Identifier
Values
[1,0] => [2,1] => [1] => 0
[1,0,1,0] => [3,1,2] => [1,2] => 0
[1,1,0,0] => [2,3,1] => [2,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,2] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [2,3,1] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,4,2] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,4,1,3] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,1,5,3,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,1,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,1,5,2] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,4,1,3,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,5,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,5,1,2,4] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,4,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,3,4,1,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,3,4,1,5] => 3
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,3,5,1,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,4,5,1,3] => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,4,5,1,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,2,3,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,2,3,6,4,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [4,1,2,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,2,5,3,4,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,6,3,4,5] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [6,1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [4,1,2,5,3,6] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,2,5,6,3,4] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,2,5,6,3] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [3,1,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [3,1,6,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [3,1,5,2,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [3,1,2,6,4,5] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,1,5,2,6,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,4,2,3,5,6] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [6,1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,5,2,3,4,6] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [6,1,2,3,4,5] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [5,1,4,2,3,6] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,4,2,6,3,5] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [1,5,2,6,3,4] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [5,1,4,2,6,3] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [3,1,4,2,5,6] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [3,1,4,6,2,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [3,1,5,2,4,6] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [3,1,6,2,4,5] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [3,1,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,4,5,2,3,6] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,4,6,2,3,5] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,6,5,2,3,4] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [6,1,4,5,2,3] => 2
>>> Load all 274 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [3,1,4,5,2,6] => 3
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [3,1,4,6,2,5] => 3
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [3,1,5,6,2,4] => 3
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [1,4,5,6,2,3] => 3
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,1,4,5,6,2] => 4
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [2,6,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [2,5,1,3,4,6] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [2,1,3,6,4,5] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [2,5,1,3,6,4] => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [2,4,1,3,5,6] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [2,4,1,6,3,5] => 2
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [2,1,5,3,4,6] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [2,1,6,3,4,5] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [2,6,1,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [2,4,1,5,3,6] => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [2,4,1,6,3,5] => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [2,1,5,6,3,4] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [2,4,1,5,6,3] => 3
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [3,1,2,4,5,6] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [5,3,1,2,4,6] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => [3,1,2,6,4,5] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [5,3,1,2,6,4] => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [4,1,2,3,5,6] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [5,1,2,3,4,6] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [5,6,1,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => [5,4,1,2,3,6] => 1
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => [4,1,2,6,3,5] => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [5,1,2,6,3,4] => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => [5,4,1,2,6,3] => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [4,3,1,2,5,6] => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => [4,3,1,6,2,5] => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [3,1,5,2,4,6] => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [7,3,1,6,2,4,5] => [3,1,6,2,4,5] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => [6,3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [4,1,5,2,3,6] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [4,1,6,2,3,5] => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [6,1,5,2,3,4] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => [6,4,1,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => [4,3,1,5,2,6] => 3
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [4,3,1,6,2,5] => 3
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => [3,1,5,6,2,4] => 3
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [4,1,5,6,2,3] => 3
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [4,3,1,5,6,2] => 4
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [2,3,1,4,5,6] => 2
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [2,3,6,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [2,3,5,1,4,6] => 3
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [2,3,1,6,4,5] => 2
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [2,3,5,1,6,4] => 3
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [2,4,1,3,5,6] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [2,6,4,1,3,5] => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [2,5,1,3,4,6] => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [2,6,1,3,4,5] => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [2,6,5,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [2,5,4,1,3,6] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [2,4,1,6,3,5] => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [2,5,1,6,3,4] => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [2,5,4,1,6,3] => 3
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => [3,4,1,2,5,6] => 2
[1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => [6,3,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [3,5,1,2,4,6] => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [6,3,7,1,2,4,5] => [6,3,1,2,4,5] => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [6,3,5,1,2,7,4] => [6,3,5,1,2,4] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [5,4,1,2,3,6] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [6,4,1,2,3,5] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [6,5,1,2,3,4] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,5,4,1,2,7,3] => [6,5,4,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,3,4,1,7,2,6] => [5,3,4,1,2,6] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => [3,4,1,6,2,5] => 3
[1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => [3,5,1,6,2,4] => 3
[1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => [5,4,1,6,2,3] => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => [5,3,4,1,6,2] => 4
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [2,3,4,1,5,6] => 3
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [2,3,4,6,1,5] => 4
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [2,3,5,1,4,6] => 3
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [2,3,6,1,4,5] => 3
[1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [2,3,6,5,1,4] => 3
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [2,4,5,1,3,6] => 3
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [2,4,6,1,3,5] => 3
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [2,6,5,1,3,4] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [2,6,4,5,1,3] => 3
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [3,4,5,1,2,6] => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [3,4,6,1,2,5] => 3
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [3,6,5,1,2,4] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [6,4,5,1,2,3] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [6,3,4,5,1,2] => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [2,3,4,5,1,6] => 4
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [2,3,4,6,1,5] => 4
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [2,3,5,6,1,4] => 4
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [2,4,5,6,1,3] => 4
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [3,4,5,6,1,2] => 4
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [2,3,4,5,6,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,1,2,3,4,7,8,5] => [6,1,2,3,4,7,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [5,1,2,3,8,4,6,7] => [5,1,2,3,4,6,7] => 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [5,1,2,3,8,7,4,6] => [5,1,2,3,7,4,6] => 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,1,2,3,6,7,8,4] => [5,1,2,3,6,7,4] => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [7,1,2,8,3,4,5,6] => [7,1,2,3,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,8,2,4,5,6,7] => [3,1,2,4,5,6,7] => 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [6,1,8,2,3,4,5,7] => [6,1,2,3,4,5,7] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [8,1,5,2,7,3,4,6] => [1,5,2,7,3,4,6] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,8,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [2,7,1,8,3,4,5,6] => [2,7,1,3,4,5,6] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [7,3,1,2,4,5,8,6] => [7,3,1,2,4,5,6] => 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [7,6,1,2,3,4,8,5] => [7,6,1,2,3,4,5] => 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [5,6,1,2,3,8,4,7] => [5,6,1,2,3,4,7] => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [5,8,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [8,6,1,2,3,7,4,5] => [6,1,2,3,7,4,5] => 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [5,6,1,2,3,7,8,4] => [5,6,1,2,3,7,4] => 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [7,3,1,8,2,4,5,6] => [7,3,1,2,4,5,6] => 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [8,4,1,6,2,3,5,7] => [4,1,6,2,3,5,7] => 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [7,4,1,8,2,3,5,6] => [7,4,1,2,3,5,6] => 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [2,8,7,1,3,4,5,6] => [2,7,1,3,4,5,6] => 2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [2,6,7,1,3,4,8,5] => [2,6,7,1,3,4,5] => 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [2,3,4,8,1,5,6,7] => [2,3,4,1,5,6,7] => 3
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [7,3,4,8,1,2,5,6] => [7,3,4,1,2,5,6] => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [7,6,4,5,1,2,8,3] => [7,6,4,5,1,2,3] => 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [2,3,4,5,8,1,6,7] => [2,3,4,5,1,6,7] => 4
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [2,3,4,5,7,1,8,6] => [2,3,4,5,7,1,6] => 5
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [2,3,4,8,7,1,5,6] => [2,3,4,7,1,5,6] => 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,7] => 5
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [2,3,4,5,8,7,1,6] => [2,3,4,5,7,1,6] => 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [2,3,4,8,6,7,1,5] => [2,3,4,6,7,1,5] => 5
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [2,3,8,5,6,7,1,4] => [2,3,5,6,7,1,4] => 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [9,1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9,1,8,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [2,9,8,1,3,4,5,6,7] => [2,8,1,3,4,5,6,7] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [9,7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => 7
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,9,11,1,10] => [2,3,4,5,6,7,8,9,1,10] => 8
[] => [1] => [] => 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,9,10,1,3,4,5,6,7,8] => [2,9,1,3,4,5,6,7,8] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [10,9,8,1,2,3,4,5,6,7] => [9,8,1,2,3,4,5,6,7] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [10,1,2,3,4,5,11,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,1,2,11,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [11,1,10,2,3,4,5,6,7,8,9] => [1,10,2,3,4,5,6,7,8,9] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [11,10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => 9
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Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ (OEIS:A000108). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ (OEIS:A000139).
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.