Identifier
Identifier
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 3
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 4
[1,4,2,3] => 4
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 3
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 3
[1,3,2,5,4] => 6
[1,3,4,2,5] => 4
[1,3,4,5,2] => 5
[1,3,5,2,4] => 5
[1,3,5,4,2] => 4
[1,4,2,3,5] => 4
[1,4,2,5,3] => 5
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 6
[1,4,5,3,2] => 5
[1,5,2,3,4] => 5
[1,5,2,4,3] => 4
[1,5,3,2,4] => 4
[1,5,3,4,2] => 3
[1,5,4,2,3] => 5
[1,5,4,3,2] => 6
[2,1,3,4,5] => 1
[2,1,3,5,4] => 4
[2,1,4,3,5] => 4
[2,1,4,5,3] => 5
[2,1,5,3,4] => 5
[2,1,5,4,3] => 4
[2,3,1,4,5] => 2
[2,3,1,5,4] => 5
[2,3,4,1,5] => 3
[2,3,4,5,1] => 4
[2,3,5,1,4] => 4
[2,3,5,4,1] => 3
[2,4,1,3,5] => 3
[2,4,1,5,3] => 4
[2,4,3,1,5] => 2
[2,4,3,5,1] => 3
[2,4,5,1,3] => 5
[2,4,5,3,1] => 4
[2,5,1,3,4] => 4
[2,5,1,4,3] => 3
[2,5,3,1,4] => 3
[2,5,3,4,1] => 2
[2,5,4,1,3] => 4
[2,5,4,3,1] => 5
[3,1,2,4,5] => 2
[3,1,2,5,4] => 5
[3,1,4,2,5] => 3
[3,1,4,5,2] => 4
[3,1,5,2,4] => 4
[3,1,5,4,2] => 3
[3,2,1,4,5] => 1
[3,2,1,5,4] => 4
[3,2,4,1,5] => 2
[3,2,4,5,1] => 3
[3,2,5,1,4] => 3
[3,2,5,4,1] => 2
[3,4,1,2,5] => 4
[3,4,1,5,2] => 5
[3,4,2,1,5] => 3
[3,4,2,5,1] => 4
[3,4,5,1,2] => 4
[3,4,5,2,1] => 5
[3,5,1,2,4] => 5
[3,5,1,4,2] => 4
[3,5,2,1,4] => 4
[3,5,2,4,1] => 3
[3,5,4,1,2] => 5
[3,5,4,2,1] => 4
[4,1,2,3,5] => 3
[4,1,2,5,3] => 4
[4,1,3,2,5] => 2
[4,1,3,5,2] => 3
[4,1,5,2,3] => 5
[4,1,5,3,2] => 4
[4,2,1,3,5] => 2
[4,2,1,5,3] => 3
[4,2,3,1,5] => 1
[4,2,3,5,1] => 2
[4,2,5,1,3] => 4
[4,2,5,3,1] => 3
[4,3,1,2,5] => 3
[4,3,1,5,2] => 4
[4,3,2,1,5] => 4
[4,3,2,5,1] => 5
[4,3,5,1,2] => 5
[4,3,5,2,1] => 4
[4,5,1,2,3] => 4
[4,5,1,3,2] => 5
[4,5,2,1,3] => 5
[4,5,2,3,1] => 4
[4,5,3,1,2] => 4
[4,5,3,2,1] => 3
[5,1,2,3,4] => 4
[5,1,2,4,3] => 3
[5,1,3,2,4] => 3
[5,1,3,4,2] => 2
[5,1,4,2,3] => 4
[5,1,4,3,2] => 5
[5,2,1,3,4] => 3
[5,2,1,4,3] => 2
[5,2,3,1,4] => 2
[5,2,3,4,1] => 1
[5,2,4,1,3] => 3
[5,2,4,3,1] => 4
[5,3,1,2,4] => 4
[5,3,1,4,2] => 3
[5,3,2,1,4] => 5
[5,3,2,4,1] => 4
[5,3,4,1,2] => 4
[5,3,4,2,1] => 5
[5,4,1,2,3] => 5
[5,4,1,3,2] => 4
[5,4,2,1,3] => 4
[5,4,2,3,1] => 5
[5,4,3,1,2] => 3
[5,4,3,2,1] => 4
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 3
[1,2,3,5,4,6] => 3
[1,2,3,5,6,4] => 4
[1,2,3,6,4,5] => 4
[1,2,3,6,5,4] => 3
[1,2,4,3,5,6] => 3
[1,2,4,3,6,5] => 6
[1,2,4,5,3,6] => 4
[1,2,4,5,6,3] => 5
[1,2,4,6,3,5] => 5
[1,2,4,6,5,3] => 4
[1,2,5,3,4,6] => 4
[1,2,5,3,6,4] => 5
[1,2,5,4,3,6] => 3
[1,2,5,4,6,3] => 4
[1,2,5,6,3,4] => 6
[1,2,5,6,4,3] => 5
[1,2,6,3,4,5] => 5
[1,2,6,3,5,4] => 4
[1,2,6,4,3,5] => 4
[1,2,6,4,5,3] => 3
[1,2,6,5,3,4] => 5
[1,2,6,5,4,3] => 6
[1,3,2,4,5,6] => 3
[1,3,2,4,6,5] => 6
[1,3,2,5,4,6] => 6
[1,3,2,5,6,4] => 7
[1,3,2,6,4,5] => 7
[1,3,2,6,5,4] => 6
[1,3,4,2,5,6] => 4
[1,3,4,2,6,5] => 7
[1,3,4,5,2,6] => 5
[1,3,4,5,6,2] => 6
[1,3,4,6,2,5] => 6
[1,3,4,6,5,2] => 5
[1,3,5,2,4,6] => 5
[1,3,5,2,6,4] => 6
[1,3,5,4,2,6] => 4
[1,3,5,4,6,2] => 5
[1,3,5,6,2,4] => 7
[1,3,5,6,4,2] => 6
[1,3,6,2,4,5] => 6
[1,3,6,2,5,4] => 5
[1,3,6,4,2,5] => 5
[1,3,6,4,5,2] => 4
[1,3,6,5,2,4] => 6
[1,3,6,5,4,2] => 7
[1,4,2,3,5,6] => 4
[1,4,2,3,6,5] => 7
[1,4,2,5,3,6] => 5
[1,4,2,5,6,3] => 6
[1,4,2,6,3,5] => 6
[1,4,2,6,5,3] => 5
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 6
[1,4,3,5,2,6] => 4
[1,4,3,5,6,2] => 5
[1,4,3,6,2,5] => 5
[1,4,3,6,5,2] => 4
[1,4,5,2,3,6] => 6
[1,4,5,2,6,3] => 7
[1,4,5,3,2,6] => 5
[1,4,5,3,6,2] => 6
[1,4,5,6,2,3] => 6
[1,4,5,6,3,2] => 7
[1,4,6,2,3,5] => 7
[1,4,6,2,5,3] => 6
[1,4,6,3,2,5] => 6
[1,4,6,3,5,2] => 5
[1,4,6,5,2,3] => 7
[1,4,6,5,3,2] => 6
[1,5,2,3,4,6] => 5
[1,5,2,3,6,4] => 6
[1,5,2,4,3,6] => 4
[1,5,2,4,6,3] => 5
[1,5,2,6,3,4] => 7
[1,5,2,6,4,3] => 6
[1,5,3,2,4,6] => 4
[1,5,3,2,6,4] => 5
[1,5,3,4,2,6] => 3
[1,5,3,4,6,2] => 4
[1,5,3,6,2,4] => 6
[1,5,3,6,4,2] => 5
[1,5,4,2,3,6] => 5
[1,5,4,2,6,3] => 6
[1,5,4,3,2,6] => 6
[1,5,4,3,6,2] => 7
[1,5,4,6,2,3] => 7
[1,5,4,6,3,2] => 6
[1,5,6,2,3,4] => 6
[1,5,6,2,4,3] => 7
[1,5,6,3,2,4] => 7
[1,5,6,3,4,2] => 6
[1,5,6,4,2,3] => 6
[1,5,6,4,3,2] => 5
[1,6,2,3,4,5] => 6
[1,6,2,3,5,4] => 5
[1,6,2,4,3,5] => 5
[1,6,2,4,5,3] => 4
[1,6,2,5,3,4] => 6
[1,6,2,5,4,3] => 7
[1,6,3,2,4,5] => 5
[1,6,3,2,5,4] => 4
[1,6,3,4,2,5] => 4
[1,6,3,4,5,2] => 3
[1,6,3,5,2,4] => 5
[1,6,3,5,4,2] => 6
[1,6,4,2,3,5] => 6
[1,6,4,2,5,3] => 5
[1,6,4,3,2,5] => 7
[1,6,4,3,5,2] => 6
[1,6,4,5,2,3] => 6
[1,6,4,5,3,2] => 7
[1,6,5,2,3,4] => 7
[1,6,5,2,4,3] => 6
[1,6,5,3,2,4] => 6
[1,6,5,3,4,2] => 7
[1,6,5,4,2,3] => 5
[1,6,5,4,3,2] => 6
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 4
[2,1,3,5,4,6] => 4
[2,1,3,5,6,4] => 5
[2,1,3,6,4,5] => 5
[2,1,3,6,5,4] => 4
[2,1,4,3,5,6] => 4
[2,1,4,3,6,5] => 7
[2,1,4,5,3,6] => 5
[2,1,4,5,6,3] => 6
[2,1,4,6,3,5] => 6
[2,1,4,6,5,3] => 5
[2,1,5,3,4,6] => 5
[2,1,5,3,6,4] => 6
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 5
[2,1,5,6,3,4] => 7
[2,1,5,6,4,3] => 6
[2,1,6,3,4,5] => 6
[2,1,6,3,5,4] => 5
[2,1,6,4,3,5] => 5
[2,1,6,4,5,3] => 4
[2,1,6,5,3,4] => 6
[2,1,6,5,4,3] => 7
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 5
[2,3,1,5,4,6] => 5
[2,3,1,5,6,4] => 6
[2,3,1,6,4,5] => 6
[2,3,1,6,5,4] => 5
[2,3,4,1,5,6] => 3
[2,3,4,1,6,5] => 6
[2,3,4,5,1,6] => 4
[2,3,4,5,6,1] => 5
[2,3,4,6,1,5] => 5
[2,3,4,6,5,1] => 4
[2,3,5,1,4,6] => 4
[2,3,5,1,6,4] => 5
[2,3,5,4,1,6] => 3
[2,3,5,4,6,1] => 4
[2,3,5,6,1,4] => 6
[2,3,5,6,4,1] => 5
[2,3,6,1,4,5] => 5
[2,3,6,1,5,4] => 4
[2,3,6,4,1,5] => 4
[2,3,6,4,5,1] => 3
[2,3,6,5,1,4] => 5
[2,3,6,5,4,1] => 6
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 6
[2,4,1,5,3,6] => 4
[2,4,1,5,6,3] => 5
[2,4,1,6,3,5] => 5
[2,4,1,6,5,3] => 4
[2,4,3,1,5,6] => 2
[2,4,3,1,6,5] => 5
[2,4,3,5,1,6] => 3
[2,4,3,5,6,1] => 4
[2,4,3,6,1,5] => 4
[2,4,3,6,5,1] => 3
[2,4,5,1,3,6] => 5
[2,4,5,1,6,3] => 6
[2,4,5,3,1,6] => 4
[2,4,5,3,6,1] => 5
[2,4,5,6,1,3] => 5
[2,4,5,6,3,1] => 6
[2,4,6,1,3,5] => 6
[2,4,6,1,5,3] => 5
[2,4,6,3,1,5] => 5
[2,4,6,3,5,1] => 4
[2,4,6,5,1,3] => 6
[2,4,6,5,3,1] => 5
[2,5,1,3,4,6] => 4
[2,5,1,3,6,4] => 5
[2,5,1,4,3,6] => 3
[2,5,1,4,6,3] => 4
[2,5,1,6,3,4] => 6
[2,5,1,6,4,3] => 5
[2,5,3,1,4,6] => 3
[2,5,3,1,6,4] => 4
[2,5,3,4,1,6] => 2
[2,5,3,4,6,1] => 3
[2,5,3,6,1,4] => 5
[2,5,3,6,4,1] => 4
[2,5,4,1,3,6] => 4
[2,5,4,1,6,3] => 5
[2,5,4,3,1,6] => 5
[2,5,4,3,6,1] => 6
[2,5,4,6,1,3] => 6
[2,5,4,6,3,1] => 5
[2,5,6,1,3,4] => 5
[2,5,6,1,4,3] => 6
[2,5,6,3,1,4] => 6
[2,5,6,3,4,1] => 5
[2,5,6,4,1,3] => 5
[2,5,6,4,3,1] => 4
[2,6,1,3,4,5] => 5
[2,6,1,3,5,4] => 4
[2,6,1,4,3,5] => 4
[2,6,1,4,5,3] => 3
[2,6,1,5,3,4] => 5
[2,6,1,5,4,3] => 6
[2,6,3,1,4,5] => 4
[2,6,3,1,5,4] => 3
[2,6,3,4,1,5] => 3
[2,6,3,4,5,1] => 2
[2,6,3,5,1,4] => 4
[2,6,3,5,4,1] => 5
[2,6,4,1,3,5] => 5
[2,6,4,1,5,3] => 4
[2,6,4,3,1,5] => 6
[2,6,4,3,5,1] => 5
[2,6,4,5,1,3] => 5
[2,6,4,5,3,1] => 6
[2,6,5,1,3,4] => 6
[2,6,5,1,4,3] => 5
[2,6,5,3,1,4] => 5
[2,6,5,3,4,1] => 6
[2,6,5,4,1,3] => 4
[2,6,5,4,3,1] => 5
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 5
[3,1,2,5,4,6] => 5
[3,1,2,5,6,4] => 6
[3,1,2,6,4,5] => 6
[3,1,2,6,5,4] => 5
[3,1,4,2,5,6] => 3
[3,1,4,2,6,5] => 6
[3,1,4,5,2,6] => 4
[3,1,4,5,6,2] => 5
[3,1,4,6,2,5] => 5
[3,1,4,6,5,2] => 4
[3,1,5,2,4,6] => 4
[3,1,5,2,6,4] => 5
[3,1,5,4,2,6] => 3
[3,1,5,4,6,2] => 4
[3,1,5,6,2,4] => 6
[3,1,5,6,4,2] => 5
[3,1,6,2,4,5] => 5
[3,1,6,2,5,4] => 4
[3,1,6,4,2,5] => 4
[3,1,6,4,5,2] => 3
[3,1,6,5,2,4] => 5
[3,1,6,5,4,2] => 6
[3,2,1,4,5,6] => 1
[3,2,1,4,6,5] => 4
[3,2,1,5,4,6] => 4
[3,2,1,5,6,4] => 5
[3,2,1,6,4,5] => 5
[3,2,1,6,5,4] => 4
[3,2,4,1,5,6] => 2
[3,2,4,1,6,5] => 5
[3,2,4,5,1,6] => 3
[3,2,4,5,6,1] => 4
[3,2,4,6,1,5] => 4
[3,2,4,6,5,1] => 3
[3,2,5,1,4,6] => 3
[3,2,5,1,6,4] => 4
[3,2,5,4,1,6] => 2
[3,2,5,4,6,1] => 3
[3,2,5,6,1,4] => 5
[3,2,5,6,4,1] => 4
[3,2,6,1,4,5] => 4
[3,2,6,1,5,4] => 3
[3,2,6,4,1,5] => 3
[3,2,6,4,5,1] => 2
[3,2,6,5,1,4] => 4
[3,2,6,5,4,1] => 5
[3,4,1,2,5,6] => 4
[3,4,1,2,6,5] => 7
[3,4,1,5,2,6] => 5
[3,4,1,5,6,2] => 6
[3,4,1,6,2,5] => 6
[3,4,1,6,5,2] => 5
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 6
[3,4,2,5,1,6] => 4
[3,4,2,5,6,1] => 5
[3,4,2,6,1,5] => 5
[3,4,2,6,5,1] => 4
[3,4,5,1,2,6] => 4
[3,4,5,1,6,2] => 5
[3,4,5,2,1,6] => 5
[3,4,5,2,6,1] => 6
[3,4,5,6,1,2] => 6
[3,4,5,6,2,1] => 5
[3,4,6,1,2,5] => 5
[3,4,6,1,5,2] => 4
[3,4,6,2,1,5] => 6
[3,4,6,2,5,1] => 5
[3,4,6,5,1,2] => 5
[3,4,6,5,2,1] => 6
[3,5,1,2,4,6] => 5
[3,5,1,2,6,4] => 6
[3,5,1,4,2,6] => 4
[3,5,1,4,6,2] => 5
[3,5,1,6,2,4] => 7
[3,5,1,6,4,2] => 6
[3,5,2,1,4,6] => 4
[3,5,2,1,6,4] => 5
[3,5,2,4,1,6] => 3
[3,5,2,4,6,1] => 4
[3,5,2,6,1,4] => 6
[3,5,2,6,4,1] => 5
[3,5,4,1,2,6] => 5
[3,5,4,1,6,2] => 6
[3,5,4,2,1,6] => 4
[3,5,4,2,6,1] => 5
[3,5,4,6,1,2] => 5
[3,5,4,6,2,1] => 6
[3,5,6,1,2,4] => 6
[3,5,6,1,4,2] => 5
[3,5,6,2,1,4] => 5
[3,5,6,2,4,1] => 6
[3,5,6,4,1,2] => 4
[3,5,6,4,2,1] => 5
[3,6,1,2,4,5] => 6
[3,6,1,2,5,4] => 5
[3,6,1,4,2,5] => 5
[3,6,1,4,5,2] => 4
[3,6,1,5,2,4] => 6
[3,6,1,5,4,2] => 7
[3,6,2,1,4,5] => 5
[3,6,2,1,5,4] => 4
[3,6,2,4,1,5] => 4
[3,6,2,4,5,1] => 3
[3,6,2,5,1,4] => 5
[3,6,2,5,4,1] => 6
[3,6,4,1,2,5] => 6
[3,6,4,1,5,2] => 5
[3,6,4,2,1,5] => 5
[3,6,4,2,5,1] => 4
[3,6,4,5,1,2] => 6
[3,6,4,5,2,1] => 5
[3,6,5,1,2,4] => 5
[3,6,5,1,4,2] => 6
[3,6,5,2,1,4] => 6
[3,6,5,2,4,1] => 5
[3,6,5,4,1,2] => 5
[3,6,5,4,2,1] => 4
[4,1,2,3,5,6] => 3
[4,1,2,3,6,5] => 6
[4,1,2,5,3,6] => 4
[4,1,2,5,6,3] => 5
[4,1,2,6,3,5] => 5
[4,1,2,6,5,3] => 4
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 5
[4,1,3,5,2,6] => 3
[4,1,3,5,6,2] => 4
[4,1,3,6,2,5] => 4
[4,1,3,6,5,2] => 3
[4,1,5,2,3,6] => 5
[4,1,5,2,6,3] => 6
[4,1,5,3,2,6] => 4
[4,1,5,3,6,2] => 5
[4,1,5,6,2,3] => 5
[4,1,5,6,3,2] => 6
[4,1,6,2,3,5] => 6
[4,1,6,2,5,3] => 5
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 4
[4,1,6,5,2,3] => 6
[4,1,6,5,3,2] => 5
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 5
[4,2,1,5,3,6] => 3
[4,2,1,5,6,3] => 4
[4,2,1,6,3,5] => 4
[4,2,1,6,5,3] => 3
[4,2,3,1,5,6] => 1
[4,2,3,1,6,5] => 4
[4,2,3,5,1,6] => 2
[4,2,3,5,6,1] => 3
[4,2,3,6,1,5] => 3
[4,2,3,6,5,1] => 2
[4,2,5,1,3,6] => 4
[4,2,5,1,6,3] => 5
[4,2,5,3,1,6] => 3
[4,2,5,3,6,1] => 4
[4,2,5,6,1,3] => 4
[4,2,5,6,3,1] => 5
[4,2,6,1,3,5] => 5
[4,2,6,1,5,3] => 4
[4,2,6,3,1,5] => 4
[4,2,6,3,5,1] => 3
[4,2,6,5,1,3] => 5
[4,2,6,5,3,1] => 4
[4,3,1,2,5,6] => 3
[4,3,1,2,6,5] => 6
[4,3,1,5,2,6] => 4
[4,3,1,5,6,2] => 5
[4,3,1,6,2,5] => 5
[4,3,1,6,5,2] => 4
[4,3,2,1,5,6] => 4
[4,3,2,1,6,5] => 7
[4,3,2,5,1,6] => 5
[4,3,2,5,6,1] => 6
[4,3,2,6,1,5] => 6
[4,3,2,6,5,1] => 5
[4,3,5,1,2,6] => 5
[4,3,5,1,6,2] => 6
[4,3,5,2,1,6] => 4
[4,3,5,2,6,1] => 5
[4,3,5,6,1,2] => 5
[4,3,5,6,2,1] => 6
[4,3,6,1,2,5] => 6
[4,3,6,1,5,2] => 5
[4,3,6,2,1,5] => 5
[4,3,6,2,5,1] => 4
[4,3,6,5,1,2] => 6
[4,3,6,5,2,1] => 5
[4,5,1,2,3,6] => 4
[4,5,1,2,6,3] => 5
[4,5,1,3,2,6] => 5
[4,5,1,3,6,2] => 6
[4,5,1,6,2,3] => 6
[4,5,1,6,3,2] => 5
[4,5,2,1,3,6] => 5
[4,5,2,1,6,3] => 6
[4,5,2,3,1,6] => 4
[4,5,2,3,6,1] => 5
[4,5,2,6,1,3] => 5
[4,5,2,6,3,1] => 6
[4,5,3,1,2,6] => 4
[4,5,3,1,6,2] => 5
[4,5,3,2,1,6] => 3
[4,5,3,2,6,1] => 4
[4,5,3,6,1,2] => 4
[4,5,3,6,2,1] => 5
[4,5,6,1,2,3] => 7
[4,5,6,1,3,2] => 6
[4,5,6,2,1,3] => 6
[4,5,6,2,3,1] => 5
[4,5,6,3,1,2] => 5
[4,5,6,3,2,1] => 6
[4,6,1,2,3,5] => 5
[4,6,1,2,5,3] => 4
[4,6,1,3,2,5] => 6
[4,6,1,3,5,2] => 5
[4,6,1,5,2,3] => 5
[4,6,1,5,3,2] => 6
[4,6,2,1,3,5] => 6
[4,6,2,1,5,3] => 5
[4,6,2,3,1,5] => 5
[4,6,2,3,5,1] => 4
[4,6,2,5,1,3] => 6
[4,6,2,5,3,1] => 5
[4,6,3,1,2,5] => 5
[4,6,3,1,5,2] => 4
[4,6,3,2,1,5] => 4
[4,6,3,2,5,1] => 3
[4,6,3,5,1,2] => 5
[4,6,3,5,2,1] => 4
[4,6,5,1,2,3] => 6
[4,6,5,1,3,2] => 7
[4,6,5,2,1,3] => 5
[4,6,5,2,3,1] => 6
[4,6,5,3,1,2] => 6
[4,6,5,3,2,1] => 5
[5,1,2,3,4,6] => 4
[5,1,2,3,6,4] => 5
[5,1,2,4,3,6] => 3
[5,1,2,4,6,3] => 4
[5,1,2,6,3,4] => 6
[5,1,2,6,4,3] => 5
[5,1,3,2,4,6] => 3
[5,1,3,2,6,4] => 4
[5,1,3,4,2,6] => 2
[5,1,3,4,6,2] => 3
[5,1,3,6,2,4] => 5
[5,1,3,6,4,2] => 4
[5,1,4,2,3,6] => 4
[5,1,4,2,6,3] => 5
[5,1,4,3,2,6] => 5
[5,1,4,3,6,2] => 6
[5,1,4,6,2,3] => 6
[5,1,4,6,3,2] => 5
[5,1,6,2,3,4] => 5
[5,1,6,2,4,3] => 6
[5,1,6,3,2,4] => 6
[5,1,6,3,4,2] => 5
[5,1,6,4,2,3] => 5
[5,1,6,4,3,2] => 4
[5,2,1,3,4,6] => 3
[5,2,1,3,6,4] => 4
[5,2,1,4,3,6] => 2
[5,2,1,4,6,3] => 3
[5,2,1,6,3,4] => 5
[5,2,1,6,4,3] => 4
[5,2,3,1,4,6] => 2
[5,2,3,1,6,4] => 3
[5,2,3,4,1,6] => 1
[5,2,3,4,6,1] => 2
[5,2,3,6,1,4] => 4
[5,2,3,6,4,1] => 3
[5,2,4,1,3,6] => 3
[5,2,4,1,6,3] => 4
[5,2,4,3,1,6] => 4
[5,2,4,3,6,1] => 5
[5,2,4,6,1,3] => 5
[5,2,4,6,3,1] => 4
[5,2,6,1,3,4] => 4
[5,2,6,1,4,3] => 5
[5,2,6,3,1,4] => 5
[5,2,6,3,4,1] => 4
[5,2,6,4,1,3] => 4
[5,2,6,4,3,1] => 3
[5,3,1,2,4,6] => 4
[5,3,1,2,6,4] => 5
[5,3,1,4,2,6] => 3
[5,3,1,4,6,2] => 4
[5,3,1,6,2,4] => 6
[5,3,1,6,4,2] => 5
[5,3,2,1,4,6] => 5
[5,3,2,1,6,4] => 6
[5,3,2,4,1,6] => 4
[5,3,2,4,6,1] => 5
[5,3,2,6,1,4] => 7
[5,3,2,6,4,1] => 6
[5,3,4,1,2,6] => 4
[5,3,4,1,6,2] => 5
[5,3,4,2,1,6] => 5
[5,3,4,2,6,1] => 6
[5,3,4,6,1,2] => 6
[5,3,4,6,2,1] => 5
[5,3,6,1,2,4] => 5
[5,3,6,1,4,2] => 6
[5,3,6,2,1,4] => 6
[5,3,6,2,4,1] => 5
[5,3,6,4,1,2] => 5
[5,3,6,4,2,1] => 4
[5,4,1,2,3,6] => 5
[5,4,1,2,6,3] => 6
[5,4,1,3,2,6] => 4
[5,4,1,3,6,2] => 5
[5,4,1,6,2,3] => 5
[5,4,1,6,3,2] => 6
[5,4,2,1,3,6] => 4
[5,4,2,1,6,3] => 5
[5,4,2,3,1,6] => 5
[5,4,2,3,6,1] => 6
[5,4,2,6,1,3] => 6
[5,4,2,6,3,1] => 5
[5,4,3,1,2,6] => 3
[5,4,3,1,6,2] => 4
[5,4,3,2,1,6] => 4
[5,4,3,2,6,1] => 5
[5,4,3,6,1,2] => 5
[5,4,3,6,2,1] => 4
[5,4,6,1,2,3] => 6
[5,4,6,1,3,2] => 5
[5,4,6,2,1,3] => 7
[5,4,6,2,3,1] => 6
[5,4,6,3,1,2] => 6
[5,4,6,3,2,1] => 5
[5,6,1,2,3,4] => 6
[5,6,1,2,4,3] => 5
[5,6,1,3,2,4] => 5
[5,6,1,3,4,2] => 6
[5,6,1,4,2,3] => 4
[5,6,1,4,3,2] => 5
[5,6,2,1,3,4] => 5
[5,6,2,1,4,3] => 6
[5,6,2,3,1,4] => 6
[5,6,2,3,4,1] => 5
[5,6,2,4,1,3] => 5
[5,6,2,4,3,1] => 4
[5,6,3,1,2,4] => 4
[5,6,3,1,4,2] => 5
[5,6,3,2,1,4] => 5
[5,6,3,2,4,1] => 4
[5,6,3,4,1,2] => 4
[5,6,3,4,2,1] => 3
[5,6,4,1,2,3] => 5
[5,6,4,1,3,2] => 6
[5,6,4,2,1,3] => 6
[5,6,4,2,3,1] => 5
[5,6,4,3,1,2] => 7
[5,6,4,3,2,1] => 6
[6,1,2,3,4,5] => 5
[6,1,2,3,5,4] => 4
[6,1,2,4,3,5] => 4
[6,1,2,4,5,3] => 3
[6,1,2,5,3,4] => 5
[6,1,2,5,4,3] => 6
[6,1,3,2,4,5] => 4
[6,1,3,2,5,4] => 3
[6,1,3,4,2,5] => 3
[6,1,3,4,5,2] => 2
[6,1,3,5,2,4] => 4
[6,1,3,5,4,2] => 5
[6,1,4,2,3,5] => 5
[6,1,4,2,5,3] => 4
[6,1,4,3,2,5] => 6
[6,1,4,3,5,2] => 5
[6,1,4,5,2,3] => 5
[6,1,4,5,3,2] => 6
[6,1,5,2,3,4] => 6
[6,1,5,2,4,3] => 5
[6,1,5,3,2,4] => 5
[6,1,5,3,4,2] => 6
[6,1,5,4,2,3] => 4
[6,1,5,4,3,2] => 5
[6,2,1,3,4,5] => 4
[6,2,1,3,5,4] => 3
[6,2,1,4,3,5] => 3
[6,2,1,4,5,3] => 2
[6,2,1,5,3,4] => 4
[6,2,1,5,4,3] => 5
[6,2,3,1,4,5] => 3
[6,2,3,1,5,4] => 2
[6,2,3,4,1,5] => 2
[6,2,3,4,5,1] => 1
[6,2,3,5,1,4] => 3
[6,2,3,5,4,1] => 4
[6,2,4,1,3,5] => 4
[6,2,4,1,5,3] => 3
[6,2,4,3,1,5] => 5
[6,2,4,3,5,1] => 4
[6,2,4,5,1,3] => 4
[6,2,4,5,3,1] => 5
[6,2,5,1,3,4] => 5
[6,2,5,1,4,3] => 4
[6,2,5,3,1,4] => 4
[6,2,5,3,4,1] => 5
[6,2,5,4,1,3] => 3
[6,2,5,4,3,1] => 4
[6,3,1,2,4,5] => 5
[6,3,1,2,5,4] => 4
[6,3,1,4,2,5] => 4
[6,3,1,4,5,2] => 3
[6,3,1,5,2,4] => 5
[6,3,1,5,4,2] => 6
[6,3,2,1,4,5] => 6
[6,3,2,1,5,4] => 5
[6,3,2,4,1,5] => 5
[6,3,2,4,5,1] => 4
[6,3,2,5,1,4] => 6
[6,3,2,5,4,1] => 7
[6,3,4,1,2,5] => 5
[6,3,4,1,5,2] => 4
[6,3,4,2,1,5] => 6
[6,3,4,2,5,1] => 5
[6,3,4,5,1,2] => 5
[6,3,4,5,2,1] => 6
[6,3,5,1,2,4] => 6
[6,3,5,1,4,2] => 5
[6,3,5,2,1,4] => 5
[6,3,5,2,4,1] => 6
[6,3,5,4,1,2] => 4
[6,3,5,4,2,1] => 5
[6,4,1,2,3,5] => 6
[6,4,1,2,5,3] => 5
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 4
[6,4,1,5,2,3] => 6
[6,4,1,5,3,2] => 5
[6,4,2,1,3,5] => 5
[6,4,2,1,5,3] => 4
[6,4,2,3,1,5] => 6
[6,4,2,3,5,1] => 5
[6,4,2,5,1,3] => 5
[6,4,2,5,3,1] => 6
[6,4,3,1,2,5] => 4
[6,4,3,1,5,2] => 3
[6,4,3,2,1,5] => 5
[6,4,3,2,5,1] => 4
[6,4,3,5,1,2] => 4
[6,4,3,5,2,1] => 5
[6,4,5,1,2,3] => 5
[6,4,5,1,3,2] => 6
[6,4,5,2,1,3] => 6
[6,4,5,2,3,1] => 7
[6,4,5,3,1,2] => 5
[6,4,5,3,2,1] => 6
[6,5,1,2,3,4] => 5
[6,5,1,2,4,3] => 6
[6,5,1,3,2,4] => 6
[6,5,1,3,4,2] => 5
[6,5,1,4,2,3] => 5
[6,5,1,4,3,2] => 4
[6,5,2,1,3,4] => 6
[6,5,2,1,4,3] => 5
[6,5,2,3,1,4] => 5
[6,5,2,3,4,1] => 6
[6,5,2,4,1,3] => 4
[6,5,2,4,3,1] => 5
[6,5,3,1,2,4] => 5
[6,5,3,1,4,2] => 4
[6,5,3,2,1,4] => 4
[6,5,3,2,4,1] => 5
[6,5,3,4,1,2] => 3
[6,5,3,4,2,1] => 4
[6,5,4,1,2,3] => 6
[6,5,4,1,3,2] => 5
[6,5,4,2,1,3] => 5
[6,5,4,2,3,1] => 6
[6,5,4,3,1,2] => 6
[6,5,4,3,2,1] => 7
[1,2,3,4,5,6,7] => 0
[1,2,3,4,5,7,6] => 3
[1,2,3,4,6,5,7] => 3
[1,2,3,4,6,7,5] => 4
[1,2,3,4,7,5,6] => 4
[1,2,3,4,7,6,5] => 3
[1,2,3,5,4,6,7] => 3
[1,2,3,5,4,7,6] => 6
[1,2,3,5,6,4,7] => 4
[1,2,3,5,6,7,4] => 5
[1,2,3,5,7,4,6] => 5
[1,2,3,5,7,6,4] => 4
[1,2,3,6,4,5,7] => 4
[1,2,3,6,4,7,5] => 5
[1,2,3,6,5,4,7] => 3
[1,2,3,6,5,7,4] => 4
[1,2,3,6,7,4,5] => 6
[1,2,3,6,7,5,4] => 5
[1,2,3,7,4,5,6] => 5
[1,2,3,7,4,6,5] => 4
[1,2,3,7,5,4,6] => 4
[1,2,3,7,5,6,4] => 3
[1,2,3,7,6,4,5] => 5
[1,2,3,7,6,5,4] => 6
[1,2,4,3,5,6,7] => 3
[1,2,4,3,5,7,6] => 6
[1,2,4,3,6,5,7] => 6
[1,2,4,3,6,7,5] => 7
[1,2,4,3,7,5,6] => 7
[1,2,4,3,7,6,5] => 6
[1,2,4,5,3,6,7] => 4
[1,2,4,5,3,7,6] => 7
[1,2,4,5,6,3,7] => 5
[1,2,4,5,6,7,3] => 6
[1,2,4,5,7,3,6] => 6
[1,2,4,5,7,6,3] => 5
[1,2,4,6,3,5,7] => 5
[1,2,4,6,3,7,5] => 6
[1,2,4,6,5,3,7] => 4
[1,2,4,6,5,7,3] => 5
[1,2,4,6,7,3,5] => 7
[1,2,4,6,7,5,3] => 6
[1,2,4,7,3,5,6] => 6
[1,2,4,7,3,6,5] => 5
[1,2,4,7,5,3,6] => 5
[1,2,4,7,5,6,3] => 4
[1,2,4,7,6,3,5] => 6
[1,2,4,7,6,5,3] => 7
[1,2,5,3,4,6,7] => 4
[1,2,5,3,4,7,6] => 7
[1,2,5,3,6,4,7] => 5
[1,2,5,3,6,7,4] => 6
[1,2,5,3,7,4,6] => 6
[1,2,5,3,7,6,4] => 5
[1,2,5,4,3,6,7] => 3
[1,2,5,4,3,7,6] => 6
[1,2,5,4,6,3,7] => 4
[1,2,5,4,6,7,3] => 5
[1,2,5,4,7,3,6] => 5
[1,2,5,4,7,6,3] => 4
[1,2,5,6,3,4,7] => 6
[1,2,5,6,3,7,4] => 7
[1,2,5,6,4,3,7] => 5
[1,2,5,6,4,7,3] => 6
[1,2,5,6,7,3,4] => 6
[1,2,5,6,7,4,3] => 7
[1,2,5,7,3,4,6] => 7
[1,2,5,7,3,6,4] => 6
[1,2,5,7,4,3,6] => 6
[1,2,5,7,4,6,3] => 5
[1,2,5,7,6,3,4] => 7
[1,2,5,7,6,4,3] => 6
[1,2,6,3,4,5,7] => 5
[1,2,6,3,4,7,5] => 6
[1,2,6,3,5,4,7] => 4
[1,2,6,3,5,7,4] => 5
[1,2,6,3,7,4,5] => 7
[1,2,6,3,7,5,4] => 6
[1,2,6,4,3,5,7] => 4
[1,2,6,4,3,7,5] => 5
[1,2,6,4,5,3,7] => 3
[1,2,6,4,5,7,3] => 4
[1,2,6,4,7,3,5] => 6
[1,2,6,4,7,5,3] => 5
[1,2,6,5,3,4,7] => 5
[1,2,6,5,3,7,4] => 6
[1,2,6,5,4,3,7] => 6
[1,2,6,5,4,7,3] => 7
[1,2,6,5,7,3,4] => 7
[1,2,6,5,7,4,3] => 6
[1,2,6,7,3,4,5] => 6
[1,2,6,7,3,5,4] => 7
[1,2,6,7,4,3,5] => 7
[1,2,6,7,4,5,3] => 6
[1,2,6,7,5,3,4] => 6
[1,2,6,7,5,4,3] => 5
[1,2,7,3,4,5,6] => 6
[1,2,7,3,4,6,5] => 5
[1,2,7,3,5,4,6] => 5
[1,2,7,3,5,6,4] => 4
[1,2,7,3,6,4,5] => 6
[1,2,7,3,6,5,4] => 7
[1,2,7,4,3,5,6] => 5
[1,2,7,4,3,6,5] => 4
[1,2,7,4,5,3,6] => 4
[1,2,7,4,5,6,3] => 3
[1,2,7,4,6,3,5] => 5
[1,2,7,4,6,5,3] => 6
[1,2,7,5,3,4,6] => 6
[1,2,7,5,3,6,4] => 5
[1,2,7,5,4,3,6] => 7
[1,2,7,5,4,6,3] => 6
[1,2,7,5,6,3,4] => 6
[1,2,7,5,6,4,3] => 7
[1,2,7,6,3,4,5] => 7
[1,2,7,6,3,5,4] => 6
[1,2,7,6,4,3,5] => 6
[1,2,7,6,4,5,3] => 7
[1,2,7,6,5,3,4] => 5
[1,2,7,6,5,4,3] => 6
[1,3,2,4,5,6,7] => 3
[1,3,2,4,5,7,6] => 6
[1,3,2,4,6,5,7] => 6
[1,3,2,4,6,7,5] => 7
[1,3,2,4,7,5,6] => 7
[1,3,2,4,7,6,5] => 6
[1,3,2,5,4,6,7] => 6
[1,3,2,5,4,7,6] => 9
[1,3,2,5,6,4,7] => 7
[1,3,2,5,6,7,4] => 8
[1,3,2,5,7,4,6] => 8
[1,3,2,5,7,6,4] => 7
[1,3,2,6,4,5,7] => 7
[1,3,2,6,4,7,5] => 8
[1,3,2,6,5,4,7] => 6
[1,3,2,6,5,7,4] => 7
[1,3,2,6,7,4,5] => 9
[1,3,2,6,7,5,4] => 8
[1,3,2,7,4,5,6] => 8
[1,3,2,7,4,6,5] => 7
[1,3,2,7,5,4,6] => 7
[1,3,2,7,5,6,4] => 6
[1,3,2,7,6,4,5] => 8
[1,3,2,7,6,5,4] => 9
[1,3,4,2,5,6,7] => 4
[1,3,4,2,5,7,6] => 7
[1,3,4,2,6,5,7] => 7
[1,3,4,2,6,7,5] => 8
[1,3,4,2,7,5,6] => 8
[1,3,4,2,7,6,5] => 7
[1,3,4,5,2,6,7] => 5
[1,3,4,5,2,7,6] => 8
[1,3,4,5,6,2,7] => 6
[1,3,4,5,6,7,2] => 7
[1,3,4,5,7,2,6] => 7
[1,3,4,5,7,6,2] => 6
[1,3,4,6,2,5,7] => 6
[1,3,4,6,2,7,5] => 7
[1,3,4,6,5,2,7] => 5
[1,3,4,6,5,7,2] => 6
[1,3,4,6,7,2,5] => 8
[1,3,4,6,7,5,2] => 7
[1,3,4,7,2,5,6] => 7
[1,3,4,7,2,6,5] => 6
[1,3,4,7,5,2,6] => 6
[1,3,4,7,5,6,2] => 5
[1,3,4,7,6,2,5] => 7
[1,3,4,7,6,5,2] => 8
[1,3,5,2,4,6,7] => 5
[1,3,5,2,4,7,6] => 8
[1,3,5,2,6,4,7] => 6
[1,3,5,2,6,7,4] => 7
[1,3,5,2,7,4,6] => 7
[1,3,5,2,7,6,4] => 6
[1,3,5,4,2,6,7] => 4
[1,3,5,4,2,7,6] => 7
[1,3,5,4,6,2,7] => 5
[1,3,5,4,6,7,2] => 6
[1,3,5,4,7,2,6] => 6
[1,3,5,4,7,6,2] => 5
[1,3,5,6,2,4,7] => 7
[1,3,5,6,2,7,4] => 8
[1,3,5,6,4,2,7] => 6
[1,3,5,6,4,7,2] => 7
[1,3,5,6,7,2,4] => 7
[1,3,5,6,7,4,2] => 8
[1,3,5,7,2,4,6] => 8
[1,3,5,7,2,6,4] => 7
[1,3,5,7,4,2,6] => 7
[1,3,5,7,4,6,2] => 6
[1,3,5,7,6,2,4] => 8
[1,3,5,7,6,4,2] => 7
[1,3,6,2,4,5,7] => 6
[1,3,6,2,4,7,5] => 7
[1,3,6,2,5,4,7] => 5
[1,3,6,2,5,7,4] => 6
[1,3,6,2,7,4,5] => 8
[1,3,6,2,7,5,4] => 7
[1,3,6,4,2,5,7] => 5
[1,3,6,4,2,7,5] => 6
[1,3,6,4,5,2,7] => 4
[1,3,6,4,5,7,2] => 5
[1,3,6,4,7,2,5] => 7
[1,3,6,4,7,5,2] => 6
[1,3,6,5,2,4,7] => 6
[1,3,6,5,2,7,4] => 7
[1,3,6,5,4,2,7] => 7
[1,3,6,5,4,7,2] => 8
[1,3,6,5,7,2,4] => 8
[1,3,6,5,7,4,2] => 7
[1,3,6,7,2,4,5] => 7
[1,3,6,7,2,5,4] => 8
[1,3,6,7,4,2,5] => 8
[1,3,6,7,4,5,2] => 7
[1,3,6,7,5,2,4] => 7
[1,3,6,7,5,4,2] => 6
[1,3,7,2,4,5,6] => 7
[1,3,7,2,4,6,5] => 6
[1,3,7,2,5,4,6] => 6
[1,3,7,2,5,6,4] => 5
[1,3,7,2,6,4,5] => 7
[1,3,7,2,6,5,4] => 8
[1,3,7,4,2,5,6] => 6
[1,3,7,4,2,6,5] => 5
[1,3,7,4,5,2,6] => 5
[1,3,7,4,5,6,2] => 4
[1,3,7,4,6,2,5] => 6
[1,3,7,4,6,5,2] => 7
[1,3,7,5,2,4,6] => 7
[1,3,7,5,2,6,4] => 6
[1,3,7,5,4,2,6] => 8
[1,3,7,5,4,6,2] => 7
[1,3,7,5,6,2,4] => 7
[1,3,7,5,6,4,2] => 8
[1,3,7,6,2,4,5] => 8
[1,3,7,6,2,5,4] => 7
[1,3,7,6,4,2,5] => 7
[1,3,7,6,4,5,2] => 8
[1,3,7,6,5,2,4] => 6
[1,3,7,6,5,4,2] => 7
[1,4,2,3,5,6,7] => 4
[1,4,2,3,5,7,6] => 7
[1,4,2,3,6,5,7] => 7
[1,4,2,3,6,7,5] => 8
[1,4,2,3,7,5,6] => 8
[1,4,2,3,7,6,5] => 7
[1,4,2,5,3,6,7] => 5
[1,4,2,5,3,7,6] => 8
[1,4,2,5,6,3,7] => 6
[1,4,2,5,6,7,3] => 7
[1,4,2,5,7,3,6] => 7
[1,4,2,5,7,6,3] => 6
[1,4,2,6,3,5,7] => 6
[1,4,2,6,3,7,5] => 7
[1,4,2,6,5,3,7] => 5
[1,4,2,6,5,7,3] => 6
[1,4,2,6,7,3,5] => 8
[1,4,2,6,7,5,3] => 7
[1,4,2,7,3,5,6] => 7
[1,4,2,7,3,6,5] => 6
[1,4,2,7,5,3,6] => 6
[1,4,2,7,5,6,3] => 5
[1,4,2,7,6,3,5] => 7
[1,4,2,7,6,5,3] => 8
[1,4,3,2,5,6,7] => 3
[1,4,3,2,5,7,6] => 6
[1,4,3,2,6,5,7] => 6
[1,4,3,2,6,7,5] => 7
[1,4,3,2,7,5,6] => 7
[1,4,3,2,7,6,5] => 6
[1,4,3,5,2,6,7] => 4
[1,4,3,5,2,7,6] => 7
[1,4,3,5,6,2,7] => 5
[1,4,3,5,6,7,2] => 6
[1,4,3,5,7,2,6] => 6
[1,4,3,5,7,6,2] => 5
[1,4,3,6,2,5,7] => 5
[1,4,3,6,2,7,5] => 6
[1,4,3,6,5,2,7] => 4
[1,4,3,6,5,7,2] => 5
[1,4,3,6,7,2,5] => 7
[1,4,3,6,7,5,2] => 6
[1,4,3,7,2,5,6] => 6
[1,4,3,7,2,6,5] => 5
[1,4,3,7,5,2,6] => 5
[1,4,3,7,5,6,2] => 4
[1,4,3,7,6,2,5] => 6
[1,4,3,7,6,5,2] => 7
[1,4,5,2,3,6,7] => 6
[1,4,5,2,3,7,6] => 9
[1,4,5,2,6,3,7] => 7
[1,4,5,2,6,7,3] => 8
[1,4,5,2,7,3,6] => 8
[1,4,5,2,7,6,3] => 7
[1,4,5,3,2,6,7] => 5
[1,4,5,3,2,7,6] => 8
[1,4,5,3,6,2,7] => 6
[1,4,5,3,6,7,2] => 7
[1,4,5,3,7,2,6] => 7
[1,4,5,3,7,6,2] => 6
[1,4,5,6,2,3,7] => 6
[1,4,5,6,2,7,3] => 7
[1,4,5,6,3,2,7] => 7
[1,4,5,6,3,7,2] => 8
[1,4,5,6,7,2,3] => 8
[1,4,5,6,7,3,2] => 7
[1,4,5,7,2,3,6] => 7
[1,4,5,7,2,6,3] => 6
[1,4,5,7,3,2,6] => 8
[1,4,5,7,3,6,2] => 7
[1,4,5,7,6,2,3] => 7
[1,4,5,7,6,3,2] => 8
[1,4,6,2,3,5,7] => 7
[1,4,6,2,3,7,5] => 8
[1,4,6,2,5,3,7] => 6
[1,4,6,2,5,7,3] => 7
[1,4,6,2,7,3,5] => 9
[1,4,6,2,7,5,3] => 8
[1,4,6,3,2,5,7] => 6
[1,4,6,3,2,7,5] => 7
[1,4,6,3,5,2,7] => 5
[1,4,6,3,5,7,2] => 6
[1,4,6,3,7,2,5] => 8
[1,4,6,3,7,5,2] => 7
[1,4,6,5,2,3,7] => 7
[1,4,6,5,2,7,3] => 8
[1,4,6,5,3,2,7] => 6
[1,4,6,5,3,7,2] => 7
click to show generating function       
Description
The prefix exchange distance of a permutation.
This is the number of star transpositions needed to write a permutation.
In symbols, for a permutation $\pi\in\mathfrak S_n$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (1,a)$ for $2 \leq a \leq n$.
[1, Lem. 2.1] shows that the this length is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$.
One may find in [2] explicit formulas for its generating function and a combinatorial proof that it is asymptotically normal.
References
[1] Pak, I. Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and $k$-ary trees MathSciNet:1691876
[2] Grusea, S., Labarre, A. Asymptotic normality and combinatorial aspects of the prefix exchange distance distribution MathSciNet:3497998
Code
def statistic(pi):
    ct = pi.cycle_tuples()
    one = None # cycle containing 1
    two = 0    # other cycles of length at least 2
    fix = 0    # other fixed points
    for c in ct:
        if one is None and 1 in c:
            one = True
        elif len(c) > 1:
            two += 1
        else:
            fix += 1

    return len(pi) + two - fix - 1

# alternative code for checking the lemma
def statistic_alt1(pi):
    def gens(n):
        S = Permutations(n)
        return [S(Permutation((1, i))) for i in range(2, n+1)]

    G = PermutationGroup(gens(len(pi)))
    pi = G(pi)
    w = gap.Factorization(G._gap_(), pi._gap_())
    return w.Length()
Created
Jan 08, 2018 at 23:01 by Martin Rubey
Updated
Aug 15, 2019 at 12:55 by Christian Stump