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%I #37 Dec 24 2023 09:38:05
%S 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,
%T 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,
%U 1,7,3,12,5,9,10,4,6,11,8,1,2,12,13,5,10,6,11,3,9,8,4,7
%N 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.
%C 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.
%H Hugo Pfoertner, <a href="/A328029/b328029.txt">Table of n, a(n) for n = 1..120</a>, rows 1..15 of triangle, flattened
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/885481/maximum-determinant-of-latin-squares">Maximum determinant of Latin squares</a>, (2014), (2016).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>.
%e The triangle starts
%e 1;
%e 2, 1;
%e 1, 2, 3;
%e 2, 1, 4, 3;
%e 1, 2, 4, 3, 5;
%e 2, 1, 6, 3, 5, 4;
%e 1, 2, 4, 6, 5, 3, 7;
%e 2, 1, 5, 4, 8, 3, 6, 7;
%e 1, 2, 4, 8, 6, 7, 5, 3, 9;
%e 1, 2, 10, 7, 8, 3, 9, 5, 4, 6;
%e .
%e 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.
%e [2, 1, 4, 3;
%e 3, 2, 1, 4;
%e 4, 3, 2, 1;
%e 1, 4, 3, 2].
%e All lexicographically earlier permutations lead to smaller determinants, with [1,2,3,4] and [1,4,3,2] producing determinants = -160.
%t 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}]);
%t 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 *)
%Y Cf. A301371, A309985, A328030, A328031, A328062.
%K nonn,tabl
%O 1,2
%A _Hugo Pfoertner_, Oct 02 2019
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