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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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17
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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
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LINKS
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Table of n, a(n) for n=0..8.
Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
Index entries for sequences related to maximal determinants
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FORMULA
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A328030(n) <= a(n) <= A328031(n). - Hugo Pfoertner, Nov 04 2019
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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
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CROSSREFS
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Cf. A085000, A309985, A328030, A328031.
Sequence in context: A052182 A309985 A328030 * A115415 A065058 A032031
Adjacent sequences: A301368 A301369 A301370 * A301372 A301373 A301374
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KEYWORD
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nonn,hard,more
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AUTHOR
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Hugo Pfoertner, Mar 21 2018
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EXTENSIONS
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a(8) from Hugo Pfoertner, Aug 26 2019
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STATUS
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approved
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