

A301371


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


18




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

Table of n, a(n) for n=0..9.
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, DeterminantOEISA3013719, program and description, 4 Dec 2019.
Index entries for sequences related to maximal determinants


FORMULA

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


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 NonLatin 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

Cf. A085000, A309985, A328030, A328031.
Sequence in context: A052182 A309985 A328030 * A115415 A065058 A032031
Adjacent sequences: A301368 A301369 A301370 * A301372 A301373 A301374


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



