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.

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

Author

Hugo Pfoertner, Oct 02 2019

Keywords

• nonn
• tabl

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)