is
[7 1 3 4 8 2 5 6]
[1 7 4 3 6 5 2 8]
[3 4 1 7 2 6 8 5]
[4 3 7 1 5 8 6 2]
[8 6 2 5 4 7 1 3]
[2 5 6 8 7 3 4 1]
[5 2 8 6 1 4 3 7]
[6 8 5 2 3 1 7 4].
An example of a 9 X 9 Latin square with maximum determinant is
[9 4 3 8 1 5 2 6 7]
[3 9 8 5 4 6 1 7 2]
[4 1 9 3 2 8 7 5 6]
[1 2 4 9 7 3 6 8 5]
[8 3 5 6 9 7 4 2 1]
[2 7 1 4 6 9 5 3 8]
[5 8 6 7 3 2 9 1 4]
[7 6 2 1 5 4 8 9 3]
[6 5 7 2 8 1 3 4 9].
An example of a 10 X 10 Latin square with abs(determinant) = 36843728625 is a circulant matrix with first row [1, 3, 7, 9, 8, 6, 5, 4, 2, 10], but it is not known if this is the best possible. - Kebbaj Mohamed Reda, Nov 27 2019 (reworded by Hugo Pfoertner)
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CROSSREFS
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Cf. A040082, A301371, A308853, A309258, A309984, A328029, A328030.
Sequence in context: A067302 A212599 A052182 * A328030 A301371 A115415
Adjacent sequences: A309982 A309983 A309984 * A309986 A309987 A309988
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KEYWORD
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nonn,hard,more
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AUTHOR
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Hugo Pfoertner, Aug 26 2019
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EXTENSIONS
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a(9) from Hugo Pfoertner, Aug 30 2019
a(0)=1 prepended by Alois P. Heinz, Oct 02 2019
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STATUS
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approved
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