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A328062
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Lexicographically earliest permutation of [1,2,...,n] minimizing the positive value of 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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3
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1, 2, 1, 1, 2, 3, 3, 1, 4, 2, 1, 2, 4, 5, 3, 1, 2, 4, 5, 6, 3, 1, 2, 4, 6, 7, 5, 3, 3, 1, 5, 4, 8, 6, 7, 2, 1, 2, 4, 6, 8, 9, 7, 5, 3, 5, 1, 7, 3, 8, 4, 10, 6, 9, 2, 1, 2, 3, 8, 6, 4, 9, 10, 11, 5, 7, 3, 1, 5, 4, 9, 8, 12, 10, 11, 6, 7, 2, 1, 2, 3, 5, 8, 7, 6, 9, 11, 12, 13, 10, 4
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OFFSET
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1,2
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LINKS
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EXAMPLE
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The triangle starts
1;
2, 1;
1, 2, 3;
3, 1, 4, 2;
1, 2, 4, 5, 3;
1, 2, 4, 5, 6, 3;
1, 2, 4, 6, 7, 5, 3;
3, 1, 5, 4, 8, 6, 7, 2;
1, 2, 4, 6, 8, 9, 7, 5, 3;
5, 1, 7, 3, 8, 4, 10, 6, 9, 2;
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The 4th row of the triangle T(4,1)..T(4,4) = a(7)..a(10) is [3,1,4,2] because this is the lexicographically earliest permutation of [1,2,3,4] producing a circulant 4 X 4 matrix with minimum positive determinant A309257(4) = 80.
[3, 1, 4, 2;
2, 3, 1, 4;
4, 2, 3, 1;
1, 4, 2, 3].
All lexicographically earlier permutations lead to the other possible determinants -160, -80, 0, 160 with [1,3,2,4], [1,4,3,2], [2,3,1,4], and [2,4,1,3] producing determinants = -80.
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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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