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A301371
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Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n.
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18
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OFFSET
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0,3
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COMMENTS
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929587995 <= a(9) <= 934173632 (upper bound from Gasper's determinant theorem). The lower bound corresponds to a Latin square provided in A309985, but it is unknown whether a larger determinant value can be achieved by an unconstrained arrangement of the matrix entries. - Hugo Pfoertner, Aug 27 2019
Oleg Vlasii found a 9 X 9 matrix significantly exceeding the determinant value achievable by a Latin square. See example and links. - Hugo Pfoertner, Nov 04 2020
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LINKS
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FORMULA
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EXAMPLE
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Matrices with maximum determinants:
a(2) = 3:
(2 1)
(1 2)
a(3) = 18:
(3 1 2)
(2 3 1)
(1 2 3)
a(4) = 160:
(4 3 2 1)
(1 4 3 2)
(3 1 4 3)
(2 2 1 4)
a(5) = 2325:
(5 3 1 2 4)
(2 5 4 1 3)
(4 1 5 3 2)
(3 4 2 5 1)
(1 2 3 4 5)
a(6) = 41895:
(6 1 4 2 3 5)
(3 6 2 1 5 4)
(4 5 6 3 2 1)
(5 3 1 6 4 2)
(1 2 5 4 6 3)
(2 4 3 5 1 6)
a(7) = 961772:
(7 2 3 5 1 4 6)
(3 7 6 4 2 1 5)
(2 1 7 6 4 5 3)
(4 5 1 7 6 3 2)
(6 3 5 1 7 2 4)
(5 6 4 2 3 7 1)
(1 4 2 3 5 6 7)
a(8) = 27296640:
(8 8 3 5 4 3 4 1)
(1 8 6 3 1 6 6 5)
(5 3 8 1 7 6 4 2)
(5 1 6 8 2 4 7 3)
(1 5 2 7 8 6 4 3)
(7 3 2 4 3 8 2 7)
(5 4 2 2 6 2 8 7)
(4 5 7 6 5 1 1 7)
a(n) is an upper bound for the determinant of an n X n Latin square. a(n) = A309985(n) for n <= 7. a(8) > A309985(8). - Hugo Pfoertner, Aug 26 2019
a(9) = 933251220, achieved by a Non-Latin square:
(9 5 5 3 3 2 2 8 8)
(4 9 2 6 7 5 3 1 8)
(4 7 9 2 1 8 6 3 5)
(6 3 7 9 4 1 8 2 5)
(6 2 8 5 9 7 1 4 3)
(7 4 1 8 2 9 5 6 3)
(7 6 3 1 8 4 9 5 2)
(1 8 6 7 5 3 4 9 2)
(1 1 4 4 6 6 7 7 9)
found by Oleg Vlasii as an answer to the IBM Ponder This Challenge November 2019. See links. (End)
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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