A328030 -id:A328030 - cp's OEIS frontend

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

Hugo Pfoertner, Mar 21 2018

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

			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)

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

Cf. A085000, A309985, A328030, A328031.

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

a(8) from Hugo Pfoertner, Aug 26 2019

a(9) from Hugo Pfoertner, Nov 04 2020

A309985 Maximum determinant of an n X n Latin square.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Aug 26 2019

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

			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_)

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

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

a(9) from Hugo Pfoertner, Aug 30 2019

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

A328029 Lexicographically earliest permutation of [1,2,...,n] maximizing the determinant of an n X n circulant matrix that uses this permutation as first row, written as triangle T(n,k), k <= n.

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 2, 1, 4, 3, 1, 2, 4, 3, 5, 2, 1, 6, 3, 5, 4, 1, 2, 4, 6, 5, 3, 7, 2, 1, 5, 4, 8, 3, 6, 7, 1, 2, 4, 8, 6, 7, 5, 3, 9, 1, 2, 10, 7, 8, 3, 9, 5, 4, 6, 1, 2, 6, 11, 7, 9, 4, 8, 5, 3, 10, 2, 1, 7, 3, 12, 5, 9, 10, 4, 6, 11, 8, 1, 2, 12, 13, 5, 10, 6, 11, 3, 9, 8, 4, 7

Offset: 1

Hugo Pfoertner, Oct 02 2019

For n <= 9 the corresponding circulant matrices are n X n Latin squares with maximum determinant A309985(n). It is conjectured that this also holds for n > 9. See Mathematics Stack Exchange link.

			The triangle starts
  1;
  2,  1;
  1,  2,  3;
  2,  1,  4,  3;
  1,  2,  4,  3,  5;
  2,  1,  6,  3,  5,  4;
  1,  2,  4,  6,  5,  3,  7;
  2,  1,  5,  4,  8,  3,  6,  7;
  1,  2,  4,  8,  6,  7,  5,  3,  9;
  1,  2, 10,  7,  8,  3,  9,  5,  4,  6;
.
The 4th row of the triangle T(4,1)..T(4,4) = a(7)..a(10) is [2,1,4,3] because this is the lexicographically earliest permutation of [1,2,3,4] producing a circulant 4 X 4 matrix with maximum determinant A328030(4) = 160.
  [2, 1, 4, 3;
   3, 2, 1, 4;
   4, 3, 2, 1;
   1, 4, 3, 2].
All lexicographically earlier permutations lead to smaller determinants, with [1,2,3,4] and [1,4,3,2] producing determinants = -160.

Hugo Pfoertner, Table of n, a(n) for n = 1..120, rows 1..15 of triangle, flattened
Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).
Wikipedia, Circulant matrix.

Cf. A301371, A309985, A328030, A328031, A328062.

f[n_] := (p = Permutations[Table[i, {i, n}]]; L = Length[p]; det = Max[Table[Det[Reverse /@ Partition[p[[i]], n, 1, {1, 1}]], {i, 1, L}]]; mat = Table[Reverse /@ Partition[p[[i]], n, 1, {1, 1}], {i, 1, L}]);
n = 1; While[n <= 10, ClearSystemCache[[]]; f[n]; triangle = Parallelize[Select[mat, Max[Det[#]] == det &]]; Print[SortBy[triangle, Less][[1]][[1]]]; n++]; (* Kebbaj Mohamed Reda, Dec 03 2019; edited by Michel Marcus, Dec 24 2023 *)

A328031 Upper bound for the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,...,n^n}.

Original entry on oeis.org

1, 1, 3, 18, 172, 2343, 42439, 976050, 27583338, 934173632, 37180409223, 1711870023666, 90007747560742, 5346164992890599, 355442084718552178, 26244000000000000000, 2137205155719002036203, 190811368062993357765186, 18577775646585813239195436, 1963166636163973976912956096

Offset: 0

Hugo Pfoertner, Oct 02 2019

Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A301371, A309985, A328030.

for(n=1,20,print1(floor(n^n*((n+1)/2)*((n+1)/12)^((n-1)/2)),", "))

a(n) = floor(n^n*((n+1)/2)*((n+1)/12)^((n-1)/2)) (Corollary 3 in M. Sigg's article).

A328062 Lexicographically earliest permutation of [1,2,...,n] minimizing the positive value of the determinant of an n X n circulant matrix that uses this permutation as first row, written as triangle T(n,k), k <= n.

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 3, 1, 4, 2, 1, 2, 4, 5, 3, 1, 2, 4, 5, 6, 3, 1, 2, 4, 6, 7, 5, 3, 3, 1, 5, 4, 8, 6, 7, 2, 1, 2, 4, 6, 8, 9, 7, 5, 3, 5, 1, 7, 3, 8, 4, 10, 6, 9, 2, 1, 2, 3, 8, 6, 4, 9, 10, 11, 5, 7, 3, 1, 5, 4, 9, 8, 12, 10, 11, 6, 7, 2, 1, 2, 3, 5, 8, 7, 6, 9, 11, 12, 13, 10, 4

Offset: 1

Hugo Pfoertner, Oct 03 2019

			The triangle starts
  1;
  2,  1;
  1,  2,  3;
  3,  1,  4,  2;
  1,  2,  4,  5,  3;
  1,  2,  4,  5,  6,  3;
  1,  2,  4,  6,  7,  5,  3;
  3,  1,  5,  4,  8,  6,  7,  2;
  1,  2,  4,  6,  8,  9,  7,  5,  3;
  5,  1,  7,  3,  8,  4, 10,  6,  9,  2;
.
The 4th row of the triangle T(4,1)..T(4,4) = a(7)..a(10) is [3,1,4,2] because this is the lexicographically earliest permutation of [1,2,3,4] producing a circulant 4 X 4 matrix with minimum positive determinant A309257(4) = 80.
  [3, 1, 4, 2;
   2, 3, 1, 4;
   4, 2, 3, 1;
   1, 4, 2, 3].
All lexicographically earlier permutations lead to the other possible determinants -160, -80, 0, 160 with [1,3,2,4], [1,4,3,2], [2,3,1,4], and [2,4,1,3] producing determinants = -80.

Hugo Pfoertner, Table of n, a(n) for n = 1..120, rows 1..15 of triangle, flattened
Wikipedia, Circulant matrix.

Cf. A309257, A328029, A328030.

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

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A309985 Maximum determinant of an n X n Latin square.

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A328029 Lexicographically earliest permutation of [1,2,...,n] maximizing the determinant of an n X n circulant matrix that uses this permutation as first row, written as triangle T(n,k), k <= n.

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A328031 Upper bound for the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,...,n^n}.

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A328062 Lexicographically earliest permutation of [1,2,...,n] minimizing the positive value of the determinant of an n X n circulant matrix that uses this permutation as first row, written as triangle T(n,k), k <= n.

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