Identifier
Values
[[1,2]] => [1,2] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [2,1] => [2,1] => 1
[[1,2,3]] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[[1,2],[3]] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[[1,2,4],[3]] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[[1,2,3],[4]] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[[1,3],[2,4]] => [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 4
[[1,2],[3,4]] => [3,4,1,2] => [1,3,4,2] => [1,4,2,3] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 5
[[1,3],[2],[4]] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 4
[[1,2],[3],[4]] => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 3
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 4
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 6
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 3
[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => 5
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,4,5,3] => [1,2,5,3,4] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 7
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,1,3,5] => [3,1,4,2,5] => 6
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 5
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => 5
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 4
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,2,1,5,4] => [3,2,1,5,4] => 8
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => [3,1,5,2,4] => 6
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,4,3,5,2] => [1,5,3,2,4] => 5
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,1,5,4,3] => [2,1,5,4,3] => 7
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,5,4,2] => [1,5,2,4,3] => 4
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 9
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => [4,3,1,5,2] => 8
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => [4,1,5,3,2] => 7
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 5
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 4
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 3
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 2
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 8
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 4
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 7
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 6
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => 3
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 6
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 5
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 4
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 9
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => 8
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => 7
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,1,5,3,4,6] => [2,1,4,5,3,6] => 7
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,3,5,2,4,6] => [1,4,2,5,3,6] => 6
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => 5
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,1,3,6,4,5] => [2,1,3,5,6,4] => 6
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => [1,3,2,5,6,4] => 5
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,4,6,3,5] => [1,2,5,3,6,4] => 4
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => 3
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 9
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,3,4,2,6,5] => [1,4,2,3,6,5] => 5
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => 7
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,3,2,5,6,4] => [1,3,2,6,4,5] => 6
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => 11
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,4,1,5,3,6] => [3,1,5,2,4,6] => 8
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,4,3,5,2,6] => [1,5,3,2,4,6] => 7
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => 10
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,3,5,4,2,6] => [1,5,2,4,3,6] => 6
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [3,2,1,4,6,5] => [3,2,1,4,6,5] => 10
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,4,1,3,6,5] => [3,1,4,2,6,5] => 9
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => 8
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,1,5,3,6,4] => [2,1,4,6,3,5] => 7
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,3,5,2,6,4] => [1,4,2,6,3,5] => 6
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,2,5,4,6,3] => [1,2,6,4,3,5] => 5
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,1,4,6,3,5] => [2,1,5,3,6,4] => 9
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,3,4,6,2,5] => [1,5,2,3,6,4] => 5
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,1,3,6,5,4] => [2,1,3,6,5,4] => 8
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => 7
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => 4
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 12
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [3,5,2,1,4,6] => [4,3,1,5,2,6] => 11
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [2,5,4,1,3,6] => [4,1,5,3,2,6] => 10
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => 9
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => [4,2,1,5,6,3] => 10
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => 9
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,4,6,3,2,5] => [1,5,4,2,6,3] => 8
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [2,1,6,5,3,4] => [2,1,5,6,4,3] => 8
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,3,6,5,2,4] => [1,5,2,6,4,3] => 7
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 6
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 12
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => [3,1,6,2,5,4] => 9
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [1,4,3,6,5,2] => [1,6,3,2,5,4] => 8
>>> Load all 193 entries. <<<
[[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [2,1,5,6,4,3] => [2,1,6,5,3,4] => 10
[[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,3,5,6,4,2] => [1,6,2,5,3,4] => 6
[[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 13
[[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => [4,3,1,6,2,5] => 11
[[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [2,5,4,1,6,3] => [4,1,6,3,2,5] => 10
[[1,2],[3,6],[4],[5]] => [5,4,3,6,1,2] => [1,5,4,3,6,2] => [1,6,4,3,2,5] => 9
[[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [3,2,6,1,5,4] => [4,2,1,6,5,3] => 12
[[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [2,4,6,1,5,3] => [4,1,6,2,5,3] => 9
[[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [1,4,6,3,5,2] => [1,6,4,2,5,3] => 8
[[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 11
[[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => 7
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 14
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [4,6,3,2,1,5] => [5,4,3,1,6,2] => 13
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [3,6,5,2,1,4] => [5,4,1,6,3,2] => 12
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [2,6,5,4,1,3] => [5,1,6,4,3,2] => 11
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 10
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 15
[[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[[1,2,4,5,6,7],[3]] => [3,1,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => 5
[[1,2,3,5,6,7],[4]] => [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => 4
[[1,2,3,4,6,7],[5]] => [5,1,2,3,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => 3
[[1,2,3,4,5,7],[6]] => [6,1,2,3,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => 2
[[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => 5
[[1,2,4,6,7],[3,5]] => [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => 8
[[1,2,3,6,7],[4,5]] => [4,5,1,2,3,6,7] => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => 4
[[1,2,4,5,7],[3,6]] => [3,6,1,2,4,5,7] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => 7
[[1,2,3,5,7],[4,6]] => [4,6,1,2,3,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => 6
[[1,2,3,4,7],[5,6]] => [5,6,1,2,3,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => 3
[[1,2,4,5,6],[3,7]] => [3,7,1,2,4,5,6] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => 6
[[1,2,3,5,6],[4,7]] => [4,7,1,2,3,5,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => 5
[[1,2,3,4,6],[5,7]] => [5,7,1,2,3,4,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => 4
[[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => 2
[[1,2,5,6,7],[3],[4]] => [4,3,1,2,5,6,7] => [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => 9
[[1,2,4,6,7],[3],[5]] => [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => 8
[[1,2,3,6,7],[4],[5]] => [5,4,1,2,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => 7
[[1,2,4,5,7],[3],[6]] => [6,3,1,2,4,5,7] => [1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => 7
[[1,2,3,5,7],[4],[6]] => [6,4,1,2,3,5,7] => [1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => 6
[[1,2,3,4,7],[5],[6]] => [6,5,1,2,3,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => 5
[[1,2,4,5,6],[3],[7]] => [7,3,1,2,4,5,6] => [1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => 6
[[1,2,3,5,6],[4],[7]] => [7,4,1,2,3,5,6] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => 5
[[1,2,3,4,6],[5],[7]] => [7,5,1,2,3,4,6] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => 4
[[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => 3
[[1,2,5,7],[3,4,6]] => [3,4,6,1,2,5,7] => [1,3,4,2,6,5,7] => [1,4,2,3,6,5,7] => 7
[[1,2,4,7],[3,5,6]] => [3,5,6,1,2,4,7] => [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => 8
[[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => 4
[[1,2,5,6],[3,4,7]] => [3,4,7,1,2,5,6] => [1,3,4,2,5,7,6] => [1,4,2,3,5,7,6] => 6
[[1,2,4,6],[3,5,7]] => [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => 9
[[1,2,3,6],[4,5,7]] => [4,5,7,1,2,3,6] => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => 5
[[1,2,4,5],[3,6,7]] => [3,6,7,1,2,4,5] => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => 7
[[1,2,3,5],[4,6,7]] => [4,6,7,1,2,3,5] => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => 6
[[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => 3
[[1,2,5,7],[3,6],[4]] => [4,3,6,1,2,5,7] => [1,4,3,2,6,5,7] => [1,4,3,2,6,5,7] => 11
[[1,2,4,7],[3,6],[5]] => [5,3,6,1,2,4,7] => [1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => 8
[[1,2,3,7],[4,6],[5]] => [5,4,6,1,2,3,7] => [1,2,5,4,6,3,7] => [1,2,6,4,3,5,7] => 7
[[1,2,4,7],[3,5],[6]] => [6,3,5,1,2,4,7] => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => 10
[[1,2,3,7],[4,5],[6]] => [6,4,5,1,2,3,7] => [1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => 6
[[1,2,5,6],[3,7],[4]] => [4,3,7,1,2,5,6] => [1,4,3,2,5,7,6] => [1,4,3,2,5,7,6] => 10
[[1,2,4,6],[3,7],[5]] => [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => [1,4,2,5,3,7,6] => 9
[[1,2,3,6],[4,7],[5]] => [5,4,7,1,2,3,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => 8
[[1,2,4,5],[3,7],[6]] => [6,3,7,1,2,4,5] => [1,3,2,6,4,7,5] => [1,3,2,5,7,4,6] => 7
[[1,2,3,5],[4,7],[6]] => [6,4,7,1,2,3,5] => [1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => 6
[[1,2,3,4],[5,7],[6]] => [6,5,7,1,2,3,4] => [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => 5
[[1,2,5,6],[3,4],[7]] => [7,3,4,1,2,5,6] => [1,3,4,2,7,5,6] => [1,4,2,3,6,7,5] => 6
[[1,2,4,6],[3,5],[7]] => [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => [1,3,2,6,4,7,5] => 9
[[1,2,3,6],[4,5],[7]] => [7,4,5,1,2,3,6] => [1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => 5
[[1,2,4,5],[3,6],[7]] => [7,3,6,1,2,4,5] => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => 8
[[1,2,3,5],[4,6],[7]] => [7,4,6,1,2,3,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => 7
[[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => 4
[[1,2,3,7],[4],[5],[6]] => [6,5,4,1,2,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => 9
[[1,2,3,6],[4],[5],[7]] => [7,5,4,1,2,3,6] => [1,2,5,7,4,3,6] => [1,2,6,5,3,7,4] => 8
[[1,2,4,5],[3],[6],[7]] => [7,6,3,1,2,4,5] => [1,3,2,7,6,4,5] => [1,3,2,6,7,5,4] => 8
[[1,2,3,5],[4],[6],[7]] => [7,6,4,1,2,3,5] => [1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => 7
[[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => 6
[[1,2,5],[3,6,7],[4]] => [4,3,6,7,1,2,5] => [1,4,3,2,6,7,5] => [1,4,3,2,7,5,6] => 11
[[1,2,4],[3,6,7],[5]] => [5,3,6,7,1,2,4] => [1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => 8
[[1,2,3],[4,6,7],[5]] => [5,4,6,7,1,2,3] => [1,2,5,4,6,7,3] => [1,2,7,4,3,5,6] => 7
[[1,2,4],[3,5,7],[6]] => [6,3,5,7,1,2,4] => [1,3,2,6,5,7,4] => [1,3,2,7,5,4,6] => 10
[[1,2,3],[4,5,7],[6]] => [6,4,5,7,1,2,3] => [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => 6
[[1,2,5],[3,4,6],[7]] => [7,3,4,6,1,2,5] => [1,3,4,2,7,6,5] => [1,4,2,3,7,6,5] => 8
[[1,2,4],[3,5,6],[7]] => [7,3,5,6,1,2,4] => [1,3,2,5,7,6,4] => [1,3,2,7,4,6,5] => 9
[[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => 5
[[1,2,5],[3,6],[4,7]] => [4,7,3,6,1,2,5] => [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => 12
[[1,2,4],[3,6],[5,7]] => [5,7,3,6,1,2,4] => [1,3,5,2,7,6,4] => [1,4,2,7,3,6,5] => 9
[[1,2,3],[4,6],[5,7]] => [5,7,4,6,1,2,3] => [1,2,5,4,7,6,3] => [1,2,7,4,3,6,5] => 8
[[1,2,4],[3,5],[6,7]] => [6,7,3,5,1,2,4] => [1,3,2,6,7,5,4] => [1,3,2,7,6,4,5] => 10
[[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => [1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => 6
[[1,2,3],[4,7],[5],[6]] => [6,5,4,7,1,2,3] => [1,2,6,5,4,7,3] => [1,2,7,5,4,3,6] => 9
[[1,2,3],[4,6],[5],[7]] => [7,5,4,6,1,2,3] => [1,2,5,7,4,6,3] => [1,2,7,5,3,6,4] => 8
[[1,2,4],[3,5],[6],[7]] => [7,6,3,5,1,2,4] => [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => 11
[[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => 7
[[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => 10
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Description
The comajor index of a permutation.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
inverse
Description
Sends a permutation to its inverse.