Identifier
Identifier
Values
[1] => 0
[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] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 3
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 4
[2,4,3,1] => 3
[3,1,2,4] => 2
[3,1,4,2] => 4
[3,2,1,4] => 1
[3,2,4,1] => 3
[3,4,1,2] => 3
[3,4,2,1] => 2
[4,1,2,3] => 2
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 4
[4,3,1,2] => 2
[4,3,2,1] => 1
[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] => 5
[1,3,4,2,5] => 3
[1,3,4,5,2] => 3
[1,3,5,2,4] => 5
[1,3,5,4,2] => 4
[1,4,2,3,5] => 3
[1,4,2,5,3] => 5
[1,4,3,2,5] => 3
[1,4,3,5,2] => 5
[1,4,5,2,3] => 3
[1,4,5,3,2] => 4
[1,5,2,3,4] => 3
[1,5,2,4,3] => 4
[1,5,3,2,4] => 5
[1,5,3,4,2] => 5
[1,5,4,2,3] => 4
[1,5,4,3,2] => 3
[2,1,3,4,5] => 1
[2,1,3,5,4] => 4
[2,1,4,3,5] => 3
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 4
[2,3,1,4,5] => 2
[2,3,1,5,4] => 4
[2,3,4,1,5] => 2
[2,3,4,5,1] => 2
[2,3,5,1,4] => 4
[2,3,5,4,1] => 3
[2,4,1,3,5] => 4
[2,4,1,5,3] => 5
[2,4,3,1,5] => 3
[2,4,3,5,1] => 5
[2,4,5,1,3] => 4
[2,4,5,3,1] => 4
[2,5,1,3,4] => 4
[2,5,1,4,3] => 5
[2,5,3,1,4] => 5
[2,5,3,4,1] => 4
[2,5,4,1,3] => 4
[2,5,4,3,1] => 3
[3,1,2,4,5] => 2
[3,1,2,5,4] => 4
[3,1,4,2,5] => 4
[3,1,4,5,2] => 4
[3,1,5,2,4] => 5
[3,1,5,4,2] => 5
[3,2,1,4,5] => 1
[3,2,1,5,4] => 3
[3,2,4,1,5] => 3
[3,2,4,5,1] => 3
[3,2,5,1,4] => 4
[3,2,5,4,1] => 4
[3,4,1,2,5] => 3
[3,4,1,5,2] => 4
[3,4,2,1,5] => 2
[3,4,2,5,1] => 4
[3,4,5,1,2] => 3
[3,4,5,2,1] => 2
[3,5,1,2,4] => 4
[3,5,1,4,2] => 5
[3,5,2,1,4] => 4
[3,5,2,4,1] => 5
[3,5,4,1,2] => 4
[3,5,4,2,1] => 3
[4,1,2,3,5] => 2
[4,1,2,5,3] => 4
[4,1,3,2,5] => 3
[4,1,3,5,2] => 5
[4,1,5,2,3] => 4
[4,1,5,3,2] => 4
[4,2,1,3,5] => 3
[4,2,1,5,3] => 4
[4,2,3,1,5] => 4
[4,2,3,5,1] => 4
[4,2,5,1,3] => 5
[4,2,5,3,1] => 5
[4,3,1,2,5] => 2
[4,3,1,5,2] => 4
[4,3,2,1,5] => 1
[4,3,2,5,1] => 3
[4,3,5,1,2] => 4
[4,3,5,2,1] => 3
[4,5,1,2,3] => 3
[4,5,1,3,2] => 4
[4,5,2,1,3] => 4
[4,5,2,3,1] => 4
[4,5,3,1,2] => 3
[4,5,3,2,1] => 2
[5,1,2,3,4] => 2
[5,1,2,4,3] => 3
[5,1,3,2,4] => 5
[5,1,3,4,2] => 4
[5,1,4,2,3] => 4
[5,1,4,3,2] => 3
[5,2,1,3,4] => 3
[5,2,1,4,3] => 4
[5,2,3,1,4] => 4
[5,2,3,4,1] => 4
[5,2,4,1,3] => 5
[5,2,4,3,1] => 4
[5,3,1,2,4] => 4
[5,3,1,4,2] => 5
[5,3,2,1,4] => 3
[5,3,2,4,1] => 4
[5,3,4,1,2] => 4
[5,3,4,2,1] => 4
[5,4,1,2,3] => 2
[5,4,1,3,2] => 3
[5,4,2,1,3] => 3
[5,4,2,3,1] => 4
[5,4,3,1,2] => 2
[5,4,3,2,1] => 1
[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] => 5
[1,2,4,5,3,6] => 4
[1,2,4,5,6,3] => 4
[1,2,4,6,3,5] => 6
[1,2,4,6,5,3] => 5
[1,2,5,3,4,6] => 4
[1,2,5,3,6,4] => 6
[1,2,5,4,3,6] => 3
[1,2,5,4,6,3] => 5
[1,2,5,6,3,4] => 4
[1,2,5,6,4,3] => 4
[1,2,6,3,4,5] => 4
[1,2,6,3,5,4] => 5
[1,2,6,4,3,5] => 5
[1,2,6,4,5,3] => 6
[1,2,6,5,3,4] => 4
[1,2,6,5,4,3] => 3
[1,3,2,4,5,6] => 3
[1,3,2,4,6,5] => 6
[1,3,2,5,4,6] => 5
[1,3,2,5,6,4] => 5
[1,3,2,6,4,5] => 5
[1,3,2,6,5,4] => 5
[1,3,4,2,5,6] => 3
[1,3,4,2,6,5] => 5
[1,3,4,5,2,6] => 3
[1,3,4,5,6,2] => 3
[1,3,4,6,2,5] => 5
[1,3,4,6,5,2] => 4
[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] => 6
[1,3,5,6,2,4] => 5
[1,3,5,6,4,2] => 5
[1,3,6,2,4,5] => 5
[1,3,6,2,5,4] => 6
[1,3,6,4,2,5] => 6
[1,3,6,4,5,2] => 5
[1,3,6,5,2,4] => 6
[1,3,6,5,4,2] => 4
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 5
[1,4,2,5,3,6] => 5
[1,4,2,5,6,3] => 5
[1,4,2,6,3,5] => 6
[1,4,2,6,5,3] => 6
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 5
[1,4,3,5,2,6] => 5
[1,4,3,5,6,2] => 5
[1,4,3,6,2,5] => 6
[1,4,3,6,5,2] => 6
[1,4,5,2,3,6] => 3
[1,4,5,2,6,3] => 5
[1,4,5,3,2,6] => 4
[1,4,5,3,6,2] => 5
[1,4,5,6,2,3] => 3
[1,4,5,6,3,2] => 4
[1,4,6,2,3,5] => 5
[1,4,6,2,5,3] => 6
[1,4,6,3,2,5] => 6
[1,4,6,3,5,2] => 6
[1,4,6,5,2,3] => 4
[1,4,6,5,3,2] => 5
[1,5,2,3,4,6] => 3
[1,5,2,3,6,4] => 5
[1,5,2,4,3,6] => 4
[1,5,2,4,6,3] => 6
[1,5,2,6,3,4] => 5
[1,5,2,6,4,3] => 6
[1,5,3,2,4,6] => 5
[1,5,3,2,6,4] => 6
[1,5,3,4,2,6] => 5
[1,5,3,4,6,2] => 5
[1,5,3,6,2,4] => 6
[1,5,3,6,4,2] => 6
[1,5,4,2,3,6] => 4
[1,5,4,2,6,3] => 6
[1,5,4,3,2,6] => 3
[1,5,4,3,6,2] => 5
[1,5,4,6,2,3] => 5
[1,5,4,6,3,2] => 5
[1,5,6,2,3,4] => 3
[1,5,6,2,4,3] => 4
[1,5,6,3,2,4] => 5
[1,5,6,3,4,2] => 5
[1,5,6,4,2,3] => 5
[1,5,6,4,3,2] => 4
[1,6,2,3,4,5] => 3
[1,6,2,3,5,4] => 4
[1,6,2,4,3,5] => 6
[1,6,2,4,5,3] => 5
[1,6,2,5,3,4] => 5
[1,6,2,5,4,3] => 4
[1,6,3,2,4,5] => 5
[1,6,3,2,5,4] => 6
[1,6,3,4,2,5] => 5
[1,6,3,4,5,2] => 5
[1,6,3,5,2,4] => 6
[1,6,3,5,4,2] => 6
[1,6,4,2,3,5] => 5
[1,6,4,2,5,3] => 6
[1,6,4,3,2,5] => 5
[1,6,4,3,5,2] => 6
[1,6,4,5,2,3] => 5
[1,6,4,5,3,2] => 5
[1,6,5,2,3,4] => 4
[1,6,5,2,4,3] => 5
[1,6,5,3,2,4] => 5
[1,6,5,3,4,2] => 5
[1,6,5,4,2,3] => 4
[1,6,5,4,3,2] => 3
[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] => 3
[2,1,4,3,6,5] => 5
[2,1,4,5,3,6] => 3
[2,1,4,5,6,3] => 3
[2,1,4,6,3,5] => 5
[2,1,4,6,5,3] => 4
[2,1,5,3,4,6] => 3
[2,1,5,3,6,4] => 5
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 5
[2,1,5,6,3,4] => 3
[2,1,5,6,4,3] => 4
[2,1,6,3,4,5] => 3
[2,1,6,3,5,4] => 4
[2,1,6,4,3,5] => 5
[2,1,6,4,5,3] => 5
[2,1,6,5,3,4] => 4
[2,1,6,5,4,3] => 4
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 5
[2,3,1,5,4,6] => 4
[2,3,1,5,6,4] => 4
[2,3,1,6,4,5] => 4
[2,3,1,6,5,4] => 5
[2,3,4,1,5,6] => 2
[2,3,4,1,6,5] => 4
[2,3,4,5,1,6] => 2
[2,3,4,5,6,1] => 2
[2,3,4,6,1,5] => 4
[2,3,4,6,5,1] => 3
[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] => 5
[2,3,5,6,1,4] => 4
[2,3,5,6,4,1] => 4
[2,3,6,1,4,5] => 4
[2,3,6,1,5,4] => 5
[2,3,6,4,1,5] => 5
[2,3,6,4,5,1] => 4
[2,3,6,5,1,4] => 5
[2,3,6,5,4,1] => 3
[2,4,1,3,5,6] => 4
[2,4,1,3,6,5] => 6
[2,4,1,5,3,6] => 5
[2,4,1,5,6,3] => 5
[2,4,1,6,3,5] => 6
[2,4,1,6,5,3] => 6
[2,4,3,1,5,6] => 3
[2,4,3,1,6,5] => 5
[2,4,3,5,1,6] => 5
[2,4,3,5,6,1] => 5
[2,4,3,6,1,5] => 5
[2,4,3,6,5,1] => 5
[2,4,5,1,3,6] => 4
[2,4,5,1,6,3] => 5
[2,4,5,3,1,6] => 4
[2,4,5,3,6,1] => 5
[2,4,5,6,1,3] => 4
[2,4,5,6,3,1] => 4
[2,4,6,1,3,5] => 6
[2,4,6,1,5,3] => 6
[2,4,6,3,1,5] => 6
[2,4,6,3,5,1] => 6
[2,4,6,5,1,3] => 5
[2,4,6,5,3,1] => 5
[2,5,1,3,4,6] => 4
[2,5,1,3,6,4] => 6
[2,5,1,4,3,6] => 5
[2,5,1,4,6,3] => 6
[2,5,1,6,3,4] => 5
[2,5,1,6,4,3] => 5
[2,5,3,1,4,6] => 5
[2,5,3,1,6,4] => 6
[2,5,3,4,1,6] => 4
[2,5,3,4,6,1] => 5
[2,5,3,6,1,4] => 6
[2,5,3,6,4,1] => 6
[2,5,4,1,3,6] => 4
[2,5,4,1,6,3] => 5
[2,5,4,3,1,6] => 3
[2,5,4,3,6,1] => 5
[2,5,4,6,1,3] => 6
[2,5,4,6,3,1] => 5
[2,5,6,1,3,4] => 4
[2,5,6,1,4,3] => 5
[2,5,6,3,1,4] => 5
[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] => 4
[2,6,1,3,5,4] => 5
[2,6,1,4,3,5] => 6
[2,6,1,4,5,3] => 5
[2,6,1,5,3,4] => 5
[2,6,1,5,4,3] => 5
[2,6,3,1,4,5] => 5
[2,6,3,1,5,4] => 6
[2,6,3,4,1,5] => 5
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 6
[2,6,3,5,4,1] => 5
[2,6,4,1,3,5] => 6
[2,6,4,1,5,3] => 6
[2,6,4,3,1,5] => 5
[2,6,4,3,5,1] => 5
[2,6,4,5,1,3] => 6
[2,6,4,5,3,1] => 6
[2,6,5,1,3,4] => 4
[2,6,5,1,4,3] => 5
[2,6,5,3,1,4] => 5
[2,6,5,3,4,1] => 4
[2,6,5,4,1,3] => 4
[2,6,5,4,3,1] => 3
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 5
[3,1,2,5,4,6] => 4
[3,1,2,5,6,4] => 4
[3,1,2,6,4,5] => 4
[3,1,2,6,5,4] => 5
[3,1,4,2,5,6] => 4
[3,1,4,2,6,5] => 6
[3,1,4,5,2,6] => 4
[3,1,4,5,6,2] => 4
[3,1,4,6,2,5] => 6
[3,1,4,6,5,2] => 5
[3,1,5,2,4,6] => 5
[3,1,5,2,6,4] => 6
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 6
[3,1,5,6,2,4] => 5
[3,1,5,6,4,2] => 5
[3,1,6,2,4,5] => 5
[3,1,6,2,5,4] => 6
[3,1,6,4,2,5] => 6
[3,1,6,4,5,2] => 5
[3,1,6,5,2,4] => 5
[3,1,6,5,4,2] => 5
[3,2,1,4,5,6] => 1
[3,2,1,4,6,5] => 4
[3,2,1,5,4,6] => 3
[3,2,1,5,6,4] => 3
[3,2,1,6,4,5] => 3
[3,2,1,6,5,4] => 4
[3,2,4,1,5,6] => 3
[3,2,4,1,6,5] => 5
[3,2,4,5,1,6] => 3
[3,2,4,5,6,1] => 3
[3,2,4,6,1,5] => 5
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 4
[3,2,5,1,6,4] => 5
[3,2,5,4,1,6] => 4
[3,2,5,4,6,1] => 5
[3,2,5,6,1,4] => 5
[3,2,5,6,4,1] => 5
[3,2,6,1,4,5] => 4
[3,2,6,1,5,4] => 5
[3,2,6,4,1,5] => 5
[3,2,6,4,5,1] => 5
[3,2,6,5,1,4] => 4
[3,2,6,5,4,1] => 4
[3,4,1,2,5,6] => 3
[3,4,1,2,6,5] => 5
[3,4,1,5,2,6] => 4
[3,4,1,5,6,2] => 4
[3,4,1,6,2,5] => 6
[3,4,1,6,5,2] => 5
[3,4,2,1,5,6] => 2
[3,4,2,1,6,5] => 4
[3,4,2,5,1,6] => 4
[3,4,2,5,6,1] => 4
[3,4,2,6,1,5] => 5
[3,4,2,6,5,1] => 5
[3,4,5,1,2,6] => 3
[3,4,5,1,6,2] => 4
[3,4,5,2,1,6] => 2
[3,4,5,2,6,1] => 4
[3,4,5,6,1,2] => 3
[3,4,5,6,2,1] => 2
[3,4,6,1,2,5] => 5
[3,4,6,1,5,2] => 5
[3,4,6,2,1,5] => 4
[3,4,6,2,5,1] => 5
[3,4,6,5,1,2] => 4
[3,4,6,5,2,1] => 3
[3,5,1,2,4,6] => 4
[3,5,1,2,6,4] => 6
[3,5,1,4,2,6] => 5
[3,5,1,4,6,2] => 6
[3,5,1,6,2,4] => 6
[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] => 5
[3,5,2,4,6,1] => 6
[3,5,2,6,1,4] => 6
[3,5,2,6,4,1] => 6
[3,5,4,1,2,6] => 4
[3,5,4,1,6,2] => 5
[3,5,4,2,1,6] => 3
[3,5,4,2,6,1] => 5
[3,5,4,6,1,2] => 6
[3,5,4,6,2,1] => 5
[3,5,6,1,2,4] => 5
[3,5,6,1,4,2] => 6
[3,5,6,2,1,4] => 4
[3,5,6,2,4,1] => 5
[3,5,6,4,1,2] => 5
[3,5,6,4,2,1] => 4
[3,6,1,2,4,5] => 4
[3,6,1,2,5,4] => 5
[3,6,1,4,2,5] => 6
[3,6,1,4,5,2] => 6
[3,6,1,5,2,4] => 6
[3,6,1,5,4,2] => 5
[3,6,2,1,4,5] => 4
[3,6,2,1,5,4] => 5
[3,6,2,4,1,5] => 6
[3,6,2,4,5,1] => 5
[3,6,2,5,1,4] => 6
[3,6,2,5,4,1] => 5
[3,6,4,1,2,5] => 6
[3,6,4,1,5,2] => 6
[3,6,4,2,1,5] => 5
[3,6,4,2,5,1] => 6
[3,6,4,5,1,2] => 5
[3,6,4,5,2,1] => 4
[3,6,5,1,2,4] => 4
[3,6,5,1,4,2] => 5
[3,6,5,2,1,4] => 5
[3,6,5,2,4,1] => 5
[3,6,5,4,1,2] => 4
[3,6,5,4,2,1] => 3
[4,1,2,3,5,6] => 2
[4,1,2,3,6,5] => 4
[4,1,2,5,3,6] => 4
[4,1,2,5,6,3] => 4
[4,1,2,6,3,5] => 5
[4,1,2,6,5,3] => 5
[4,1,3,2,5,6] => 3
[4,1,3,2,6,5] => 5
[4,1,3,5,2,6] => 5
[4,1,3,5,6,2] => 5
[4,1,3,6,2,5] => 6
[4,1,3,6,5,2] => 6
[4,1,5,2,3,6] => 4
[4,1,5,2,6,3] => 6
[4,1,5,3,2,6] => 4
[4,1,5,3,6,2] => 6
[4,1,5,6,2,3] => 4
[4,1,5,6,3,2] => 4
[4,1,6,2,3,5] => 5
[4,1,6,2,5,3] => 6
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 6
[4,1,6,5,2,3] => 5
[4,1,6,5,3,2] => 5
[4,2,1,3,5,6] => 3
[4,2,1,3,6,5] => 5
[4,2,1,5,3,6] => 4
[4,2,1,5,6,3] => 4
[4,2,1,6,3,5] => 5
[4,2,1,6,5,3] => 5
[4,2,3,1,5,6] => 4
[4,2,3,1,6,5] => 6
[4,2,3,5,1,6] => 4
[4,2,3,5,6,1] => 4
[4,2,3,6,1,5] => 6
[4,2,3,6,5,1] => 5
[4,2,5,1,3,6] => 5
[4,2,5,1,6,3] => 6
[4,2,5,3,1,6] => 5
[4,2,5,3,6,1] => 6
[4,2,5,6,1,3] => 5
[4,2,5,6,3,1] => 5
[4,2,6,1,3,5] => 6
[4,2,6,1,5,3] => 6
[4,2,6,3,1,5] => 6
[4,2,6,3,5,1] => 6
[4,2,6,5,1,3] => 5
[4,2,6,5,3,1] => 5
[4,3,1,2,5,6] => 2
[4,3,1,2,6,5] => 4
[4,3,1,5,2,6] => 4
[4,3,1,5,6,2] => 4
[4,3,1,6,2,5] => 5
[4,3,1,6,5,2] => 5
[4,3,2,1,5,6] => 1
[4,3,2,1,6,5] => 3
[4,3,2,5,1,6] => 3
[4,3,2,5,6,1] => 3
[4,3,2,6,1,5] => 4
[4,3,2,6,5,1] => 4
[4,3,5,1,2,6] => 4
[4,3,5,1,6,2] => 5
[4,3,5,2,1,6] => 3
[4,3,5,2,6,1] => 5
[4,3,5,6,1,2] => 4
[4,3,5,6,2,1] => 3
[4,3,6,1,2,5] => 4
[4,3,6,1,5,2] => 6
[4,3,6,2,1,5] => 5
[4,3,6,2,5,1] => 5
[4,3,6,5,1,2] => 4
[4,3,6,5,2,1] => 4
[4,5,1,2,3,6] => 3
[4,5,1,2,6,3] => 5
[4,5,1,3,2,6] => 4
[4,5,1,3,6,2] => 6
[4,5,1,6,2,3] => 5
[4,5,1,6,3,2] => 4
[4,5,2,1,3,6] => 4
[4,5,2,1,6,3] => 4
[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] => 5
[4,5,3,1,2,6] => 3
[4,5,3,1,6,2] => 5
[4,5,3,2,1,6] => 2
[4,5,3,2,6,1] => 4
[4,5,3,6,1,2] => 5
[4,5,3,6,2,1] => 4
[4,5,6,1,2,3] => 3
[4,5,6,1,3,2] => 4
[4,5,6,2,1,3] => 4
[4,5,6,2,3,1] => 4
[4,5,6,3,1,2] => 3
[4,5,6,3,2,1] => 2
[4,6,1,2,3,5] => 4
[4,6,1,2,5,3] => 5
[4,6,1,3,2,5] => 5
[4,6,1,3,5,2] => 6
[4,6,1,5,2,3] => 6
[4,6,1,5,3,2] => 5
[4,6,2,1,3,5] => 5
[4,6,2,1,5,3] => 6
[4,6,2,3,1,5] => 6
[4,6,2,3,5,1] => 6
[4,6,2,5,1,3] => 7
[4,6,2,5,3,1] => 6
[4,6,3,1,2,5] => 5
[4,6,3,1,5,2] => 6
[4,6,3,2,1,5] => 4
[4,6,3,2,5,1] => 5
[4,6,3,5,1,2] => 5
[4,6,3,5,2,1] => 5
[4,6,5,1,2,3] => 4
[4,6,5,1,3,2] => 5
[4,6,5,2,1,3] => 5
[4,6,5,2,3,1] => 5
[4,6,5,3,1,2] => 4
[4,6,5,3,2,1] => 3
[5,1,2,3,4,6] => 2
[5,1,2,3,6,4] => 4
[5,1,2,4,3,6] => 3
[5,1,2,4,6,3] => 5
[5,1,2,6,3,4] => 4
[5,1,2,6,4,3] => 5
[5,1,3,2,4,6] => 5
[5,1,3,2,6,4] => 5
[5,1,3,4,2,6] => 4
[5,1,3,4,6,2] => 5
[5,1,3,6,2,4] => 6
[5,1,3,6,4,2] => 6
[5,1,4,2,3,6] => 4
[5,1,4,2,6,3] => 6
[5,1,4,3,2,6] => 3
[5,1,4,3,6,2] => 5
[5,1,4,6,2,3] => 5
[5,1,4,6,3,2] => 5
[5,1,6,2,3,4] => 4
[5,1,6,2,4,3] => 5
[5,1,6,3,2,4] => 6
[5,1,6,3,4,2] => 6
[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] => 5
[5,2,1,4,3,6] => 4
[5,2,1,4,6,3] => 5
[5,2,1,6,3,4] => 5
[5,2,1,6,4,3] => 4
[5,2,3,1,4,6] => 4
[5,2,3,1,6,4] => 6
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 4
[5,2,3,6,1,4] => 6
[5,2,3,6,4,1] => 5
[5,2,4,1,3,6] => 5
[5,2,4,1,6,3] => 6
[5,2,4,3,1,6] => 4
[5,2,4,3,6,1] => 5
[5,2,4,6,1,3] => 6
[5,2,4,6,3,1] => 6
[5,2,6,1,3,4] => 5
[5,2,6,1,4,3] => 5
[5,2,6,3,1,4] => 6
[5,2,6,3,4,1] => 6
[5,2,6,4,1,3] => 6
[5,2,6,4,3,1] => 5
[5,3,1,2,4,6] => 4
[5,3,1,2,6,4] => 5
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 6
[5,3,1,6,2,4] => 6
[5,3,1,6,4,2] => 6
[5,3,2,1,4,6] => 3
[5,3,2,1,6,4] => 4
[5,3,2,4,1,6] => 4
[5,3,2,4,6,1] => 5
[5,3,2,6,1,4] => 5
[5,3,2,6,4,1] => 5
[5,3,4,1,2,6] => 4
[5,3,4,1,6,2] => 6
[5,3,4,2,1,6] => 4
[5,3,4,2,6,1] => 6
[5,3,4,6,1,2] => 5
[5,3,4,6,2,1] => 4
[5,3,6,1,2,4] => 5
[5,3,6,1,4,2] => 7
[5,3,6,2,1,4] => 5
[5,3,6,2,4,1] => 6
[5,3,6,4,1,2] => 5
[5,3,6,4,2,1] => 5
[5,4,1,2,3,6] => 2
[5,4,1,2,6,3] => 4
[5,4,1,3,2,6] => 3
[5,4,1,3,6,2] => 5
[5,4,1,6,2,3] => 4
[5,4,1,6,3,2] => 5
[5,4,2,1,3,6] => 3
[5,4,2,1,6,3] => 5
[5,4,2,3,1,6] => 4
[5,4,2,3,6,1] => 4
[5,4,2,6,1,3] => 5
[5,4,2,6,3,1] => 5
[5,4,3,1,2,6] => 2
[5,4,3,1,6,2] => 4
[5,4,3,2,1,6] => 1
[5,4,3,2,6,1] => 3
[5,4,3,6,1,2] => 4
[5,4,3,6,2,1] => 3
[5,4,6,1,2,3] => 4
[5,4,6,1,3,2] => 5
[5,4,6,2,1,3] => 5
[5,4,6,2,3,1] => 5
[5,4,6,3,1,2] => 4
[5,4,6,3,2,1] => 3
[5,6,1,2,3,4] => 3
[5,6,1,2,4,3] => 4
[5,6,1,3,2,4] => 6
[5,6,1,3,4,2] => 5
[5,6,1,4,2,3] => 5
[5,6,1,4,3,2] => 4
[5,6,2,1,3,4] => 4
[5,6,2,1,4,3] => 4
[5,6,2,3,1,4] => 5
[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] => 5
[5,6,3,1,4,2] => 5
[5,6,3,2,1,4] => 4
[5,6,3,2,4,1] => 4
[5,6,3,4,1,2] => 5
[5,6,3,4,2,1] => 4
[5,6,4,1,2,3] => 3
[5,6,4,1,3,2] => 4
[5,6,4,2,1,3] => 4
[5,6,4,2,3,1] => 5
[5,6,4,3,1,2] => 3
[5,6,4,3,2,1] => 2
[6,1,2,3,4,5] => 2
[6,1,2,3,5,4] => 3
[6,1,2,4,3,5] => 5
[6,1,2,4,5,3] => 4
[6,1,2,5,3,4] => 4
[6,1,2,5,4,3] => 3
[6,1,3,2,4,5] => 5
[6,1,3,2,5,4] => 5
[6,1,3,4,2,5] => 5
[6,1,3,4,5,2] => 4
[6,1,3,5,2,4] => 6
[6,1,3,5,4,2] => 5
[6,1,4,2,3,5] => 5
[6,1,4,2,5,3] => 6
[6,1,4,3,2,5] => 5
[6,1,4,3,5,2] => 5
[6,1,4,5,2,3] => 5
[6,1,4,5,3,2] => 4
[6,1,5,2,3,4] => 4
[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] => 3
[6,2,1,3,4,5] => 3
[6,2,1,3,5,4] => 4
[6,2,1,4,3,5] => 5
[6,2,1,4,5,3] => 5
[6,2,1,5,3,4] => 5
[6,2,1,5,4,3] => 4
[6,2,3,1,4,5] => 4
[6,2,3,1,5,4] => 5
[6,2,3,4,1,5] => 4
[6,2,3,4,5,1] => 4
[6,2,3,5,1,4] => 5
[6,2,3,5,4,1] => 5
[6,2,4,1,3,5] => 6
[6,2,4,1,5,3] => 6
[6,2,4,3,1,5] => 5
[6,2,4,3,5,1] => 6
[6,2,4,5,1,3] => 6
[6,2,4,5,3,1] => 5
[6,2,5,1,3,4] => 5
[6,2,5,1,4,3] => 5
[6,2,5,3,1,4] => 6
[6,2,5,3,4,1] => 5
[6,2,5,4,1,3] => 5
[6,2,5,4,3,1] => 4
[6,3,1,2,4,5] => 4
[6,3,1,2,5,4] => 5
[6,3,1,4,2,5] => 6
[6,3,1,4,5,2] => 5
[6,3,1,5,2,4] => 6
[6,3,1,5,4,2] => 5
[6,3,2,1,4,5] => 3
[6,3,2,1,5,4] => 4
[6,3,2,4,1,5] => 5
[6,3,2,4,5,1] => 5
[6,3,2,5,1,4] => 5
[6,3,2,5,4,1] => 4
[6,3,4,1,2,5] => 5
[6,3,4,1,5,2] => 6
[6,3,4,2,1,5] => 4
[6,3,4,2,5,1] => 5
[6,3,4,5,1,2] => 5
[6,3,4,5,2,1] => 4
[6,3,5,1,2,4] => 5
[6,3,5,1,4,2] => 6
[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] => 4
[6,4,1,2,3,5] => 4
[6,4,1,2,5,3] => 5
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 6
[6,4,1,5,2,3] => 5
[6,4,1,5,3,2] => 5
[6,4,2,1,3,5] => 5
[6,4,2,1,5,3] => 5
[6,4,2,3,1,5] => 6
[6,4,2,3,5,1] => 5
[6,4,2,5,1,3] => 6
[6,4,2,5,3,1] => 6
[6,4,3,1,2,5] => 4
[6,4,3,1,5,2] => 5
[6,4,3,2,1,5] => 3
[6,4,3,2,5,1] => 4
[6,4,3,5,1,2] => 4
[6,4,3,5,2,1] => 4
[6,4,5,1,2,3] => 4
[6,4,5,1,3,2] => 5
[6,4,5,2,1,3] => 5
[6,4,5,2,3,1] => 6
[6,4,5,3,1,2] => 5
[6,4,5,3,2,1] => 4
[6,5,1,2,3,4] => 2
[6,5,1,2,4,3] => 3
[6,5,1,3,2,4] => 5
[6,5,1,3,4,2] => 4
[6,5,1,4,2,3] => 4
[6,5,1,4,3,2] => 3
[6,5,2,1,3,4] => 3
[6,5,2,1,4,3] => 4
[6,5,2,3,1,4] => 4
[6,5,2,3,4,1] => 4
[6,5,2,4,1,3] => 5
[6,5,2,4,3,1] => 4
[6,5,3,1,2,4] => 4
[6,5,3,1,4,2] => 5
[6,5,3,2,1,4] => 3
[6,5,3,2,4,1] => 4
[6,5,3,4,1,2] => 4
[6,5,3,4,2,1] => 4
[6,5,4,1,2,3] => 2
[6,5,4,1,3,2] => 3
[6,5,4,2,1,3] => 3
[6,5,4,2,3,1] => 4
[6,5,4,3,1,2] => 2
[6,5,4,3,2,1] => 1
[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] => 5
[1,2,3,5,6,4,7] => 4
[1,2,3,5,6,7,4] => 4
[1,2,3,5,7,4,6] => 6
[1,2,3,5,7,6,4] => 5
[1,2,3,6,4,5,7] => 4
[1,2,3,6,4,7,5] => 6
[1,2,3,6,5,4,7] => 3
[1,2,3,6,5,7,4] => 5
[1,2,3,6,7,4,5] => 4
[1,2,3,6,7,5,4] => 4
[1,2,3,7,4,5,6] => 4
[1,2,3,7,4,6,5] => 5
[1,2,3,7,5,4,6] => 5
[1,2,3,7,5,6,4] => 6
[1,2,3,7,6,4,5] => 4
[1,2,3,7,6,5,4] => 3
[1,2,4,3,5,6,7] => 3
[1,2,4,3,5,7,6] => 6
[1,2,4,3,6,5,7] => 5
[1,2,4,3,6,7,5] => 5
[1,2,4,3,7,5,6] => 5
[1,2,4,3,7,6,5] => 5
[1,2,4,5,3,6,7] => 4
[1,2,4,5,3,7,6] => 5
[1,2,4,5,6,3,7] => 4
[1,2,4,5,6,7,3] => 4
[1,2,4,5,7,3,6] => 6
[1,2,4,5,7,6,3] => 5
[1,2,4,6,3,5,7] => 6
[1,2,4,6,3,7,5] => 7
[1,2,4,6,5,3,7] => 5
[1,2,4,6,5,7,3] => 7
[1,2,4,6,7,3,5] => 6
[1,2,4,6,7,5,3] => 6
[1,2,4,7,3,5,6] => 6
[1,2,4,7,3,6,5] => 6
[1,2,4,7,5,3,6] => 7
[1,2,4,7,5,6,3] => 6
[1,2,4,7,6,3,5] => 6
[1,2,4,7,6,5,3] => 5
[1,2,5,3,4,6,7] => 4
[1,2,5,3,4,7,6] => 5
[1,2,5,3,6,4,7] => 6
[1,2,5,3,6,7,4] => 6
[1,2,5,3,7,4,6] => 7
[1,2,5,3,7,6,4] => 6
[1,2,5,4,3,6,7] => 3
[1,2,5,4,3,7,6] => 5
[1,2,5,4,6,3,7] => 5
[1,2,5,4,6,7,3] => 5
[1,2,5,4,7,3,6] => 6
[1,2,5,4,7,6,3] => 6
[1,2,5,6,3,4,7] => 4
[1,2,5,6,3,7,4] => 6
[1,2,5,6,4,3,7] => 4
[1,2,5,6,4,7,3] => 6
[1,2,5,6,7,3,4] => 4
[1,2,5,6,7,4,3] => 4
[1,2,5,7,3,4,6] => 6
[1,2,5,7,3,6,4] => 7
[1,2,5,7,4,3,6] => 6
[1,2,5,7,4,6,3] => 7
[1,2,5,7,6,3,4] => 5
[1,2,5,7,6,4,3] => 5
[1,2,6,3,4,5,7] => 4
[1,2,6,3,4,7,5] => 6
[1,2,6,3,5,4,7] => 5
[1,2,6,3,5,7,4] => 7
[1,2,6,3,7,4,5] => 6
[1,2,6,3,7,5,4] => 6
[1,2,6,4,3,5,7] => 5
[1,2,6,4,3,7,5] => 6
[1,2,6,4,5,3,7] => 6
[1,2,6,4,5,7,3] => 6
[1,2,6,4,7,3,5] => 7
[1,2,6,4,7,5,3] => 7
[1,2,6,5,3,4,7] => 4
[1,2,6,5,3,7,4] => 6
[1,2,6,5,4,3,7] => 3
[1,2,6,5,4,7,3] => 5
[1,2,6,5,7,3,4] => 6
[1,2,6,5,7,4,3] => 5
[1,2,6,7,3,4,5] => 4
[1,2,6,7,3,5,4] => 5
[1,2,6,7,4,3,5] => 6
[1,2,6,7,4,5,3] => 6
[1,2,6,7,5,3,4] => 5
[1,2,6,7,5,4,3] => 4
[1,2,7,3,4,5,6] => 4
[1,2,7,3,4,6,5] => 5
[1,2,7,3,5,4,6] => 7
[1,2,7,3,5,6,4] => 6
[1,2,7,3,6,4,5] => 6
[1,2,7,3,6,5,4] => 5
[1,2,7,4,3,5,6] => 5
[1,2,7,4,3,6,5] => 6
[1,2,7,4,5,3,6] => 6
[1,2,7,4,5,6,3] => 6
[1,2,7,4,6,3,5] => 7
[1,2,7,4,6,5,3] => 6
[1,2,7,5,3,4,6] => 6
[1,2,7,5,3,6,4] => 7
[1,2,7,5,4,3,6] => 5
[1,2,7,5,4,6,3] => 6
[1,2,7,5,6,3,4] => 6
[1,2,7,5,6,4,3] => 6
[1,2,7,6,3,4,5] => 4
[1,2,7,6,3,5,4] => 5
[1,2,7,6,4,3,5] => 5
[1,2,7,6,4,5,3] => 6
[1,2,7,6,5,3,4] => 4
[1,2,7,6,5,4,3] => 3
[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] => 5
[1,3,2,5,4,7,6] => 7
[1,3,2,5,6,4,7] => 5
[1,3,2,5,6,7,4] => 5
[1,3,2,5,7,4,6] => 7
[1,3,2,5,7,6,4] => 6
[1,3,2,6,4,5,7] => 5
[1,3,2,6,4,7,5] => 7
[1,3,2,6,5,4,7] => 5
[1,3,2,6,5,7,4] => 7
[1,3,2,6,7,4,5] => 5
[1,3,2,6,7,5,4] => 6
[1,3,2,7,4,5,6] => 5
[1,3,2,7,4,6,5] => 6
[1,3,2,7,5,4,6] => 7
[1,3,2,7,5,6,4] => 7
[1,3,2,7,6,4,5] => 6
[1,3,2,7,6,5,4] => 5
[1,3,4,2,5,6,7] => 3
[1,3,4,2,5,7,6] => 6
[1,3,4,2,6,5,7] => 5
[1,3,4,2,6,7,5] => 5
[1,3,4,2,7,5,6] => 5
[1,3,4,2,7,6,5] => 5
[1,3,4,5,2,6,7] => 3
[1,3,4,5,2,7,6] => 5
[1,3,4,5,6,2,7] => 3
[1,3,4,5,6,7,2] => 3
[1,3,4,5,7,2,6] => 5
[1,3,4,5,7,6,2] => 4
[1,3,4,6,2,5,7] => 5
[1,3,4,6,2,7,5] => 6
[1,3,4,6,5,2,7] => 4
[1,3,4,6,5,7,2] => 6
[1,3,4,6,7,2,5] => 5
[1,3,4,6,7,5,2] => 5
[1,3,4,7,2,5,6] => 5
[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] => 6
[1,3,4,7,6,5,2] => 4
[1,3,5,2,4,6,7] => 5
[1,3,5,2,4,7,6] => 7
[1,3,5,2,6,4,7] => 6
[1,3,5,2,6,7,4] => 6
[1,3,5,2,7,4,6] => 7
[1,3,5,2,7,6,4] => 7
[1,3,5,4,2,6,7] => 4
[1,3,5,4,2,7,6] => 6
[1,3,5,4,6,2,7] => 6
[1,3,5,4,6,7,2] => 6
[1,3,5,4,7,2,6] => 6
[1,3,5,4,7,6,2] => 6
[1,3,5,6,2,4,7] => 5
[1,3,5,6,2,7,4] => 7
[1,3,5,6,4,2,7] => 5
[1,3,5,6,4,7,2] => 6
[1,3,5,6,7,2,4] => 5
[1,3,5,6,7,4,2] => 5
[1,3,5,7,2,4,6] => 7
[1,3,5,7,2,6,4] => 7
[1,3,5,7,4,2,6] => 7
[1,3,5,7,4,6,2] => 7
[1,3,5,7,6,2,4] => 6
[1,3,5,7,6,4,2] => 6
[1,3,6,2,4,5,7] => 5
[1,3,6,2,4,7,5] => 7
[1,3,6,2,5,4,7] => 6
[1,3,6,2,5,7,4] => 7
[1,3,6,2,7,4,5] => 7
[1,3,6,2,7,5,4] => 6
[1,3,6,4,2,5,7] => 6
[1,3,6,4,2,7,5] => 7
[1,3,6,4,5,2,7] => 5
[1,3,6,4,5,7,2] => 6
[1,3,6,4,7,2,5] => 7
[1,3,6,4,7,5,2] => 7
[1,3,6,5,2,4,7] => 6
[1,3,6,5,2,7,4] => 6
[1,3,6,5,4,2,7] => 4
[1,3,6,5,4,7,2] => 6
[1,3,6,5,7,2,4] => 7
[1,3,6,5,7,4,2] => 6
[1,3,6,7,2,4,5] => 5
[1,3,6,7,2,5,4] => 6
[1,3,6,7,4,2,5] => 7
[1,3,6,7,4,5,2] => 6
[1,3,6,7,5,2,4] => 7
[1,3,6,7,5,4,2] => 5
[1,3,7,2,4,5,6] => 5
[1,3,7,2,4,6,5] => 6
[1,3,7,2,5,4,6] => 7
[1,3,7,2,5,6,4] => 7
[1,3,7,2,6,4,5] => 7
[1,3,7,2,6,5,4] => 6
[1,3,7,4,2,5,6] => 6
[1,3,7,4,2,6,5] => 7
[1,3,7,4,5,2,6] => 7
[1,3,7,4,5,6,2] => 5
[1,3,7,4,6,2,5] => 7
[1,3,7,4,6,5,2] => 6
[1,3,7,5,2,4,6] => 7
[1,3,7,5,2,6,4] => 8
[1,3,7,5,4,2,6] => 6
[1,3,7,5,4,6,2] => 6
[1,3,7,5,6,2,4] => 7
[1,3,7,5,6,4,2] => 7
[1,3,7,6,2,4,5] => 6
[1,3,7,6,2,5,4] => 6
[1,3,7,6,4,2,5] => 6
[1,3,7,6,4,5,2] => 5
[1,3,7,6,5,2,4] => 6
[1,3,7,6,5,4,2] => 4
[1,4,2,3,5,6,7] => 3
[1,4,2,3,5,7,6] => 6
[1,4,2,3,6,5,7] => 5
[1,4,2,3,6,7,5] => 5
[1,4,2,3,7,5,6] => 5
[1,4,2,3,7,6,5] => 5
[1,4,2,5,3,6,7] => 5
[1,4,2,5,3,7,6] => 7
[1,4,2,5,6,3,7] => 5
[1,4,2,5,6,7,3] => 5
[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] => 6
[1,4,2,6,5,7,3] => 7
[1,4,2,6,7,3,5] => 7
[1,4,2,6,7,5,3] => 7
[1,4,2,7,3,5,6] => 6
[1,4,2,7,3,6,5] => 7
[1,4,2,7,5,3,6] => 7
[1,4,2,7,5,6,3] => 7
[1,4,2,7,6,3,5] => 6
[1,4,2,7,6,5,3] => 6
[1,4,3,2,5,6,7] => 3
[1,4,3,2,5,7,6] => 6
[1,4,3,2,6,5,7] => 5
[1,4,3,2,6,7,5] => 5
[1,4,3,2,7,5,6] => 5
[1,4,3,2,7,6,5] => 6
[1,4,3,5,2,6,7] => 5
[1,4,3,5,2,7,6] => 6
[1,4,3,5,6,2,7] => 5
[1,4,3,5,6,7,2] => 5
[1,4,3,5,7,2,6] => 7
[1,4,3,5,7,6,2] => 6
[1,4,3,6,2,5,7] => 6
[1,4,3,6,2,7,5] => 7
[1,4,3,6,5,2,7] => 6
[1,4,3,6,5,7,2] => 7
[1,4,3,6,7,2,5] => 6
[1,4,3,6,7,5,2] => 7
[1,4,3,7,2,5,6] => 6
[1,4,3,7,2,6,5] => 6
[1,4,3,7,5,2,6] => 7
[1,4,3,7,5,6,2] => 7
[1,4,3,7,6,2,5] => 6
[1,4,3,7,6,5,2] => 6
[1,4,5,2,3,6,7] => 3
[1,4,5,2,3,7,6] => 5
[1,4,5,2,6,3,7] => 5
[1,4,5,2,6,7,3] => 5
[1,4,5,2,7,3,6] => 6
[1,4,5,2,7,6,3] => 6
[1,4,5,3,2,6,7] => 4
[1,4,5,3,2,7,6] => 6
[1,4,5,3,6,2,7] => 5
[1,4,5,3,6,7,2] => 5
[1,4,5,3,7,2,6] => 7
[1,4,5,3,7,6,2] => 6
[1,4,5,6,2,3,7] => 3
[1,4,5,6,2,7,3] => 5
[1,4,5,6,3,2,7] => 4
[1,4,5,6,3,7,2] => 5
[1,4,5,6,7,2,3] => 3
[1,4,5,6,7,3,2] => 4
[1,4,5,7,2,3,6] => 5
[1,4,5,7,2,6,3] => 6
[1,4,5,7,3,2,6] => 6
[1,4,5,7,3,6,2] => 6
[1,4,5,7,6,2,3] => 4
[1,4,5,7,6,3,2] => 5
[1,4,6,2,3,5,7] => 5
[1,4,6,2,3,7,5] => 6
[1,4,6,2,5,3,7] => 6
[1,4,6,2,5,7,3] => 7
[1,4,6,2,7,3,5] => 7
[1,4,6,2,7,5,3] => 7
[1,4,6,3,2,5,7] => 6
[1,4,6,3,2,7,5] => 7
[1,4,6,3,5,2,7] => 6
[1,4,6,3,5,7,2] => 7
[1,4,6,3,7,2,5] => 7
[1,4,6,3,7,5,2] => 7
[1,4,6,5,2,3,7] => 4
[1,4,6,5,2,7,3] => 6
[1,4,6,5,3,2,7] => 5
click to show generating function       
Description
The pancake length of a permutation.
This is the minimal number of pancake moves needed to generate a permutation where a pancake move is a reversal of a prefix in a permutation.
References
[1] Blanco, S. A., Buehrle, C. Some relations on prefix reversal generators of the symmetric and hyperoctahedral group arXiv:1803.01760
Code
@cached_function
def generate_group_from_generators(parent, gens):
    gens = list(gens)
    inds = range(len(gens))

    assert all( gen.parent() == parent for gen in gens )

    one = parent.one()
    level_set_cur  = set([one])
    found_elements = set([one])

    level_sets = []

    while level_set_cur:
        level_sets.append(level_set_cur)
        level_set_new = set()
        for x in level_set_cur:
            for i in inds:
                y = x * gens[i]
                if y not in found_elements:
                    found_elements.add(y)
                    level_set_new.add(y)
        level_set_cur = level_set_new
    return level_sets

@cached_function
def pancake_gens(n):
    return tuple(Permutations(n)(range(k,0,-1)+range(k+1,n+1)) for k in [2 .. n])

def statistic(sigma):
    n = len(sigma)
    level_sets = generate_group_from_generators(Permutations(n), pancake_gens(n))
    i = 0
    for level in level_sets:
        if sigma not in level:
            i += 1
        else:
            return i

Created
Mar 25, 2019 at 14:37 by Christian Stump
Updated
Mar 25, 2019 at 14:37 by Christian Stump