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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

EXAMPLE

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.

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 *)

(* alternate program *)

n1 = DialogInput[{name = ""},

  Column[{"Input n :", InputField[Dynamic[name], String],

    Button["Proceed", DialogReturn[name], ImageSize -> Automatic]}]];

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; sequance = {}; While[n <= ToExpression[n1], ClearSystemCache[[]];

f[n]; triangle = Parallelize[Select[mat, Max[Det[#]] == det &]];

AppendTo[sequance, SortBy[triangle, Less][[1]][[1]]];

n++]; Flatten[sequance] (* Kebbaj Mohamed Reda, Dec 03 2019 *)

CROSSREFS

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

Sequence in context: A005794 A280860 A208993 * A201384 A238348 A143066

Adjacent sequences:  A328026 A328027 A328028 * A328030 A328031 A328032

KEYWORD

nonn,tabl,changed

AUTHOR

Hugo Pfoertner, Oct 02 2019

STATUS

approved

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Last modified December 14 07:03 EST 2019. Contains 329978 sequences. (Running on oeis4.)