Identifier
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 4
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 4
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 4
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => 5
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => 4
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => 5
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [5,1,3,4,2] => [5,1,2,4,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => 7
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => 5
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => 6
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => 4
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [5,2,3,1,4] => [5,1,3,2,4] => 5
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [5,2,3,4,1] => [5,1,2,4,3] => 5
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [5,2,4,3,1] => [5,1,4,3,2] => 7
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 6
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => 7
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [5,3,2,4,1] => [5,2,1,4,3] => 6
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [5,4,2,3,1] => [5,4,1,3,2] => 8
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [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] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,6,4,3,5] => [1,2,6,4,3,5] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [1,2,6,3,5,4] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => [1,6,2,4,5,3] => [1,6,2,3,5,4] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => [1,6,2,5,4,3] => 7
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,5,3,2,4,6] => [1,5,3,2,4,6] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,6,3,2,4,5] => [1,6,3,2,4,5] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,6,3,2,5,4] => [1,6,3,2,5,4] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,5,2,4,3,6] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [1,6,3,4,2,5] => [1,6,2,4,3,5] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => [1,6,2,3,5,4] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [1,6,3,5,4,2] => [1,6,2,5,4,3] => 7
>>> Load all 262 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => 6
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,6,4,3,2,5] => [1,6,4,3,2,5] => 7
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,6,4,3,5,2] => [1,6,4,3,5,2] => [1,6,3,2,5,4] => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => [1,6,5,3,4,2] => [1,6,5,2,4,3] => 8
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 10
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => [2,1,3,6,4,5] => 3
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [2,1,3,6,5,4] => [2,1,3,6,5,4] => 4
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [2,1,6,3,4,5] => [2,1,6,3,4,5] => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => [2,1,6,3,5,4] => [2,1,6,3,5,4] => 5
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => [2,1,6,4,3,5] => [2,1,6,4,3,5] => 5
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,6,4,5,3] => [2,1,6,4,5,3] => [2,1,6,3,5,4] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 7
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [3,1,2,4,6,5] => [3,1,2,4,6,5] => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => 4
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [3,1,2,6,5,4] => [3,1,2,6,5,4] => 5
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => 3
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => [4,1,2,3,6,5] => [4,1,2,3,6,5] => 4
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => 4
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 5
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => 6
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => 5
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [6,1,2,4,3,5] => [6,1,2,4,3,5] => 6
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [6,1,2,4,5,3] => [6,1,2,3,5,4] => 6
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [6,1,2,5,4,3] => [6,1,2,5,4,3] => 8
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => 4
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => [4,1,3,2,6,5] => [4,1,3,2,6,5] => 5
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => [5,1,3,2,4,6] => [5,1,3,2,4,6] => 5
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [6,1,3,2,4,5] => [6,1,3,2,4,5] => 6
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => [6,1,3,2,5,4] => [6,1,3,2,5,4] => 7
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,5,3,4,1,6] => [5,1,3,4,2,6] => [5,1,2,4,3,6] => 5
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,5,3,4,6,1] => [6,1,3,4,2,5] => [6,1,2,4,3,5] => 6
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [6,1,3,4,5,2] => [6,1,2,3,5,4] => 6
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => [6,1,3,5,4,2] => [6,1,2,5,4,3] => 8
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [5,1,4,3,2,6] => [5,1,4,3,2,6] => 7
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [6,1,4,3,2,5] => [6,1,4,3,2,5] => 8
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => [6,1,4,3,5,2] => [6,1,3,2,5,4] => 7
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,6,4,5,3,1] => [6,1,5,3,4,2] => [6,1,5,2,4,3] => 9
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [6,1,5,4,3,2] => [6,1,5,4,3,2] => 11
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 3
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [3,2,1,4,6,5] => [3,2,1,4,6,5] => 4
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => 4
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => [3,2,1,6,4,5] => [3,2,1,6,4,5] => 5
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 6
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [4,2,1,3,5,6] => [4,2,1,3,5,6] => 4
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => [4,2,1,3,6,5] => [4,2,1,3,6,5] => 5
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [5,2,1,3,4,6] => [5,2,1,3,4,6] => 5
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [6,2,1,3,4,5] => [6,2,1,3,4,5] => 6
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [6,2,1,3,5,4] => [6,2,1,3,5,4] => 7
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => [5,2,1,4,3,6] => [5,2,1,4,3,6] => 6
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [6,2,1,4,3,5] => [6,2,1,4,3,5] => 7
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => [6,2,1,4,5,3] => [6,2,1,3,5,4] => 7
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [6,2,1,5,4,3] => [6,2,1,5,4,3] => 9
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,2,3,1,5,6] => [4,2,3,1,5,6] => [4,1,3,2,5,6] => 4
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,2,3,1,6,5] => [4,2,3,1,6,5] => [4,1,3,2,6,5] => 5
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,2,3,5,1,6] => [5,2,3,1,4,6] => [5,1,3,2,4,6] => 5
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [6,2,3,1,4,5] => [6,1,3,2,4,5] => 6
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,2,3,6,5,1] => [6,2,3,1,5,4] => [6,1,3,2,5,4] => 7
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,3,4,1,6] => [5,2,3,4,1,6] => [5,1,2,4,3,6] => 5
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [6,2,3,4,1,5] => [6,1,2,4,3,5] => 6
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => [6,1,2,3,5,4] => 6
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [6,2,3,5,4,1] => [6,1,2,5,4,3] => 8
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,2,4,3,1,6] => [5,2,4,3,1,6] => [5,1,4,3,2,6] => 7
[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => [6,2,4,3,1,5] => [6,1,4,3,2,5] => 8
[1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => [6,2,4,3,5,1] => [6,1,3,2,5,4] => 7
[1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [6,2,5,3,4,1] => [6,1,5,2,4,3] => 9
[1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => [6,1,5,4,3,2] => 11
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 6
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 7
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [5,3,2,1,4,6] => [5,3,2,1,4,6] => 7
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [6,3,2,1,4,5] => [6,3,2,1,4,5] => 8
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => 9
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,3,2,4,1,6] => [5,3,2,4,1,6] => [5,2,1,4,3,6] => 6
[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [6,3,2,4,1,5] => [6,2,1,4,3,5] => 7
[1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => [6,3,2,4,5,1] => [6,2,1,3,5,4] => 7
[1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => [6,3,2,5,4,1] => [6,2,1,5,4,3] => 9
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,3,4,2,1,6] => [5,4,2,3,1,6] => [5,4,1,3,2,6] => 8
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => [6,4,2,3,1,5] => [6,4,1,3,2,5] => 9
[1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => [6,4,2,3,5,1] => [6,3,1,2,5,4] => 8
[1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [6,5,2,3,4,1] => [6,5,1,2,4,3] => 10
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => [6,5,2,4,3,1] => [6,5,1,4,3,2] => 12
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 10
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [6,4,3,2,1,5] => [6,4,3,2,1,5] => 11
[1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => [6,4,3,2,5,1] => [6,3,2,1,5,4] => 9
[1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [6,5,3,2,4,1] => [6,5,2,1,4,3] => 11
[1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => [6,5,1,4,3,2] => 12
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 15
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => 5
[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => [1,2,7,3,4,6,5] => 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,2,7,3,6,5,4] => 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,2,5,4,6,3,7] => [1,2,6,4,3,5,7] => [1,2,6,4,3,5,7] => 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => [1,2,7,4,3,5,6] => [1,2,7,4,3,5,6] => 5
[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,2,5,4,7,6,3] => [1,2,7,4,3,6,5] => [1,2,7,4,3,6,5] => 6
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,2,6,4,5,3,7] => [1,2,6,4,5,3,7] => [1,2,6,3,5,4,7] => 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,6,4,5,7,3] => [1,2,7,4,5,3,6] => [1,2,7,3,5,4,6] => 5
[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,7,4,5,6,3] => [1,2,7,4,5,6,3] => [1,2,7,3,4,6,5] => 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,7,4,6,5,3] => [1,2,7,4,6,5,3] => [1,2,7,3,6,5,4] => 7
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => [1,2,7,5,4,3,6] => [1,2,7,5,4,3,6] => 7
[1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [1,2,7,5,4,6,3] => [1,2,7,5,4,6,3] => [1,2,7,4,3,6,5] => 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,7,5,6,4,3] => [1,2,7,6,4,5,3] => [1,2,7,6,3,5,4] => 8
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => 10
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,2,5,7,6,4] => [1,3,2,7,4,6,5] => [1,3,2,7,4,6,5] => 5
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => 4
[1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,5,7,4] => [1,3,2,7,5,4,6] => [1,3,2,7,5,4,6] => 5
[1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,2,7,5,6,4] => [1,3,2,7,5,6,4] => [1,3,2,7,4,6,5] => 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => 7
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5,7,6] => [1,4,2,3,5,7,6] => [1,4,2,3,5,7,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [1,3,4,2,6,5,7] => [1,4,2,3,6,5,7] => [1,4,2,3,6,5,7] => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,2,6,7,5] => [1,4,2,3,7,5,6] => [1,4,2,3,7,5,6] => 4
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [1,3,4,2,7,6,5] => [1,4,2,3,7,6,5] => [1,4,2,3,7,6,5] => 5
[1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => [1,4,3,2,5,7,6] => [1,4,3,2,5,7,6] => 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => [1,4,3,2,6,5,7] => [1,4,3,2,6,5,7] => 4
[1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [1,4,3,2,6,7,5] => [1,4,3,2,7,5,6] => [1,4,3,2,7,5,6] => 5
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => 6
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Description
The number of transpositions that are smaller or equal to a permutation in Bruhat order.
A statistic is known to be smooth if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern $4231$ and $3412$.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..