OFFSET
0,3
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
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.
FORMULA
EXAMPLE
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, 2020 Nov 04: (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)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Hugo Pfoertner, Mar 21 2018
EXTENSIONS
a(8) from Hugo Pfoertner, Aug 26 2019
a(9) from Hugo Pfoertner, Nov 04 2020
STATUS
approved