Identifier
Values
[[1]] => 0
[[1,2]] => 0
[[1],[2]] => 0
[[1,2,3]] => 0
[[1,3],[2]] => 1
[[1,2],[3]] => 0
[[1],[2],[3]] => 0
[[1,2,3,4]] => 0
[[1,3,4],[2]] => 2
[[1,2,4],[3]] => 1
[[1,2,3],[4]] => 0
[[1,3],[2,4]] => 2
[[1,2],[3,4]] => 0
[[1,4],[2],[3]] => 1
[[1,3],[2],[4]] => 2
[[1,2],[3],[4]] => 0
[[1],[2],[3],[4]] => 0
[[1,2,3,4,5]] => 0
[[1,3,4,5],[2]] => 3
[[1,2,4,5],[3]] => 2
[[1,2,3,5],[4]] => 1
[[1,2,3,4],[5]] => 0
[[1,3,5],[2,4]] => 4
[[1,2,5],[3,4]] => 1
[[1,3,4],[2,5]] => 3
[[1,2,4],[3,5]] => 2
[[1,2,3],[4,5]] => 0
[[1,4,5],[2],[3]] => 2
[[1,3,5],[2],[4]] => 4
[[1,2,5],[3],[4]] => 1
[[1,3,4],[2],[5]] => 3
[[1,2,4],[3],[5]] => 2
[[1,2,3],[4],[5]] => 0
[[1,4],[2,5],[3]] => 2
[[1,3],[2,5],[4]] => 4
[[1,2],[3,5],[4]] => 1
[[1,3],[2,4],[5]] => 3
[[1,2],[3,4],[5]] => 0
[[1,5],[2],[3],[4]] => 1
[[1,4],[2],[3],[5]] => 2
[[1,3],[2],[4],[5]] => 3
[[1,2],[3],[4],[5]] => 0
[[1],[2],[3],[4],[5]] => 0
[[1,2,3,4,5,6]] => 0
[[1,3,4,5,6],[2]] => 4
[[1,2,4,5,6],[3]] => 3
[[1,2,3,5,6],[4]] => 2
[[1,2,3,4,6],[5]] => 1
[[1,2,3,4,5],[6]] => 0
[[1,3,5,6],[2,4]] => 6
[[1,2,5,6],[3,4]] => 2
[[1,3,4,6],[2,5]] => 5
[[1,2,4,6],[3,5]] => 4
[[1,2,3,6],[4,5]] => 1
[[1,3,4,5],[2,6]] => 4
[[1,2,4,5],[3,6]] => 3
[[1,2,3,5],[4,6]] => 2
[[1,2,3,4],[5,6]] => 0
[[1,4,5,6],[2],[3]] => 3
[[1,3,5,6],[2],[4]] => 6
[[1,2,5,6],[3],[4]] => 2
[[1,3,4,6],[2],[5]] => 5
[[1,2,4,6],[3],[5]] => 4
[[1,2,3,6],[4],[5]] => 1
[[1,3,4,5],[2],[6]] => 4
[[1,2,4,5],[3],[6]] => 3
[[1,2,3,5],[4],[6]] => 2
[[1,2,3,4],[5],[6]] => 0
[[1,3,5],[2,4,6]] => 6
[[1,2,5],[3,4,6]] => 2
[[1,3,4],[2,5,6]] => 4
[[1,2,4],[3,5,6]] => 3
[[1,2,3],[4,5,6]] => 0
[[1,4,6],[2,5],[3]] => 4
[[1,3,6],[2,5],[4]] => 7
[[1,2,6],[3,5],[4]] => 3
[[1,3,6],[2,4],[5]] => 5
[[1,2,6],[3,4],[5]] => 1
[[1,4,5],[2,6],[3]] => 3
[[1,3,5],[2,6],[4]] => 6
[[1,2,5],[3,6],[4]] => 2
[[1,3,4],[2,6],[5]] => 5
[[1,2,4],[3,6],[5]] => 4
[[1,2,3],[4,6],[5]] => 1
[[1,3,5],[2,4],[6]] => 6
[[1,2,5],[3,4],[6]] => 2
[[1,3,4],[2,5],[6]] => 4
[[1,2,4],[3,5],[6]] => 3
[[1,2,3],[4,5],[6]] => 0
[[1,5,6],[2],[3],[4]] => 2
[[1,4,6],[2],[3],[5]] => 4
[[1,3,6],[2],[4],[5]] => 5
[[1,2,6],[3],[4],[5]] => 1
[[1,4,5],[2],[3],[6]] => 3
[[1,3,5],[2],[4],[6]] => 6
[[1,2,5],[3],[4],[6]] => 2
[[1,3,4],[2],[5],[6]] => 4
[[1,2,4],[3],[5],[6]] => 3
[[1,2,3],[4],[5],[6]] => 0
[[1,4],[2,5],[3,6]] => 3
[[1,3],[2,5],[4,6]] => 6
>>> Load all 1115 entries. <<<[[1,2],[3,5],[4,6]] => 2
[[1,3],[2,4],[5,6]] => 4
[[1,2],[3,4],[5,6]] => 0
[[1,5],[2,6],[3],[4]] => 2
[[1,4],[2,6],[3],[5]] => 4
[[1,3],[2,6],[4],[5]] => 5
[[1,2],[3,6],[4],[5]] => 1
[[1,4],[2,5],[3],[6]] => 3
[[1,3],[2,5],[4],[6]] => 6
[[1,2],[3,5],[4],[6]] => 2
[[1,3],[2,4],[5],[6]] => 4
[[1,2],[3,4],[5],[6]] => 0
[[1,6],[2],[3],[4],[5]] => 1
[[1,5],[2],[3],[4],[6]] => 2
[[1,4],[2],[3],[5],[6]] => 3
[[1,3],[2],[4],[5],[6]] => 4
[[1,2],[3],[4],[5],[6]] => 0
[[1],[2],[3],[4],[5],[6]] => 0
[[1,2,3,4,5,6,7]] => 0
[[1,3,4,5,6,7],[2]] => 5
[[1,2,4,5,6,7],[3]] => 4
[[1,2,3,5,6,7],[4]] => 3
[[1,2,3,4,6,7],[5]] => 2
[[1,2,3,4,5,7],[6]] => 1
[[1,2,3,4,5,6],[7]] => 0
[[1,3,5,6,7],[2,4]] => 8
[[1,2,5,6,7],[3,4]] => 3
[[1,3,4,6,7],[2,5]] => 7
[[1,2,4,6,7],[3,5]] => 6
[[1,2,3,6,7],[4,5]] => 2
[[1,3,4,5,7],[2,6]] => 6
[[1,2,4,5,7],[3,6]] => 5
[[1,2,3,5,7],[4,6]] => 4
[[1,2,3,4,7],[5,6]] => 1
[[1,3,4,5,6],[2,7]] => 5
[[1,2,4,5,6],[3,7]] => 4
[[1,2,3,5,6],[4,7]] => 3
[[1,2,3,4,6],[5,7]] => 2
[[1,2,3,4,5],[6,7]] => 0
[[1,4,5,6,7],[2],[3]] => 4
[[1,3,5,6,7],[2],[4]] => 8
[[1,2,5,6,7],[3],[4]] => 3
[[1,3,4,6,7],[2],[5]] => 7
[[1,2,4,6,7],[3],[5]] => 6
[[1,2,3,6,7],[4],[5]] => 2
[[1,3,4,5,7],[2],[6]] => 6
[[1,2,4,5,7],[3],[6]] => 5
[[1,2,3,5,7],[4],[6]] => 4
[[1,2,3,4,7],[5],[6]] => 1
[[1,3,4,5,6],[2],[7]] => 5
[[1,2,4,5,6],[3],[7]] => 4
[[1,2,3,5,6],[4],[7]] => 3
[[1,2,3,4,6],[5],[7]] => 2
[[1,2,3,4,5],[6],[7]] => 0
[[1,3,5,7],[2,4,6]] => 9
[[1,2,5,7],[3,4,6]] => 4
[[1,3,4,7],[2,5,6]] => 6
[[1,2,4,7],[3,5,6]] => 5
[[1,2,3,7],[4,5,6]] => 1
[[1,3,5,6],[2,4,7]] => 8
[[1,2,5,6],[3,4,7]] => 3
[[1,3,4,6],[2,5,7]] => 7
[[1,2,4,6],[3,5,7]] => 6
[[1,2,3,6],[4,5,7]] => 2
[[1,3,4,5],[2,6,7]] => 5
[[1,2,4,5],[3,6,7]] => 4
[[1,2,3,5],[4,6,7]] => 3
[[1,2,3,4],[5,6,7]] => 0
[[1,4,6,7],[2,5],[3]] => 6
[[1,3,6,7],[2,5],[4]] => 10
[[1,2,6,7],[3,5],[4]] => 5
[[1,3,6,7],[2,4],[5]] => 7
[[1,2,6,7],[3,4],[5]] => 2
[[1,4,5,7],[2,6],[3]] => 5
[[1,3,5,7],[2,6],[4]] => 9
[[1,2,5,7],[3,6],[4]] => 4
[[1,3,4,7],[2,6],[5]] => 8
[[1,2,4,7],[3,6],[5]] => 7
[[1,2,3,7],[4,6],[5]] => 3
[[1,3,5,7],[2,4],[6]] => 9
[[1,2,5,7],[3,4],[6]] => 4
[[1,3,4,7],[2,5],[6]] => 6
[[1,2,4,7],[3,5],[6]] => 5
[[1,2,3,7],[4,5],[6]] => 1
[[1,4,5,6],[2,7],[3]] => 4
[[1,3,5,6],[2,7],[4]] => 8
[[1,2,5,6],[3,7],[4]] => 3
[[1,3,4,6],[2,7],[5]] => 7
[[1,2,4,6],[3,7],[5]] => 6
[[1,2,3,6],[4,7],[5]] => 2
[[1,3,4,5],[2,7],[6]] => 6
[[1,2,4,5],[3,7],[6]] => 5
[[1,2,3,5],[4,7],[6]] => 4
[[1,2,3,4],[5,7],[6]] => 1
[[1,3,5,6],[2,4],[7]] => 8
[[1,2,5,6],[3,4],[7]] => 3
[[1,3,4,6],[2,5],[7]] => 7
[[1,2,4,6],[3,5],[7]] => 6
[[1,2,3,6],[4,5],[7]] => 2
[[1,3,4,5],[2,6],[7]] => 5
[[1,2,4,5],[3,6],[7]] => 4
[[1,2,3,5],[4,6],[7]] => 3
[[1,2,3,4],[5,6],[7]] => 0
[[1,5,6,7],[2],[3],[4]] => 3
[[1,4,6,7],[2],[3],[5]] => 6
[[1,3,6,7],[2],[4],[5]] => 7
[[1,2,6,7],[3],[4],[5]] => 2
[[1,4,5,7],[2],[3],[6]] => 5
[[1,3,5,7],[2],[4],[6]] => 9
[[1,2,5,7],[3],[4],[6]] => 4
[[1,3,4,7],[2],[5],[6]] => 6
[[1,2,4,7],[3],[5],[6]] => 5
[[1,2,3,7],[4],[5],[6]] => 1
[[1,4,5,6],[2],[3],[7]] => 4
[[1,3,5,6],[2],[4],[7]] => 8
[[1,2,5,6],[3],[4],[7]] => 3
[[1,3,4,6],[2],[5],[7]] => 7
[[1,2,4,6],[3],[5],[7]] => 6
[[1,2,3,6],[4],[5],[7]] => 2
[[1,3,4,5],[2],[6],[7]] => 5
[[1,2,4,5],[3],[6],[7]] => 4
[[1,2,3,5],[4],[6],[7]] => 3
[[1,2,3,4],[5],[6],[7]] => 0
[[1,4,6],[2,5,7],[3]] => 6
[[1,3,6],[2,5,7],[4]] => 10
[[1,2,6],[3,5,7],[4]] => 5
[[1,3,6],[2,4,7],[5]] => 7
[[1,2,6],[3,4,7],[5]] => 2
[[1,4,5],[2,6,7],[3]] => 4
[[1,3,5],[2,6,7],[4]] => 8
[[1,2,5],[3,6,7],[4]] => 3
[[1,3,4],[2,6,7],[5]] => 7
[[1,2,4],[3,6,7],[5]] => 6
[[1,2,3],[4,6,7],[5]] => 2
[[1,3,5],[2,4,7],[6]] => 9
[[1,2,5],[3,4,7],[6]] => 4
[[1,3,4],[2,5,7],[6]] => 6
[[1,2,4],[3,5,7],[6]] => 5
[[1,2,3],[4,5,7],[6]] => 1
[[1,3,5],[2,4,6],[7]] => 8
[[1,2,5],[3,4,6],[7]] => 3
[[1,3,4],[2,5,6],[7]] => 5
[[1,2,4],[3,5,6],[7]] => 4
[[1,2,3],[4,5,6],[7]] => 0
[[1,4,7],[2,5],[3,6]] => 5
[[1,3,7],[2,5],[4,6]] => 9
[[1,2,7],[3,5],[4,6]] => 4
[[1,3,7],[2,4],[5,6]] => 6
[[1,2,7],[3,4],[5,6]] => 1
[[1,4,6],[2,5],[3,7]] => 6
[[1,3,6],[2,5],[4,7]] => 10
[[1,2,6],[3,5],[4,7]] => 5
[[1,3,6],[2,4],[5,7]] => 7
[[1,2,6],[3,4],[5,7]] => 2
[[1,4,5],[2,6],[3,7]] => 4
[[1,3,5],[2,6],[4,7]] => 8
[[1,2,5],[3,6],[4,7]] => 3
[[1,3,4],[2,6],[5,7]] => 7
[[1,2,4],[3,6],[5,7]] => 6
[[1,2,3],[4,6],[5,7]] => 2
[[1,3,5],[2,4],[6,7]] => 8
[[1,2,5],[3,4],[6,7]] => 3
[[1,3,4],[2,5],[6,7]] => 5
[[1,2,4],[3,5],[6,7]] => 4
[[1,2,3],[4,5],[6,7]] => 0
[[1,5,7],[2,6],[3],[4]] => 4
[[1,4,7],[2,6],[3],[5]] => 7
[[1,3,7],[2,6],[4],[5]] => 8
[[1,2,7],[3,6],[4],[5]] => 3
[[1,4,7],[2,5],[3],[6]] => 5
[[1,3,7],[2,5],[4],[6]] => 9
[[1,2,7],[3,5],[4],[6]] => 4
[[1,3,7],[2,4],[5],[6]] => 6
[[1,2,7],[3,4],[5],[6]] => 1
[[1,5,6],[2,7],[3],[4]] => 3
[[1,4,6],[2,7],[3],[5]] => 6
[[1,3,6],[2,7],[4],[5]] => 7
[[1,2,6],[3,7],[4],[5]] => 2
[[1,4,5],[2,7],[3],[6]] => 5
[[1,3,5],[2,7],[4],[6]] => 9
[[1,2,5],[3,7],[4],[6]] => 4
[[1,3,4],[2,7],[5],[6]] => 6
[[1,2,4],[3,7],[5],[6]] => 5
[[1,2,3],[4,7],[5],[6]] => 1
[[1,4,6],[2,5],[3],[7]] => 6
[[1,3,6],[2,5],[4],[7]] => 10
[[1,2,6],[3,5],[4],[7]] => 5
[[1,3,6],[2,4],[5],[7]] => 7
[[1,2,6],[3,4],[5],[7]] => 2
[[1,4,5],[2,6],[3],[7]] => 4
[[1,3,5],[2,6],[4],[7]] => 8
[[1,2,5],[3,6],[4],[7]] => 3
[[1,3,4],[2,6],[5],[7]] => 7
[[1,2,4],[3,6],[5],[7]] => 6
[[1,2,3],[4,6],[5],[7]] => 2
[[1,3,5],[2,4],[6],[7]] => 8
[[1,2,5],[3,4],[6],[7]] => 3
[[1,3,4],[2,5],[6],[7]] => 5
[[1,2,4],[3,5],[6],[7]] => 4
[[1,2,3],[4,5],[6],[7]] => 0
[[1,6,7],[2],[3],[4],[5]] => 2
[[1,5,7],[2],[3],[4],[6]] => 4
[[1,4,7],[2],[3],[5],[6]] => 5
[[1,3,7],[2],[4],[5],[6]] => 6
[[1,2,7],[3],[4],[5],[6]] => 1
[[1,5,6],[2],[3],[4],[7]] => 3
[[1,4,6],[2],[3],[5],[7]] => 6
[[1,3,6],[2],[4],[5],[7]] => 7
[[1,2,6],[3],[4],[5],[7]] => 2
[[1,4,5],[2],[3],[6],[7]] => 4
[[1,3,5],[2],[4],[6],[7]] => 8
[[1,2,5],[3],[4],[6],[7]] => 3
[[1,3,4],[2],[5],[6],[7]] => 5
[[1,2,4],[3],[5],[6],[7]] => 4
[[1,2,3],[4],[5],[6],[7]] => 0
[[1,5],[2,6],[3,7],[4]] => 3
[[1,4],[2,6],[3,7],[5]] => 6
[[1,3],[2,6],[4,7],[5]] => 7
[[1,2],[3,6],[4,7],[5]] => 2
[[1,4],[2,5],[3,7],[6]] => 5
[[1,3],[2,5],[4,7],[6]] => 9
[[1,2],[3,5],[4,7],[6]] => 4
[[1,3],[2,4],[5,7],[6]] => 6
[[1,2],[3,4],[5,7],[6]] => 1
[[1,4],[2,5],[3,6],[7]] => 4
[[1,3],[2,5],[4,6],[7]] => 8
[[1,2],[3,5],[4,6],[7]] => 3
[[1,3],[2,4],[5,6],[7]] => 5
[[1,2],[3,4],[5,6],[7]] => 0
[[1,6],[2,7],[3],[4],[5]] => 2
[[1,5],[2,7],[3],[4],[6]] => 4
[[1,4],[2,7],[3],[5],[6]] => 5
[[1,3],[2,7],[4],[5],[6]] => 6
[[1,2],[3,7],[4],[5],[6]] => 1
[[1,5],[2,6],[3],[4],[7]] => 3
[[1,4],[2,6],[3],[5],[7]] => 6
[[1,3],[2,6],[4],[5],[7]] => 7
[[1,2],[3,6],[4],[5],[7]] => 2
[[1,4],[2,5],[3],[6],[7]] => 4
[[1,3],[2,5],[4],[6],[7]] => 8
[[1,2],[3,5],[4],[6],[7]] => 3
[[1,3],[2,4],[5],[6],[7]] => 5
[[1,2],[3,4],[5],[6],[7]] => 0
[[1,7],[2],[3],[4],[5],[6]] => 1
[[1,6],[2],[3],[4],[5],[7]] => 2
[[1,5],[2],[3],[4],[6],[7]] => 3
[[1,4],[2],[3],[5],[6],[7]] => 4
[[1,3],[2],[4],[5],[6],[7]] => 5
[[1,2],[3],[4],[5],[6],[7]] => 0
[[1],[2],[3],[4],[5],[6],[7]] => 0
[[1,2,3,4,5,6,7,8]] => 0
[[1,3,4,5,6,7,8],[2]] => 6
[[1,2,4,5,6,7,8],[3]] => 5
[[1,2,3,5,6,7,8],[4]] => 4
[[1,2,3,4,6,7,8],[5]] => 3
[[1,2,3,4,5,7,8],[6]] => 2
[[1,2,3,4,5,6,8],[7]] => 1
[[1,2,3,4,5,6,7],[8]] => 0
[[1,3,5,6,7,8],[2,4]] => 10
[[1,2,5,6,7,8],[3,4]] => 4
[[1,3,4,6,7,8],[2,5]] => 9
[[1,2,4,6,7,8],[3,5]] => 8
[[1,2,3,6,7,8],[4,5]] => 3
[[1,3,4,5,7,8],[2,6]] => 8
[[1,2,4,5,7,8],[3,6]] => 7
[[1,2,3,5,7,8],[4,6]] => 6
[[1,2,3,4,7,8],[5,6]] => 2
[[1,3,4,5,6,8],[2,7]] => 7
[[1,2,4,5,6,8],[3,7]] => 6
[[1,2,3,5,6,8],[4,7]] => 5
[[1,2,3,4,6,8],[5,7]] => 4
[[1,2,3,4,5,8],[6,7]] => 1
[[1,3,4,5,6,7],[2,8]] => 6
[[1,2,4,5,6,7],[3,8]] => 5
[[1,2,3,5,6,7],[4,8]] => 4
[[1,2,3,4,6,7],[5,8]] => 3
[[1,2,3,4,5,7],[6,8]] => 2
[[1,2,3,4,5,6],[7,8]] => 0
[[1,4,5,6,7,8],[2],[3]] => 5
[[1,3,5,6,7,8],[2],[4]] => 10
[[1,2,5,6,7,8],[3],[4]] => 4
[[1,3,4,6,7,8],[2],[5]] => 9
[[1,2,4,6,7,8],[3],[5]] => 8
[[1,2,3,6,7,8],[4],[5]] => 3
[[1,3,4,5,7,8],[2],[6]] => 8
[[1,2,4,5,7,8],[3],[6]] => 7
[[1,2,3,5,7,8],[4],[6]] => 6
[[1,2,3,4,7,8],[5],[6]] => 2
[[1,3,4,5,6,8],[2],[7]] => 7
[[1,2,4,5,6,8],[3],[7]] => 6
[[1,2,3,5,6,8],[4],[7]] => 5
[[1,2,3,4,6,8],[5],[7]] => 4
[[1,2,3,4,5,8],[6],[7]] => 1
[[1,3,4,5,6,7],[2],[8]] => 6
[[1,2,4,5,6,7],[3],[8]] => 5
[[1,2,3,5,6,7],[4],[8]] => 4
[[1,2,3,4,6,7],[5],[8]] => 3
[[1,2,3,4,5,7],[6],[8]] => 2
[[1,2,3,4,5,6],[7],[8]] => 0
[[1,3,5,7,8],[2,4,6]] => 12
[[1,2,5,7,8],[3,4,6]] => 6
[[1,3,4,7,8],[2,5,6]] => 8
[[1,2,4,7,8],[3,5,6]] => 7
[[1,2,3,7,8],[4,5,6]] => 2
[[1,3,5,6,8],[2,4,7]] => 11
[[1,2,5,6,8],[3,4,7]] => 5
[[1,3,4,6,8],[2,5,7]] => 10
[[1,2,4,6,8],[3,5,7]] => 9
[[1,2,3,6,8],[4,5,7]] => 4
[[1,3,4,5,8],[2,6,7]] => 7
[[1,2,4,5,8],[3,6,7]] => 6
[[1,2,3,5,8],[4,6,7]] => 5
[[1,2,3,4,8],[5,6,7]] => 1
[[1,3,5,6,7],[2,4,8]] => 10
[[1,2,5,6,7],[3,4,8]] => 4
[[1,3,4,6,7],[2,5,8]] => 9
[[1,2,4,6,7],[3,5,8]] => 8
[[1,2,3,6,7],[4,5,8]] => 3
[[1,3,4,5,7],[2,6,8]] => 8
[[1,2,4,5,7],[3,6,8]] => 7
[[1,2,3,5,7],[4,6,8]] => 6
[[1,2,3,4,7],[5,6,8]] => 2
[[1,3,4,5,6],[2,7,8]] => 6
[[1,2,4,5,6],[3,7,8]] => 5
[[1,2,3,5,6],[4,7,8]] => 4
[[1,2,3,4,6],[5,7,8]] => 3
[[1,2,3,4,5],[6,7,8]] => 0
[[1,4,6,7,8],[2,5],[3]] => 8
[[1,3,6,7,8],[2,5],[4]] => 13
[[1,2,6,7,8],[3,5],[4]] => 7
[[1,3,6,7,8],[2,4],[5]] => 9
[[1,2,6,7,8],[3,4],[5]] => 3
[[1,4,5,7,8],[2,6],[3]] => 7
[[1,3,5,7,8],[2,6],[4]] => 12
[[1,2,5,7,8],[3,6],[4]] => 6
[[1,3,4,7,8],[2,6],[5]] => 11
[[1,2,4,7,8],[3,6],[5]] => 10
[[1,2,3,7,8],[4,6],[5]] => 5
[[1,3,5,7,8],[2,4],[6]] => 12
[[1,2,5,7,8],[3,4],[6]] => 6
[[1,3,4,7,8],[2,5],[6]] => 8
[[1,2,4,7,8],[3,5],[6]] => 7
[[1,2,3,7,8],[4,5],[6]] => 2
[[1,4,5,6,8],[2,7],[3]] => 6
[[1,3,5,6,8],[2,7],[4]] => 11
[[1,2,5,6,8],[3,7],[4]] => 5
[[1,3,4,6,8],[2,7],[5]] => 10
[[1,2,4,6,8],[3,7],[5]] => 9
[[1,2,3,6,8],[4,7],[5]] => 4
[[1,3,4,5,8],[2,7],[6]] => 9
[[1,2,4,5,8],[3,7],[6]] => 8
[[1,2,3,5,8],[4,7],[6]] => 7
[[1,2,3,4,8],[5,7],[6]] => 3
[[1,3,5,6,8],[2,4],[7]] => 11
[[1,2,5,6,8],[3,4],[7]] => 5
[[1,3,4,6,8],[2,5],[7]] => 10
[[1,2,4,6,8],[3,5],[7]] => 9
[[1,2,3,6,8],[4,5],[7]] => 4
[[1,3,4,5,8],[2,6],[7]] => 7
[[1,2,4,5,8],[3,6],[7]] => 6
[[1,2,3,5,8],[4,6],[7]] => 5
[[1,2,3,4,8],[5,6],[7]] => 1
[[1,4,5,6,7],[2,8],[3]] => 5
[[1,3,5,6,7],[2,8],[4]] => 10
[[1,2,5,6,7],[3,8],[4]] => 4
[[1,3,4,6,7],[2,8],[5]] => 9
[[1,2,4,6,7],[3,8],[5]] => 8
[[1,2,3,6,7],[4,8],[5]] => 3
[[1,3,4,5,7],[2,8],[6]] => 8
[[1,2,4,5,7],[3,8],[6]] => 7
[[1,2,3,5,7],[4,8],[6]] => 6
[[1,2,3,4,7],[5,8],[6]] => 2
[[1,3,4,5,6],[2,8],[7]] => 7
[[1,2,4,5,6],[3,8],[7]] => 6
[[1,2,3,5,6],[4,8],[7]] => 5
[[1,2,3,4,6],[5,8],[7]] => 4
[[1,2,3,4,5],[6,8],[7]] => 1
[[1,3,5,6,7],[2,4],[8]] => 10
[[1,2,5,6,7],[3,4],[8]] => 4
[[1,3,4,6,7],[2,5],[8]] => 9
[[1,2,4,6,7],[3,5],[8]] => 8
[[1,2,3,6,7],[4,5],[8]] => 3
[[1,3,4,5,7],[2,6],[8]] => 8
[[1,2,4,5,7],[3,6],[8]] => 7
[[1,2,3,5,7],[4,6],[8]] => 6
[[1,2,3,4,7],[5,6],[8]] => 2
[[1,3,4,5,6],[2,7],[8]] => 6
[[1,2,4,5,6],[3,7],[8]] => 5
[[1,2,3,5,6],[4,7],[8]] => 4
[[1,2,3,4,6],[5,7],[8]] => 3
[[1,2,3,4,5],[6,7],[8]] => 0
[[1,5,6,7,8],[2],[3],[4]] => 4
[[1,4,6,7,8],[2],[3],[5]] => 8
[[1,3,6,7,8],[2],[4],[5]] => 9
[[1,2,6,7,8],[3],[4],[5]] => 3
[[1,4,5,7,8],[2],[3],[6]] => 7
[[1,3,5,7,8],[2],[4],[6]] => 12
[[1,2,5,7,8],[3],[4],[6]] => 6
[[1,3,4,7,8],[2],[5],[6]] => 8
[[1,2,4,7,8],[3],[5],[6]] => 7
[[1,2,3,7,8],[4],[5],[6]] => 2
[[1,4,5,6,8],[2],[3],[7]] => 6
[[1,3,5,6,8],[2],[4],[7]] => 11
[[1,2,5,6,8],[3],[4],[7]] => 5
[[1,3,4,6,8],[2],[5],[7]] => 10
[[1,2,4,6,8],[3],[5],[7]] => 9
[[1,2,3,6,8],[4],[5],[7]] => 4
[[1,3,4,5,8],[2],[6],[7]] => 7
[[1,2,4,5,8],[3],[6],[7]] => 6
[[1,2,3,5,8],[4],[6],[7]] => 5
[[1,2,3,4,8],[5],[6],[7]] => 1
[[1,4,5,6,7],[2],[3],[8]] => 5
[[1,3,5,6,7],[2],[4],[8]] => 10
[[1,2,5,6,7],[3],[4],[8]] => 4
[[1,3,4,6,7],[2],[5],[8]] => 9
[[1,2,4,6,7],[3],[5],[8]] => 8
[[1,2,3,6,7],[4],[5],[8]] => 3
[[1,3,4,5,7],[2],[6],[8]] => 8
[[1,2,4,5,7],[3],[6],[8]] => 7
[[1,2,3,5,7],[4],[6],[8]] => 6
[[1,2,3,4,7],[5],[6],[8]] => 2
[[1,3,4,5,6],[2],[7],[8]] => 6
[[1,2,4,5,6],[3],[7],[8]] => 5
[[1,2,3,5,6],[4],[7],[8]] => 4
[[1,2,3,4,6],[5],[7],[8]] => 3
[[1,2,3,4,5],[6],[7],[8]] => 0
[[1,3,5,7],[2,4,6,8]] => 12
[[1,2,5,7],[3,4,6,8]] => 6
[[1,3,4,7],[2,5,6,8]] => 8
[[1,2,4,7],[3,5,6,8]] => 7
[[1,2,3,7],[4,5,6,8]] => 2
[[1,3,5,6],[2,4,7,8]] => 10
[[1,2,5,6],[3,4,7,8]] => 4
[[1,3,4,6],[2,5,7,8]] => 9
[[1,2,4,6],[3,5,7,8]] => 8
[[1,2,3,6],[4,5,7,8]] => 3
[[1,3,4,5],[2,6,7,8]] => 6
[[1,2,4,5],[3,6,7,8]] => 5
[[1,2,3,5],[4,6,7,8]] => 4
[[1,2,3,4],[5,6,7,8]] => 0
[[1,4,6,8],[2,5,7],[3]] => 9
[[1,3,6,8],[2,5,7],[4]] => 14
[[1,2,6,8],[3,5,7],[4]] => 8
[[1,3,6,8],[2,4,7],[5]] => 10
[[1,2,6,8],[3,4,7],[5]] => 4
[[1,4,5,8],[2,6,7],[3]] => 6
[[1,3,5,8],[2,6,7],[4]] => 11
[[1,2,5,8],[3,6,7],[4]] => 5
[[1,3,4,8],[2,6,7],[5]] => 10
[[1,2,4,8],[3,6,7],[5]] => 9
[[1,2,3,8],[4,6,7],[5]] => 4
[[1,3,5,8],[2,4,7],[6]] => 13
[[1,2,5,8],[3,4,7],[6]] => 7
[[1,3,4,8],[2,5,7],[6]] => 9
[[1,2,4,8],[3,5,7],[6]] => 8
[[1,2,3,8],[4,5,7],[6]] => 3
[[1,3,5,8],[2,4,6],[7]] => 11
[[1,2,5,8],[3,4,6],[7]] => 5
[[1,3,4,8],[2,5,6],[7]] => 7
[[1,2,4,8],[3,5,6],[7]] => 6
[[1,2,3,8],[4,5,6],[7]] => 1
[[1,4,6,7],[2,5,8],[3]] => 8
[[1,3,6,7],[2,5,8],[4]] => 13
[[1,2,6,7],[3,5,8],[4]] => 7
[[1,3,6,7],[2,4,8],[5]] => 9
[[1,2,6,7],[3,4,8],[5]] => 3
[[1,4,5,7],[2,6,8],[3]] => 7
[[1,3,5,7],[2,6,8],[4]] => 12
[[1,2,5,7],[3,6,8],[4]] => 6
[[1,3,4,7],[2,6,8],[5]] => 11
[[1,2,4,7],[3,6,8],[5]] => 10
[[1,2,3,7],[4,6,8],[5]] => 5
[[1,3,5,7],[2,4,8],[6]] => 12
[[1,2,5,7],[3,4,8],[6]] => 6
[[1,3,4,7],[2,5,8],[6]] => 8
[[1,2,4,7],[3,5,8],[6]] => 7
[[1,2,3,7],[4,5,8],[6]] => 2
[[1,4,5,6],[2,7,8],[3]] => 5
[[1,3,5,6],[2,7,8],[4]] => 10
[[1,2,5,6],[3,7,8],[4]] => 4
[[1,3,4,6],[2,7,8],[5]] => 9
[[1,2,4,6],[3,7,8],[5]] => 8
[[1,2,3,6],[4,7,8],[5]] => 3
[[1,3,4,5],[2,7,8],[6]] => 8
[[1,2,4,5],[3,7,8],[6]] => 7
[[1,2,3,5],[4,7,8],[6]] => 6
[[1,2,3,4],[5,7,8],[6]] => 2
[[1,3,5,6],[2,4,8],[7]] => 11
[[1,2,5,6],[3,4,8],[7]] => 5
[[1,3,4,6],[2,5,8],[7]] => 10
[[1,2,4,6],[3,5,8],[7]] => 9
[[1,2,3,6],[4,5,8],[7]] => 4
[[1,3,4,5],[2,6,8],[7]] => 7
[[1,2,4,5],[3,6,8],[7]] => 6
[[1,2,3,5],[4,6,8],[7]] => 5
[[1,2,3,4],[5,6,8],[7]] => 1
[[1,3,5,7],[2,4,6],[8]] => 12
[[1,2,5,7],[3,4,6],[8]] => 6
[[1,3,4,7],[2,5,6],[8]] => 8
[[1,2,4,7],[3,5,6],[8]] => 7
[[1,2,3,7],[4,5,6],[8]] => 2
[[1,3,5,6],[2,4,7],[8]] => 10
[[1,2,5,6],[3,4,7],[8]] => 4
[[1,3,4,6],[2,5,7],[8]] => 9
[[1,2,4,6],[3,5,7],[8]] => 8
[[1,2,3,6],[4,5,7],[8]] => 3
[[1,3,4,5],[2,6,7],[8]] => 6
[[1,2,4,5],[3,6,7],[8]] => 5
[[1,2,3,5],[4,6,7],[8]] => 4
[[1,2,3,4],[5,6,7],[8]] => 0
[[1,4,7,8],[2,5],[3,6]] => 7
[[1,3,7,8],[2,5],[4,6]] => 12
[[1,2,7,8],[3,5],[4,6]] => 6
[[1,3,7,8],[2,4],[5,6]] => 8
[[1,2,7,8],[3,4],[5,6]] => 2
[[1,4,6,8],[2,5],[3,7]] => 9
[[1,3,6,8],[2,5],[4,7]] => 14
[[1,2,6,8],[3,5],[4,7]] => 8
[[1,3,6,8],[2,4],[5,7]] => 10
[[1,2,6,8],[3,4],[5,7]] => 4
[[1,4,5,8],[2,6],[3,7]] => 6
[[1,3,5,8],[2,6],[4,7]] => 11
[[1,2,5,8],[3,6],[4,7]] => 5
[[1,3,4,8],[2,6],[5,7]] => 10
[[1,2,4,8],[3,6],[5,7]] => 9
[[1,2,3,8],[4,6],[5,7]] => 4
[[1,3,5,8],[2,4],[6,7]] => 11
[[1,2,5,8],[3,4],[6,7]] => 5
[[1,3,4,8],[2,5],[6,7]] => 7
[[1,2,4,8],[3,5],[6,7]] => 6
[[1,2,3,8],[4,5],[6,7]] => 1
[[1,4,6,7],[2,5],[3,8]] => 8
[[1,3,6,7],[2,5],[4,8]] => 13
[[1,2,6,7],[3,5],[4,8]] => 7
[[1,3,6,7],[2,4],[5,8]] => 9
[[1,2,6,7],[3,4],[5,8]] => 3
[[1,4,5,7],[2,6],[3,8]] => 7
[[1,3,5,7],[2,6],[4,8]] => 12
[[1,2,5,7],[3,6],[4,8]] => 6
[[1,3,4,7],[2,6],[5,8]] => 11
[[1,2,4,7],[3,6],[5,8]] => 10
[[1,2,3,7],[4,6],[5,8]] => 5
[[1,3,5,7],[2,4],[6,8]] => 12
[[1,2,5,7],[3,4],[6,8]] => 6
[[1,3,4,7],[2,5],[6,8]] => 8
[[1,2,4,7],[3,5],[6,8]] => 7
[[1,2,3,7],[4,5],[6,8]] => 2
[[1,4,5,6],[2,7],[3,8]] => 5
[[1,3,5,6],[2,7],[4,8]] => 10
[[1,2,5,6],[3,7],[4,8]] => 4
[[1,3,4,6],[2,7],[5,8]] => 9
[[1,2,4,6],[3,7],[5,8]] => 8
[[1,2,3,6],[4,7],[5,8]] => 3
[[1,3,4,5],[2,7],[6,8]] => 8
[[1,2,4,5],[3,7],[6,8]] => 7
[[1,2,3,5],[4,7],[6,8]] => 6
[[1,2,3,4],[5,7],[6,8]] => 2
[[1,3,5,6],[2,4],[7,8]] => 10
[[1,2,5,6],[3,4],[7,8]] => 4
[[1,3,4,6],[2,5],[7,8]] => 9
[[1,2,4,6],[3,5],[7,8]] => 8
[[1,2,3,6],[4,5],[7,8]] => 3
[[1,3,4,5],[2,6],[7,8]] => 6
[[1,2,4,5],[3,6],[7,8]] => 5
[[1,2,3,5],[4,6],[7,8]] => 4
[[1,2,3,4],[5,6],[7,8]] => 0
[[1,5,7,8],[2,6],[3],[4]] => 6
[[1,4,7,8],[2,6],[3],[5]] => 10
[[1,3,7,8],[2,6],[4],[5]] => 11
[[1,2,7,8],[3,6],[4],[5]] => 5
[[1,4,7,8],[2,5],[3],[6]] => 7
[[1,3,7,8],[2,5],[4],[6]] => 12
[[1,2,7,8],[3,5],[4],[6]] => 6
[[1,3,7,8],[2,4],[5],[6]] => 8
[[1,2,7,8],[3,4],[5],[6]] => 2
[[1,5,6,8],[2,7],[3],[4]] => 5
[[1,4,6,8],[2,7],[3],[5]] => 9
[[1,3,6,8],[2,7],[4],[5]] => 10
[[1,2,6,8],[3,7],[4],[5]] => 4
[[1,4,5,8],[2,7],[3],[6]] => 8
[[1,3,5,8],[2,7],[4],[6]] => 13
[[1,2,5,8],[3,7],[4],[6]] => 7
[[1,3,4,8],[2,7],[5],[6]] => 9
[[1,2,4,8],[3,7],[5],[6]] => 8
[[1,2,3,8],[4,7],[5],[6]] => 3
[[1,4,6,8],[2,5],[3],[7]] => 9
[[1,3,6,8],[2,5],[4],[7]] => 14
[[1,2,6,8],[3,5],[4],[7]] => 8
[[1,3,6,8],[2,4],[5],[7]] => 10
[[1,2,6,8],[3,4],[5],[7]] => 4
[[1,4,5,8],[2,6],[3],[7]] => 6
[[1,3,5,8],[2,6],[4],[7]] => 11
[[1,2,5,8],[3,6],[4],[7]] => 5
[[1,3,4,8],[2,6],[5],[7]] => 10
[[1,2,4,8],[3,6],[5],[7]] => 9
[[1,2,3,8],[4,6],[5],[7]] => 4
[[1,3,5,8],[2,4],[6],[7]] => 11
[[1,2,5,8],[3,4],[6],[7]] => 5
[[1,3,4,8],[2,5],[6],[7]] => 7
[[1,2,4,8],[3,5],[6],[7]] => 6
[[1,2,3,8],[4,5],[6],[7]] => 1
[[1,5,6,7],[2,8],[3],[4]] => 4
[[1,4,6,7],[2,8],[3],[5]] => 8
[[1,3,6,7],[2,8],[4],[5]] => 9
[[1,2,6,7],[3,8],[4],[5]] => 3
[[1,4,5,7],[2,8],[3],[6]] => 7
[[1,3,5,7],[2,8],[4],[6]] => 12
[[1,2,5,7],[3,8],[4],[6]] => 6
[[1,3,4,7],[2,8],[5],[6]] => 8
[[1,2,4,7],[3,8],[5],[6]] => 7
[[1,2,3,7],[4,8],[5],[6]] => 2
[[1,4,5,6],[2,8],[3],[7]] => 6
[[1,3,5,6],[2,8],[4],[7]] => 11
[[1,2,5,6],[3,8],[4],[7]] => 5
[[1,3,4,6],[2,8],[5],[7]] => 10
[[1,2,4,6],[3,8],[5],[7]] => 9
[[1,2,3,6],[4,8],[5],[7]] => 4
[[1,3,4,5],[2,8],[6],[7]] => 7
[[1,2,4,5],[3,8],[6],[7]] => 6
[[1,2,3,5],[4,8],[6],[7]] => 5
[[1,2,3,4],[5,8],[6],[7]] => 1
[[1,4,6,7],[2,5],[3],[8]] => 8
[[1,3,6,7],[2,5],[4],[8]] => 13
[[1,2,6,7],[3,5],[4],[8]] => 7
[[1,3,6,7],[2,4],[5],[8]] => 9
[[1,2,6,7],[3,4],[5],[8]] => 3
[[1,4,5,7],[2,6],[3],[8]] => 7
[[1,3,5,7],[2,6],[4],[8]] => 12
[[1,2,5,7],[3,6],[4],[8]] => 6
[[1,3,4,7],[2,6],[5],[8]] => 11
[[1,2,4,7],[3,6],[5],[8]] => 10
[[1,2,3,7],[4,6],[5],[8]] => 5
[[1,3,5,7],[2,4],[6],[8]] => 12
[[1,2,5,7],[3,4],[6],[8]] => 6
[[1,3,4,7],[2,5],[6],[8]] => 8
[[1,2,4,7],[3,5],[6],[8]] => 7
[[1,2,3,7],[4,5],[6],[8]] => 2
[[1,4,5,6],[2,7],[3],[8]] => 5
[[1,3,5,6],[2,7],[4],[8]] => 10
[[1,2,5,6],[3,7],[4],[8]] => 4
[[1,3,4,6],[2,7],[5],[8]] => 9
[[1,2,4,6],[3,7],[5],[8]] => 8
[[1,2,3,6],[4,7],[5],[8]] => 3
[[1,3,4,5],[2,7],[6],[8]] => 8
[[1,2,4,5],[3,7],[6],[8]] => 7
[[1,2,3,5],[4,7],[6],[8]] => 6
[[1,2,3,4],[5,7],[6],[8]] => 2
[[1,3,5,6],[2,4],[7],[8]] => 10
[[1,2,5,6],[3,4],[7],[8]] => 4
[[1,3,4,6],[2,5],[7],[8]] => 9
[[1,2,4,6],[3,5],[7],[8]] => 8
[[1,2,3,6],[4,5],[7],[8]] => 3
[[1,3,4,5],[2,6],[7],[8]] => 6
[[1,2,4,5],[3,6],[7],[8]] => 5
[[1,2,3,5],[4,6],[7],[8]] => 4
[[1,2,3,4],[5,6],[7],[8]] => 0
[[1,6,7,8],[2],[3],[4],[5]] => 3
[[1,5,7,8],[2],[3],[4],[6]] => 6
[[1,4,7,8],[2],[3],[5],[6]] => 7
[[1,3,7,8],[2],[4],[5],[6]] => 8
[[1,2,7,8],[3],[4],[5],[6]] => 2
[[1,5,6,8],[2],[3],[4],[7]] => 5
[[1,4,6,8],[2],[3],[5],[7]] => 9
[[1,3,6,8],[2],[4],[5],[7]] => 10
[[1,2,6,8],[3],[4],[5],[7]] => 4
[[1,4,5,8],[2],[3],[6],[7]] => 6
[[1,3,5,8],[2],[4],[6],[7]] => 11
[[1,2,5,8],[3],[4],[6],[7]] => 5
[[1,3,4,8],[2],[5],[6],[7]] => 7
[[1,2,4,8],[3],[5],[6],[7]] => 6
[[1,2,3,8],[4],[5],[6],[7]] => 1
[[1,5,6,7],[2],[3],[4],[8]] => 4
[[1,4,6,7],[2],[3],[5],[8]] => 8
[[1,3,6,7],[2],[4],[5],[8]] => 9
[[1,2,6,7],[3],[4],[5],[8]] => 3
[[1,4,5,7],[2],[3],[6],[8]] => 7
[[1,3,5,7],[2],[4],[6],[8]] => 12
[[1,2,5,7],[3],[4],[6],[8]] => 6
[[1,3,4,7],[2],[5],[6],[8]] => 8
[[1,2,4,7],[3],[5],[6],[8]] => 7
[[1,2,3,7],[4],[5],[6],[8]] => 2
[[1,4,5,6],[2],[3],[7],[8]] => 5
[[1,3,5,6],[2],[4],[7],[8]] => 10
[[1,2,5,6],[3],[4],[7],[8]] => 4
[[1,3,4,6],[2],[5],[7],[8]] => 9
[[1,2,4,6],[3],[5],[7],[8]] => 8
[[1,2,3,6],[4],[5],[7],[8]] => 3
[[1,3,4,5],[2],[6],[7],[8]] => 6
[[1,2,4,5],[3],[6],[7],[8]] => 5
[[1,2,3,5],[4],[6],[7],[8]] => 4
[[1,2,3,4],[5],[6],[7],[8]] => 0
[[1,4,7],[2,5,8],[3,6]] => 7
[[1,3,7],[2,5,8],[4,6]] => 12
[[1,2,7],[3,5,8],[4,6]] => 6
[[1,3,7],[2,4,8],[5,6]] => 8
[[1,2,7],[3,4,8],[5,6]] => 2
[[1,4,6],[2,5,8],[3,7]] => 9
[[1,3,6],[2,5,8],[4,7]] => 14
[[1,2,6],[3,5,8],[4,7]] => 8
[[1,3,6],[2,4,8],[5,7]] => 10
[[1,2,6],[3,4,8],[5,7]] => 4
[[1,4,5],[2,6,8],[3,7]] => 6
[[1,3,5],[2,6,8],[4,7]] => 11
[[1,2,5],[3,6,8],[4,7]] => 5
[[1,3,4],[2,6,8],[5,7]] => 10
[[1,2,4],[3,6,8],[5,7]] => 9
[[1,2,3],[4,6,8],[5,7]] => 4
[[1,3,5],[2,4,8],[6,7]] => 11
[[1,2,5],[3,4,8],[6,7]] => 5
[[1,3,4],[2,5,8],[6,7]] => 7
[[1,2,4],[3,5,8],[6,7]] => 6
[[1,2,3],[4,5,8],[6,7]] => 1
[[1,4,6],[2,5,7],[3,8]] => 8
[[1,3,6],[2,5,7],[4,8]] => 13
[[1,2,6],[3,5,7],[4,8]] => 7
[[1,3,6],[2,4,7],[5,8]] => 9
[[1,2,6],[3,4,7],[5,8]] => 3
[[1,4,5],[2,6,7],[3,8]] => 5
[[1,3,5],[2,6,7],[4,8]] => 10
[[1,2,5],[3,6,7],[4,8]] => 4
[[1,3,4],[2,6,7],[5,8]] => 9
[[1,2,4],[3,6,7],[5,8]] => 8
[[1,2,3],[4,6,7],[5,8]] => 3
[[1,3,5],[2,4,7],[6,8]] => 12
[[1,2,5],[3,4,7],[6,8]] => 6
[[1,3,4],[2,5,7],[6,8]] => 8
[[1,2,4],[3,5,7],[6,8]] => 7
[[1,2,3],[4,5,7],[6,8]] => 2
[[1,3,5],[2,4,6],[7,8]] => 10
[[1,2,5],[3,4,6],[7,8]] => 4
[[1,3,4],[2,5,6],[7,8]] => 6
[[1,2,4],[3,5,6],[7,8]] => 5
[[1,2,3],[4,5,6],[7,8]] => 0
[[1,5,7],[2,6,8],[3],[4]] => 6
[[1,4,7],[2,6,8],[3],[5]] => 10
[[1,3,7],[2,6,8],[4],[5]] => 11
[[1,2,7],[3,6,8],[4],[5]] => 5
[[1,4,7],[2,5,8],[3],[6]] => 7
[[1,3,7],[2,5,8],[4],[6]] => 12
[[1,2,7],[3,5,8],[4],[6]] => 6
[[1,3,7],[2,4,8],[5],[6]] => 8
[[1,2,7],[3,4,8],[5],[6]] => 2
[[1,5,6],[2,7,8],[3],[4]] => 4
[[1,4,6],[2,7,8],[3],[5]] => 8
[[1,3,6],[2,7,8],[4],[5]] => 9
[[1,2,6],[3,7,8],[4],[5]] => 3
[[1,4,5],[2,7,8],[3],[6]] => 7
[[1,3,5],[2,7,8],[4],[6]] => 12
[[1,2,5],[3,7,8],[4],[6]] => 6
[[1,3,4],[2,7,8],[5],[6]] => 8
[[1,2,4],[3,7,8],[5],[6]] => 7
[[1,2,3],[4,7,8],[5],[6]] => 2
[[1,4,6],[2,5,8],[3],[7]] => 9
[[1,3,6],[2,5,8],[4],[7]] => 14
[[1,2,6],[3,5,8],[4],[7]] => 8
[[1,3,6],[2,4,8],[5],[7]] => 10
[[1,2,6],[3,4,8],[5],[7]] => 4
[[1,4,5],[2,6,8],[3],[7]] => 6
[[1,3,5],[2,6,8],[4],[7]] => 11
[[1,2,5],[3,6,8],[4],[7]] => 5
[[1,3,4],[2,6,8],[5],[7]] => 10
[[1,2,4],[3,6,8],[5],[7]] => 9
[[1,2,3],[4,6,8],[5],[7]] => 4
[[1,3,5],[2,4,8],[6],[7]] => 11
[[1,2,5],[3,4,8],[6],[7]] => 5
[[1,3,4],[2,5,8],[6],[7]] => 7
[[1,2,4],[3,5,8],[6],[7]] => 6
[[1,2,3],[4,5,8],[6],[7]] => 1
[[1,4,6],[2,5,7],[3],[8]] => 8
[[1,3,6],[2,5,7],[4],[8]] => 13
[[1,2,6],[3,5,7],[4],[8]] => 7
[[1,3,6],[2,4,7],[5],[8]] => 9
[[1,2,6],[3,4,7],[5],[8]] => 3
[[1,4,5],[2,6,7],[3],[8]] => 5
[[1,3,5],[2,6,7],[4],[8]] => 10
[[1,2,5],[3,6,7],[4],[8]] => 4
[[1,3,4],[2,6,7],[5],[8]] => 9
[[1,2,4],[3,6,7],[5],[8]] => 8
[[1,2,3],[4,6,7],[5],[8]] => 3
[[1,3,5],[2,4,7],[6],[8]] => 12
[[1,2,5],[3,4,7],[6],[8]] => 6
[[1,3,4],[2,5,7],[6],[8]] => 8
[[1,2,4],[3,5,7],[6],[8]] => 7
[[1,2,3],[4,5,7],[6],[8]] => 2
[[1,3,5],[2,4,6],[7],[8]] => 10
[[1,2,5],[3,4,6],[7],[8]] => 4
[[1,3,4],[2,5,6],[7],[8]] => 6
[[1,2,4],[3,5,6],[7],[8]] => 5
[[1,2,3],[4,5,6],[7],[8]] => 0
[[1,5,8],[2,6],[3,7],[4]] => 5
[[1,4,8],[2,6],[3,7],[5]] => 9
[[1,3,8],[2,6],[4,7],[5]] => 10
[[1,2,8],[3,6],[4,7],[5]] => 4
[[1,4,8],[2,5],[3,7],[6]] => 8
[[1,3,8],[2,5],[4,7],[6]] => 13
[[1,2,8],[3,5],[4,7],[6]] => 7
[[1,3,8],[2,4],[5,7],[6]] => 9
[[1,2,8],[3,4],[5,7],[6]] => 3
[[1,4,8],[2,5],[3,6],[7]] => 6
[[1,3,8],[2,5],[4,6],[7]] => 11
[[1,2,8],[3,5],[4,6],[7]] => 5
[[1,3,8],[2,4],[5,6],[7]] => 7
[[1,2,8],[3,4],[5,6],[7]] => 1
[[1,5,7],[2,6],[3,8],[4]] => 6
[[1,4,7],[2,6],[3,8],[5]] => 10
[[1,3,7],[2,6],[4,8],[5]] => 11
[[1,2,7],[3,6],[4,8],[5]] => 5
[[1,4,7],[2,5],[3,8],[6]] => 7
[[1,3,7],[2,5],[4,8],[6]] => 12
[[1,2,7],[3,5],[4,8],[6]] => 6
[[1,3,7],[2,4],[5,8],[6]] => 8
[[1,2,7],[3,4],[5,8],[6]] => 2
[[1,5,6],[2,7],[3,8],[4]] => 4
[[1,4,6],[2,7],[3,8],[5]] => 8
[[1,3,6],[2,7],[4,8],[5]] => 9
[[1,2,6],[3,7],[4,8],[5]] => 3
[[1,4,5],[2,7],[3,8],[6]] => 7
[[1,3,5],[2,7],[4,8],[6]] => 12
[[1,2,5],[3,7],[4,8],[6]] => 6
[[1,3,4],[2,7],[5,8],[6]] => 8
[[1,2,4],[3,7],[5,8],[6]] => 7
[[1,2,3],[4,7],[5,8],[6]] => 2
[[1,4,6],[2,5],[3,8],[7]] => 9
[[1,3,6],[2,5],[4,8],[7]] => 14
[[1,2,6],[3,5],[4,8],[7]] => 8
[[1,3,6],[2,4],[5,8],[7]] => 10
[[1,2,6],[3,4],[5,8],[7]] => 4
[[1,4,5],[2,6],[3,8],[7]] => 6
[[1,3,5],[2,6],[4,8],[7]] => 11
[[1,2,5],[3,6],[4,8],[7]] => 5
[[1,3,4],[2,6],[5,8],[7]] => 10
[[1,2,4],[3,6],[5,8],[7]] => 9
[[1,2,3],[4,6],[5,8],[7]] => 4
[[1,3,5],[2,4],[6,8],[7]] => 11
[[1,2,5],[3,4],[6,8],[7]] => 5
[[1,3,4],[2,5],[6,8],[7]] => 7
[[1,2,4],[3,5],[6,8],[7]] => 6
[[1,2,3],[4,5],[6,8],[7]] => 1
[[1,4,7],[2,5],[3,6],[8]] => 7
[[1,3,7],[2,5],[4,6],[8]] => 12
[[1,2,7],[3,5],[4,6],[8]] => 6
[[1,3,7],[2,4],[5,6],[8]] => 8
[[1,2,7],[3,4],[5,6],[8]] => 2
[[1,4,6],[2,5],[3,7],[8]] => 8
[[1,3,6],[2,5],[4,7],[8]] => 13
[[1,2,6],[3,5],[4,7],[8]] => 7
[[1,3,6],[2,4],[5,7],[8]] => 9
[[1,2,6],[3,4],[5,7],[8]] => 3
[[1,4,5],[2,6],[3,7],[8]] => 5
[[1,3,5],[2,6],[4,7],[8]] => 10
[[1,2,5],[3,6],[4,7],[8]] => 4
[[1,3,4],[2,6],[5,7],[8]] => 9
[[1,2,4],[3,6],[5,7],[8]] => 8
[[1,2,3],[4,6],[5,7],[8]] => 3
[[1,3,5],[2,4],[6,7],[8]] => 10
[[1,2,5],[3,4],[6,7],[8]] => 4
[[1,3,4],[2,5],[6,7],[8]] => 6
[[1,2,4],[3,5],[6,7],[8]] => 5
[[1,2,3],[4,5],[6,7],[8]] => 0
[[1,6,8],[2,7],[3],[4],[5]] => 4
[[1,5,8],[2,7],[3],[4],[6]] => 7
[[1,4,8],[2,7],[3],[5],[6]] => 8
[[1,3,8],[2,7],[4],[5],[6]] => 9
[[1,2,8],[3,7],[4],[5],[6]] => 3
[[1,5,8],[2,6],[3],[4],[7]] => 5
[[1,4,8],[2,6],[3],[5],[7]] => 9
[[1,3,8],[2,6],[4],[5],[7]] => 10
[[1,2,8],[3,6],[4],[5],[7]] => 4
[[1,4,8],[2,5],[3],[6],[7]] => 6
[[1,3,8],[2,5],[4],[6],[7]] => 11
[[1,2,8],[3,5],[4],[6],[7]] => 5
[[1,3,8],[2,4],[5],[6],[7]] => 7
[[1,2,8],[3,4],[5],[6],[7]] => 1
[[1,6,7],[2,8],[3],[4],[5]] => 3
[[1,5,7],[2,8],[3],[4],[6]] => 6
[[1,4,7],[2,8],[3],[5],[6]] => 7
[[1,3,7],[2,8],[4],[5],[6]] => 8
[[1,2,7],[3,8],[4],[5],[6]] => 2
[[1,5,6],[2,8],[3],[4],[7]] => 5
[[1,4,6],[2,8],[3],[5],[7]] => 9
[[1,3,6],[2,8],[4],[5],[7]] => 10
[[1,2,6],[3,8],[4],[5],[7]] => 4
[[1,4,5],[2,8],[3],[6],[7]] => 6
[[1,3,5],[2,8],[4],[6],[7]] => 11
[[1,2,5],[3,8],[4],[6],[7]] => 5
[[1,3,4],[2,8],[5],[6],[7]] => 7
[[1,2,4],[3,8],[5],[6],[7]] => 6
[[1,2,3],[4,8],[5],[6],[7]] => 1
[[1,5,7],[2,6],[3],[4],[8]] => 6
[[1,4,7],[2,6],[3],[5],[8]] => 10
[[1,3,7],[2,6],[4],[5],[8]] => 11
[[1,2,7],[3,6],[4],[5],[8]] => 5
[[1,4,7],[2,5],[3],[6],[8]] => 7
[[1,3,7],[2,5],[4],[6],[8]] => 12
[[1,2,7],[3,5],[4],[6],[8]] => 6
[[1,3,7],[2,4],[5],[6],[8]] => 8
[[1,2,7],[3,4],[5],[6],[8]] => 2
[[1,5,6],[2,7],[3],[4],[8]] => 4
[[1,4,6],[2,7],[3],[5],[8]] => 8
[[1,3,6],[2,7],[4],[5],[8]] => 9
[[1,2,6],[3,7],[4],[5],[8]] => 3
[[1,4,5],[2,7],[3],[6],[8]] => 7
[[1,3,5],[2,7],[4],[6],[8]] => 12
[[1,2,5],[3,7],[4],[6],[8]] => 6
[[1,3,4],[2,7],[5],[6],[8]] => 8
[[1,2,4],[3,7],[5],[6],[8]] => 7
[[1,2,3],[4,7],[5],[6],[8]] => 2
[[1,4,6],[2,5],[3],[7],[8]] => 8
[[1,3,6],[2,5],[4],[7],[8]] => 13
[[1,2,6],[3,5],[4],[7],[8]] => 7
[[1,3,6],[2,4],[5],[7],[8]] => 9
[[1,2,6],[3,4],[5],[7],[8]] => 3
[[1,4,5],[2,6],[3],[7],[8]] => 5
[[1,3,5],[2,6],[4],[7],[8]] => 10
[[1,2,5],[3,6],[4],[7],[8]] => 4
[[1,3,4],[2,6],[5],[7],[8]] => 9
[[1,2,4],[3,6],[5],[7],[8]] => 8
[[1,2,3],[4,6],[5],[7],[8]] => 3
[[1,3,5],[2,4],[6],[7],[8]] => 10
[[1,2,5],[3,4],[6],[7],[8]] => 4
[[1,3,4],[2,5],[6],[7],[8]] => 6
[[1,2,4],[3,5],[6],[7],[8]] => 5
[[1,2,3],[4,5],[6],[7],[8]] => 0
[[1,7,8],[2],[3],[4],[5],[6]] => 2
[[1,6,8],[2],[3],[4],[5],[7]] => 4
[[1,5,8],[2],[3],[4],[6],[7]] => 5
[[1,4,8],[2],[3],[5],[6],[7]] => 6
[[1,3,8],[2],[4],[5],[6],[7]] => 7
[[1,2,8],[3],[4],[5],[6],[7]] => 1
[[1,6,7],[2],[3],[4],[5],[8]] => 3
[[1,5,7],[2],[3],[4],[6],[8]] => 6
[[1,4,7],[2],[3],[5],[6],[8]] => 7
[[1,3,7],[2],[4],[5],[6],[8]] => 8
[[1,2,7],[3],[4],[5],[6],[8]] => 2
[[1,5,6],[2],[3],[4],[7],[8]] => 4
[[1,4,6],[2],[3],[5],[7],[8]] => 8
[[1,3,6],[2],[4],[5],[7],[8]] => 9
[[1,2,6],[3],[4],[5],[7],[8]] => 3
[[1,4,5],[2],[3],[6],[7],[8]] => 5
[[1,3,5],[2],[4],[6],[7],[8]] => 10
[[1,2,5],[3],[4],[6],[7],[8]] => 4
[[1,3,4],[2],[5],[6],[7],[8]] => 6
[[1,2,4],[3],[5],[6],[7],[8]] => 5
[[1,2,3],[4],[5],[6],[7],[8]] => 0
[[1,5],[2,6],[3,7],[4,8]] => 4
[[1,4],[2,6],[3,7],[5,8]] => 8
[[1,3],[2,6],[4,7],[5,8]] => 9
[[1,2],[3,6],[4,7],[5,8]] => 3
[[1,4],[2,5],[3,7],[6,8]] => 7
[[1,3],[2,5],[4,7],[6,8]] => 12
[[1,2],[3,5],[4,7],[6,8]] => 6
[[1,3],[2,4],[5,7],[6,8]] => 8
[[1,2],[3,4],[5,7],[6,8]] => 2
[[1,4],[2,5],[3,6],[7,8]] => 5
[[1,3],[2,5],[4,6],[7,8]] => 10
[[1,2],[3,5],[4,6],[7,8]] => 4
[[1,3],[2,4],[5,6],[7,8]] => 6
[[1,2],[3,4],[5,6],[7,8]] => 0
[[1,6],[2,7],[3,8],[4],[5]] => 3
[[1,5],[2,7],[3,8],[4],[6]] => 6
[[1,4],[2,7],[3,8],[5],[6]] => 7
[[1,3],[2,7],[4,8],[5],[6]] => 8
[[1,2],[3,7],[4,8],[5],[6]] => 2
[[1,5],[2,6],[3,8],[4],[7]] => 5
[[1,4],[2,6],[3,8],[5],[7]] => 9
[[1,3],[2,6],[4,8],[5],[7]] => 10
[[1,2],[3,6],[4,8],[5],[7]] => 4
[[1,4],[2,5],[3,8],[6],[7]] => 6
[[1,3],[2,5],[4,8],[6],[7]] => 11
[[1,2],[3,5],[4,8],[6],[7]] => 5
[[1,3],[2,4],[5,8],[6],[7]] => 7
[[1,2],[3,4],[5,8],[6],[7]] => 1
[[1,5],[2,6],[3,7],[4],[8]] => 4
[[1,4],[2,6],[3,7],[5],[8]] => 8
[[1,3],[2,6],[4,7],[5],[8]] => 9
[[1,2],[3,6],[4,7],[5],[8]] => 3
[[1,4],[2,5],[3,7],[6],[8]] => 7
[[1,3],[2,5],[4,7],[6],[8]] => 12
[[1,2],[3,5],[4,7],[6],[8]] => 6
[[1,3],[2,4],[5,7],[6],[8]] => 8
[[1,2],[3,4],[5,7],[6],[8]] => 2
[[1,4],[2,5],[3,6],[7],[8]] => 5
[[1,3],[2,5],[4,6],[7],[8]] => 10
[[1,2],[3,5],[4,6],[7],[8]] => 4
[[1,3],[2,4],[5,6],[7],[8]] => 6
[[1,2],[3,4],[5,6],[7],[8]] => 0
[[1,7],[2,8],[3],[4],[5],[6]] => 2
[[1,6],[2,8],[3],[4],[5],[7]] => 4
[[1,5],[2,8],[3],[4],[6],[7]] => 5
[[1,4],[2,8],[3],[5],[6],[7]] => 6
[[1,3],[2,8],[4],[5],[6],[7]] => 7
[[1,2],[3,8],[4],[5],[6],[7]] => 1
[[1,6],[2,7],[3],[4],[5],[8]] => 3
[[1,5],[2,7],[3],[4],[6],[8]] => 6
[[1,4],[2,7],[3],[5],[6],[8]] => 7
[[1,3],[2,7],[4],[5],[6],[8]] => 8
[[1,2],[3,7],[4],[5],[6],[8]] => 2
[[1,5],[2,6],[3],[4],[7],[8]] => 4
[[1,4],[2,6],[3],[5],[7],[8]] => 8
[[1,3],[2,6],[4],[5],[7],[8]] => 9
[[1,2],[3,6],[4],[5],[7],[8]] => 3
[[1,4],[2,5],[3],[6],[7],[8]] => 5
[[1,3],[2,5],[4],[6],[7],[8]] => 10
[[1,2],[3,5],[4],[6],[7],[8]] => 4
[[1,3],[2,4],[5],[6],[7],[8]] => 6
[[1,2],[3,4],[5],[6],[7],[8]] => 0
[[1,8],[2],[3],[4],[5],[6],[7]] => 1
[[1,7],[2],[3],[4],[5],[6],[8]] => 2
[[1,6],[2],[3],[4],[5],[7],[8]] => 3
[[1,5],[2],[3],[4],[6],[7],[8]] => 4
[[1,4],[2],[3],[5],[6],[7],[8]] => 5
[[1,3],[2],[4],[5],[6],[7],[8]] => 6
[[1,2],[3],[4],[5],[6],[7],[8]] => 0
[[1],[2],[3],[4],[5],[6],[7],[8]] => 0
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Generating function
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Search the OEIS for these generating functions
Search the Online Encyclopedia of Integer
Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
3,1 5,2,3 7,5,6,5,3 11,8,14,13,15,6,8,1 15,15,24,29,35,33,31,20,17,9,4 22,23,44,53,78,79,98,82,86,65,58,31,28,11,6
$F_{1} = 1$
$F_{2} = 2$
$F_{3} = 3 + q$
$F_{4} = 5 + 2\ q + 3\ q^{2}$
$F_{5} = 7 + 5\ q + 6\ q^{2} + 5\ q^{3} + 3\ q^{4}$
$F_{6} = 11 + 8\ q + 14\ q^{2} + 13\ q^{3} + 15\ q^{4} + 6\ q^{5} + 8\ q^{6} + q^{7}$
$F_{7} = 15 + 15\ q + 24\ q^{2} + 29\ q^{3} + 35\ q^{4} + 33\ q^{5} + 31\ q^{6} + 20\ q^{7} + 17\ q^{8} + 9\ q^{9} + 4\ q^{10}$
$F_{8} = 22 + 23\ q + 44\ q^{2} + 53\ q^{3} + 78\ q^{4} + 79\ q^{5} + 98\ q^{6} + 82\ q^{7} + 86\ q^{8} + 65\ q^{9} + 58\ q^{10} + 31\ q^{11} + 28\ q^{12} + 11\ q^{13} + 6\ q^{14}$
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
References
[1] Hopkins, S. Two majs for standard Young tableaux? MathOverflow:385374
Code
def natural_descents(T):
n = T.size()
D = []
for i in range(1, n):
for row in T:
if row.count(i):
break
if row.count(i+1):
D.append(i)
break
return D
def statistic(T):
n = T.size()
D = natural_descents(T)
return sum(n-d for d in D)
Created
Mar 04, 2021 at 08:46 by Martin Rubey
Updated
Mar 04, 2021 at 08:46 by Martin Rubey
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