A309985 Maximum determinant of an n X n Latin square.

1, 1, 3, 18, 160, 2325, 41895, 961772, 26978400, 929587995

Offset: 0

Author

Hugo Pfoertner, Aug 26 2019

Keywords

• nonn
• hard
• more

Comments

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

Examples

			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_)
		

Links

• Brendan McKay, Latin squares.
• Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).

Crossrefs

Cf. A040082, A301371, A308853, A309258, A309984, A328029, A328030.

Extensions

a(9) from Hugo Pfoertner, Aug 30 2019

a(0)=1 prepended by Alois P. Heinz, Oct 02 2019