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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.

%I A328029 #37 Dec 24 2023 09:38:05
%S A328029 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 A328029 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 A328029 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 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.
%C A328029 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 A328029 Hugo Pfoertner, <a href="/A328029/b328029.txt">Table of n, a(n) for n = 1..120</a>, rows 1..15 of triangle, flattened
%H A328029 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 A328029 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>.
%e A328029 The triangle starts
%e A328029   1;
%e A328029   2,  1;
%e A328029   1,  2,  3;
%e A328029   2,  1,  4,  3;
%e A328029   1,  2,  4,  3,  5;
%e A328029   2,  1,  6,  3,  5,  4;
%e A328029   1,  2,  4,  6,  5,  3,  7;
%e A328029   2,  1,  5,  4,  8,  3,  6,  7;
%e A328029   1,  2,  4,  8,  6,  7,  5,  3,  9;
%e A328029   1,  2, 10,  7,  8,  3,  9,  5,  4,  6;
%e A328029 .
%e A328029 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 A328029   [2, 1, 4, 3;
%e A328029    3, 2, 1, 4;
%e A328029    4, 3, 2, 1;
%e A328029    1, 4, 3, 2].
%e A328029 All lexicographically earlier permutations lead to smaller determinants, with [1,2,3,4] and [1,4,3,2] producing determinants = -160.
%t A328029 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 A328029 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 A328029 Cf. A301371, A309985, A328030, A328031, A328062.
%K A328029 nonn,tabl
%O A328029 1,2
%A A328029 _Hugo Pfoertner_, Oct 02 2019