A328062 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.
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
Offset: 1
Examples
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; . 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.
Links
- Hugo Pfoertner, Table of n, a(n) for n = 1..120, rows 1..15 of triangle, flattened
- Wikipedia, Circulant matrix.