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A301371 Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n. 17

%I

%S 1,1,3,18,160,2325,41895,961772,27296640

%N Maximum determinant of an n X n matrix with n copies of the numbers 1 .. n.

%C 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

%H Ortwin Gasper, Hugo Pfoertner and Markus Sigg, <a href="http://www.emis.de/journals/JIPAM/article1119.html">An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum</a>, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.

%H Markus Sigg, <a href="https://arxiv.org/abs/1804.02897">Gasper's determinant theorem, revisited</a>, arXiv:1804.02897 [math.CO], 2018.

%H <a href="/index/De#determinants">Index entries for sequences related to maximal determinants</a>

%F A328030(n) <= a(n) <= A328031(n). - _Hugo Pfoertner_, Nov 04 2019

%e Matrices with maximum determinants:

%e a(2) = 3:

%e (2 1)

%e (1 2)

%e a(3) = 18:

%e (3 1 2)

%e (2 3 1)

%e (1 2 3)

%e a(4) = 160:

%e (4 3 2 1)

%e (1 4 3 2)

%e (3 1 4 3)

%e (2 2 1 4)

%e a(5) = 2325:

%e (5 3 1 2 4)

%e (2 5 4 1 3)

%e (4 1 5 3 2)

%e (3 4 2 5 1)

%e (1 2 3 4 5)

%e a(6) = 41895:

%e (6 1 4 2 3 5)

%e (3 6 2 1 5 4)

%e (4 5 6 3 2 1)

%e (5 3 1 6 4 2)

%e (1 2 5 4 6 3)

%e (2 4 3 5 1 6)

%e a(7) = 961772:

%e (7 2 3 5 1 4 6)

%e (3 7 6 4 2 1 5)

%e (2 1 7 6 4 5 3)

%e (4 5 1 7 6 3 2)

%e (6 3 5 1 7 2 4)

%e (5 6 4 2 3 7 1)

%e (1 4 2 3 5 6 7)

%e a(8) = 27296640:

%e (8 8 3 5 4 3 4 1)

%e (1 8 6 3 1 6 6 5)

%e (5 3 8 1 7 6 4 2)

%e (5 1 6 8 2 4 7 3)

%e (1 5 2 7 8 6 4 3)

%e (7 3 2 4 3 8 2 7)

%e (5 4 2 2 6 2 8 7)

%e (4 5 7 6 5 1 1 7)

%e 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

%Y Cf. A085000, A309985, A328030, A328031.

%K nonn,hard,more

%O 0,3

%A _Hugo Pfoertner_, Mar 21 2018

%E a(8) from _Hugo Pfoertner_, Aug 26 2019

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Last modified August 14 07:04 EDT 2020. Contains 336477 sequences. (Running on oeis4.)