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

1, 1, 3, 18, 160, 2325, 41895, 961772, 27296640, 933251220

Offset: 0

Author

Hugo Pfoertner, Mar 21 2018

Keywords

• nonn
• hard
• more

Comments

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

Examples

			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
From _Hugo Pfoertner_, Nov 04 2020: (Start)
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)
		

Links

• 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.
• IBM Research, Large 9x9 determinant, Ponder This Challenge November 2019.
• Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
• Oleg Vlasii, Determinant-OEIS-A301371-9, program and description, 4 Dec 2019.
• Index entries for sequences related to maximal determinants

Crossrefs

Cf. A085000, A309985, A328030, A328031.

Formula

A328030(n) <= a(n) <= A328031(n). - Hugo Pfoertner, Nov 04 2019

Extensions

a(8) from Hugo Pfoertner, Aug 26 2019

a(9) from Hugo Pfoertner, Nov 04 2020