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A328029
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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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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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;
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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