Identifier
Values
[[1]] => [1] => [1] => 0
[[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [2,1] => 0
[[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [2,1,3] => 0
[[1,2],[3]] => [3,1,2] => [2,3,1] => 1
[[1],[2],[3]] => [3,2,1] => [3,2,1] => 0
[[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] => 0
[[1,2,4],[3]] => [3,1,2,4] => [2,3,1,4] => 1
[[1,2,3],[4]] => [4,1,2,3] => [2,3,4,1] => 1
[[1,3],[2,4]] => [2,4,1,3] => [1,3,4,2] => 1
[[1,2],[3,4]] => [3,4,1,2] => [3,1,4,2] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [3,2,4,1] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [3,4,2,1] => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 0
[[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] => 0
[[1,2,4,5],[3]] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [1,3,4,2,5] => 1
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,1,4,2,5] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,5,2] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [3,1,4,5,2] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [3,4,1,5,2] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 2
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [2,1,4,5,3] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,2,1,5,3] => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,4,5,3,1] => 2
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,2,5,3,1] => 3
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,3,5,2,1] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [4,5,3,2,1] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[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] => 0
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [2,3,1,4,5,6] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [2,3,4,1,5,6] => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [1,3,4,2,5,6] => 1
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [3,1,4,2,5,6] => 2
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,3,4,5,2,6] => 1
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [3,1,4,5,2,6] => 2
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [3,4,1,5,2,6] => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,6,2] => 1
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [3,1,4,5,6,2] => 2
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [3,4,1,5,6,2] => 2
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [3,4,5,1,6,2] => 2
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 0
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [3,2,4,1,5,6] => 1
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [3,4,2,1,5,6] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [3,2,4,5,1,6] => 1
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [3,4,2,5,1,6] => 2
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [3,4,5,2,1,6] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [3,2,4,5,6,1] => 1
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 2
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => 2
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [3,4,5,6,2,1] => 1
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [4,1,2,5,6,3] => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,4,2,5,6,3] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [4,5,1,2,6,3] => 2
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,4,5,2,6,3] => 2
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 3
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [2,1,4,5,3,6] => 1
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,4,1,5,3,6] => 2
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [4,2,1,5,3,6] => 2
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,4,5,3,1,6] => 2
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [4,2,5,3,1,6] => 3
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [2,1,4,5,6,3] => 1
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,4,1,5,6,3] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [4,2,1,5,6,3] => 2
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,4,5,1,6,3] => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [4,2,5,1,6,3] => 3
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [4,5,2,1,6,3] => 2
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,4,5,3,6,1] => 2
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [4,2,5,3,6,1] => 3
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,4,5,6,3,1] => 2
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [4,2,5,6,3,1] => 3
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [4,5,2,6,3,1] => 3
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 0
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [4,3,2,5,1,6] => 1
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [4,3,5,2,1,6] => 1
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [4,5,3,2,1,6] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [4,3,2,5,6,1] => 1
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [4,3,5,2,6,1] => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [4,5,3,2,6,1] => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [4,3,5,6,2,1] => 1
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [4,5,3,6,2,1] => 2
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => 1
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [1,3,5,6,4,2] => 2
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,1,5,6,4,2] => 3
>>> Load all 236 entries. <<<
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [1,5,3,6,4,2] => 3
[[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [3,5,1,6,4,2] => 3
[[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 4
[[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [3,2,1,5,6,4] => 1
[[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 2
[[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [3,5,2,1,6,4] => 2
[[1,2],[3,6],[4],[5]] => [5,4,3,6,1,2] => [5,3,2,1,6,4] => 2
[[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [3,2,5,6,4,1] => 2
[[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [3,5,2,6,4,1] => 3
[[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [5,3,2,6,4,1] => 3
[[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [3,5,6,4,2,1] => 2
[[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [5,3,6,4,2,1] => 3
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 0
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 1
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 1
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 1
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 1
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0
[[1,2,4,5,6,7],[3]] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => 1
[[1,2,3,5,6,7],[4]] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => 1
[[1,2,3,4,6,7],[5]] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => 1
[[1,2,3,4,5,7],[6]] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => 1
[[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => 1
[[1,3,5,6,7],[2,4]] => [2,4,1,3,5,6,7] => [1,3,4,2,5,6,7] => 1
[[1,3,4,6,7],[2,5]] => [2,5,1,3,4,6,7] => [1,3,4,5,2,6,7] => 1
[[1,3,4,5,7],[2,6]] => [2,6,1,3,4,5,7] => [1,3,4,5,6,2,7] => 1
[[1,3,4,5,6],[2,7]] => [2,7,1,3,4,5,6] => [1,3,4,5,6,7,2] => 1
[[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 0
[[1,3,4,5,6],[2],[7]] => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => 1
[[1,2,4,5,6],[3],[7]] => [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => 2
[[1,2,3,5,6],[4],[7]] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => 2
[[1,2,3,4,6],[5],[7]] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => 2
[[1,2,5,7],[3,4,6]] => [3,4,6,1,2,5,7] => [1,4,2,5,6,3,7] => 2
[[1,2,4,7],[3,5,6]] => [3,5,6,1,2,4,7] => [1,4,5,2,6,3,7] => 2
[[1,2,5,6],[3,4,7]] => [3,4,7,1,2,5,6] => [1,4,2,5,6,7,3] => 2
[[1,2,4,6],[3,5,7]] => [3,5,7,1,2,4,6] => [1,4,5,2,6,7,3] => 2
[[1,2,4,5],[3,6,7]] => [3,6,7,1,2,4,5] => [1,4,5,6,2,7,3] => 2
[[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 0
[[1,2,4,7],[3],[5],[6]] => [6,5,3,1,2,4,7] => [4,5,3,6,2,1,7] => 2
[[1,2,3,7],[4],[5],[6]] => [6,5,4,1,2,3,7] => [4,5,6,3,2,1,7] => 1
[[1,4,5,6],[2],[3],[7]] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => 1
[[1,3,5,6],[2],[4],[7]] => [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => 2
[[1,3,5],[2,6,7],[4]] => [4,2,6,7,1,3,5] => [5,1,6,2,3,7,4] => 3
[[1,4,7],[2,5],[3,6]] => [3,6,2,5,1,4,7] => [1,3,5,6,4,2,7] => 2
[[1,4,6],[2,5],[3,7]] => [3,7,2,5,1,4,6] => [1,3,5,6,4,7,2] => 2
[[1,4,5],[2,6],[3,7]] => [3,7,2,6,1,4,5] => [1,3,5,6,7,4,2] => 2
[[1,3,7],[2,4],[5],[6]] => [6,5,2,4,1,3,7] => [3,5,6,4,2,1,7] => 2
[[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 0
[[1,5,7],[2],[3],[4],[6]] => [6,4,3,2,1,5,7] => [5,4,3,2,6,1,7] => 1
[[1,4,7],[2],[3],[5],[6]] => [6,5,3,2,1,4,7] => [5,4,3,6,2,1,7] => 1
[[1,3,7],[2],[4],[5],[6]] => [6,5,4,2,1,3,7] => [5,4,6,3,2,1,7] => 1
[[1,2,7],[3],[4],[5],[6]] => [6,5,4,3,1,2,7] => [5,6,4,3,2,1,7] => 1
[[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 0
[[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 0
[[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 0
[[1,2,3,4,6,7,8],[5]] => [5,1,2,3,4,6,7,8] => [2,3,4,5,1,6,7,8] => 1
[[1,2,3,4,5,7,8],[6]] => [6,1,2,3,4,5,7,8] => [2,3,4,5,6,1,7,8] => 1
[[1,2,3,4,5,6,8],[7]] => [7,1,2,3,4,5,6,8] => [2,3,4,5,6,7,1,8] => 1
[[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => 1
[[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 0
[[1,3,4,5,6,7],[2],[8]] => [8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => 1
[[1,2,4,5,6,7],[3],[8]] => [8,3,1,2,4,5,6,7] => [3,4,2,5,6,7,8,1] => 2
[[1,2,3,5,6,7],[4],[8]] => [8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => 2
[[1,3,4,5,6],[2,8],[7]] => [7,2,8,1,3,4,5,6] => [2,4,5,6,7,1,8,3] => 2
[[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 0
[[1,4,5,6,7],[2],[3],[8]] => [8,3,2,1,4,5,6,7] => [4,3,2,5,6,7,8,1] => 1
[[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 4
[[1,2,3,5],[4,7,8],[6]] => [6,4,7,8,1,2,3,5] => [5,2,6,1,7,3,8,4] => 4
[[1,2,3,6],[4,5,8],[7]] => [7,4,5,8,1,2,3,6] => [5,2,6,3,7,1,8,4] => 4
[[1,2,3,7],[4,5,6],[8]] => [8,4,5,6,1,2,3,7] => [5,2,6,3,7,4,8,1] => 4
[[1,2,4,7],[3,5],[6,8]] => [6,8,3,5,1,2,4,7] => [5,3,6,1,7,4,8,2] => 4
[[1,2,5,6],[3,4],[7,8]] => [7,8,3,4,1,2,5,6] => [5,3,6,4,7,1,8,2] => 4
[[1,2,4,6],[3,8],[5],[7]] => [7,5,3,8,1,2,4,6] => [5,3,6,2,7,1,8,4] => 4
[[1,2,4,7],[3,6],[5],[8]] => [8,5,3,6,1,2,4,7] => [5,3,6,2,7,4,8,1] => 4
[[1,2,5,7],[3,4],[6],[8]] => [8,6,3,4,1,2,5,7] => [5,3,6,4,7,2,8,1] => 4
[[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 0
[[1,2,3,8],[4],[5],[6],[7]] => [7,6,5,4,1,2,3,8] => [5,6,7,4,3,2,1,8] => 1
[[1,3,5,7],[2],[4],[6],[8]] => [8,6,4,2,1,3,5,7] => [5,4,6,3,7,2,8,1] => 3
[[1,2,6],[3,4,8],[5,7]] => [5,7,3,4,8,1,2,6] => [3,1,6,4,7,2,8,5] => 4
[[1,2,4],[3,5,6],[7,8]] => [7,8,3,5,6,1,2,4] => [3,6,7,4,1,8,5,2] => 4
[[1,5,7],[2,6,8],[3],[4]] => [4,3,2,6,8,1,5,7] => [2,1,6,3,4,7,8,5] => 2
[[1,4,5],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5] => [2,6,7,3,1,4,8,5] => 3
[[1,2,5],[3,7,8],[4],[6]] => [6,4,3,7,8,1,2,5] => [3,2,6,1,7,4,8,5] => 3
[[1,2,6],[3,5,8],[4],[7]] => [7,4,3,5,8,1,2,6] => [3,2,6,4,7,1,8,5] => 3
[[1,3,7],[2,5],[4,8],[6]] => [6,4,8,2,5,1,3,7] => [4,2,6,1,7,5,8,3] => 4
[[1,4,6],[2,8],[3],[5],[7]] => [7,5,3,2,8,1,4,6] => [4,3,6,2,7,1,8,5] => 3
[[1,4,7],[2,6],[3],[5],[8]] => [8,5,3,2,6,1,4,7] => [4,3,6,2,7,5,8,1] => 3
[[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 0
[[1,4,8],[2],[3],[5],[6],[7]] => [7,6,5,3,2,1,4,8] => [6,5,4,7,3,2,1,8] => 1
[[1,3,8],[2],[4],[5],[6],[7]] => [7,6,5,4,2,1,3,8] => [6,5,7,4,3,2,1,8] => 1
[[1,2,8],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2,8] => [6,7,5,4,3,2,1,8] => 1
[[1,5,7],[2],[3],[4],[6],[8]] => [8,6,4,3,2,1,5,7] => [6,5,4,3,7,2,8,1] => 2
[[1,3,7],[2],[4],[5],[6],[8]] => [8,6,5,4,2,1,3,7] => [6,5,7,4,3,2,8,1] => 2
[[1,3,5],[2],[4],[6],[7],[8]] => [8,7,6,4,2,1,3,5] => [6,5,7,4,8,3,2,1] => 2
[[1,5],[2,6],[3,7],[4,8]] => [4,8,3,7,2,6,1,5] => [1,3,5,7,8,6,4,2] => 3
[[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 6
[[1,4],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => 4
[[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => 2
[[1,3],[2,8],[4],[5],[6],[7]] => [7,6,5,4,2,8,1,3] => [5,7,4,3,2,1,8,6] => 2
[[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 0
[[1,7],[2],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,1,7] => [7,6,5,4,3,2,8,1] => 1
[[1,5],[2],[3],[4],[6],[7],[8]] => [8,7,6,4,3,2,1,5] => [7,6,5,4,8,3,2,1] => 1
[[1,3],[2],[4],[5],[6],[7],[8]] => [8,7,6,5,4,2,1,3] => [7,6,8,5,4,3,2,1] => 1
[[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => 1
[[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 0
[[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,10,1] => 1
[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 0
[[1,4,5,6,7,8,9],[2],[3]] => [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 0
[[1,5,6,7,8,9],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 0
[[1,6,7,8,9],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 0
[[1,7,8,9],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 0
[[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 0
[[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 0
[[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 0
[[1,4,5,6,7,8,9,10],[2],[3]] => [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 0
[[1,5,6,7,8,9,10],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 0
[[1,6,7,8,9,10],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 0
[[1,7,8,9,10],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 0
[[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 0
[[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9,10] => 0
[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 0
[[1,2,3,4,5,6,7,9],[8]] => [8,1,2,3,4,5,6,7,9] => [2,3,4,5,6,7,8,1,9] => 1
[[1,2,9],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2,9] => [7,8,6,5,4,3,2,1,9] => 1
[[1,2,3,4,5,6,7,8,10],[9]] => [9,1,2,3,4,5,6,7,8,10] => [2,3,4,5,6,7,8,9,1,10] => 1
[[1,2,10],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,1,2,10] => [8,9,7,6,5,4,3,2,1,10] => 1
[] => [] => [] => 0
[[1,3,4,5,6,7,8],[2],[9]] => [9,2,1,3,4,5,6,7,8] => [3,2,4,5,6,7,8,9,1] => 1
[[1,3,4,5,6,7,8,9],[2],[10]] => [10,2,1,3,4,5,6,7,8,9] => [3,2,4,5,6,7,8,9,10,1] => 1
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Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation Mp00090cycle-as-one-line notation, see [Theorem 2.5, 1].
For the number of small descents, see St000214The number of adjacencies of a permutation..
Map
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
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.