A328062 -id:A328062 - cp's OEIS frontend

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

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 *)

A309257 a(n) is the minimum positive value of the determinants of circulant order n Latin squares.

Original entry on oeis.org

1, 3, 18, 80, 75, 3087, 196, 1440, 405, 6325, 726, 7488, 1183, 11025, 1800

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 19 2019

These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.

			For n=2, a(2)=3 is the minimum absolute value determinant of a back circulant Latin square of order 2. An example of one such matrix is [[2 1], [1 2]].
For n=5, a(5)= 75 is the minimum absolute value determinant of a back circulant Latin square of order 5. An example of one such matrix is [[1, 2, 4, 5, 3], [3, 1, 2, 4, 5], [5, 3, 1, 2, 4], [4, 5, 3, 1, 2], [2, 4, 5, 3, 1]] has determinant 75.

Froylan Maldonado, Minimum absolute value determinant of back circulant Latin squares code

Cf. A301371, A308853, A309088, A328062.

Sage
```
See Maldonado link.
```

Modified title and a(8)-a(13) from Hugo Pfoertner, Oct 01 2019
a(14) from Hugo Pfoertner, Oct 07 2019
a(15) from Hugo Pfoertner, Oct 13 2019

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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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