Identifier
Identifier
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[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] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[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] => 2
[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] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 3
[1,5,2,3,4] => 3
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 1
[1,5,4,2,3] => 3
[1,5,4,3,2] => 2
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 2
[2,3,1,4,5] => 2
[2,3,1,5,4] => 3
[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] => 3
[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] => 3
[3,1,2,4,5] => 2
[3,1,2,5,4] => 3
[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] => 2
[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] => 2
[3,4,1,5,2] => 3
[3,4,2,1,5] => 3
[3,4,2,5,1] => 4
[3,4,5,1,2] => 4
[3,4,5,2,1] => 3
[3,5,1,2,4] => 3
[3,5,1,4,2] => 2
[3,5,2,1,4] => 4
[3,5,2,4,1] => 3
[3,5,4,1,2] => 3
[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] => 3
[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] => 2
[4,2,5,3,1] => 3
[4,3,1,2,5] => 3
[4,3,1,5,2] => 4
[4,3,2,1,5] => 2
[4,3,2,5,1] => 3
[4,3,5,1,2] => 3
[4,3,5,2,1] => 4
[4,5,1,2,3] => 4
[4,5,1,3,2] => 3
[4,5,2,1,3] => 3
[4,5,2,3,1] => 4
[4,5,3,1,2] => 2
[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] => 3
[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] => 2
[5,3,1,2,4] => 4
[5,3,1,4,2] => 3
[5,3,2,1,4] => 3
[5,3,2,4,1] => 2
[5,3,4,1,2] => 4
[5,3,4,2,1] => 3
[5,4,1,2,3] => 3
[5,4,1,3,2] => 4
[5,4,2,1,3] => 4
[5,4,2,3,1] => 3
[5,4,3,1,2] => 3
[5,4,3,2,1] => 2
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 1
[1,2,3,5,4,6] => 1
[1,2,3,5,6,4] => 2
[1,2,3,6,4,5] => 2
[1,2,3,6,5,4] => 1
[1,2,4,3,5,6] => 1
[1,2,4,3,6,5] => 2
[1,2,4,5,3,6] => 2
[1,2,4,5,6,3] => 3
[1,2,4,6,3,5] => 3
[1,2,4,6,5,3] => 2
[1,2,5,3,4,6] => 2
[1,2,5,3,6,4] => 3
[1,2,5,4,3,6] => 1
[1,2,5,4,6,3] => 2
[1,2,5,6,3,4] => 2
[1,2,5,6,4,3] => 3
[1,2,6,3,4,5] => 3
[1,2,6,3,5,4] => 2
[1,2,6,4,3,5] => 2
[1,2,6,4,5,3] => 1
[1,2,6,5,3,4] => 3
[1,2,6,5,4,3] => 2
[1,3,2,4,5,6] => 1
[1,3,2,4,6,5] => 2
[1,3,2,5,4,6] => 2
[1,3,2,5,6,4] => 3
[1,3,2,6,4,5] => 3
[1,3,2,6,5,4] => 2
[1,3,4,2,5,6] => 2
[1,3,4,2,6,5] => 3
[1,3,4,5,2,6] => 3
[1,3,4,5,6,2] => 4
[1,3,4,6,2,5] => 4
[1,3,4,6,5,2] => 3
[1,3,5,2,4,6] => 3
[1,3,5,2,6,4] => 4
[1,3,5,4,2,6] => 2
[1,3,5,4,6,2] => 3
[1,3,5,6,2,4] => 3
[1,3,5,6,4,2] => 4
[1,3,6,2,4,5] => 4
[1,3,6,2,5,4] => 3
[1,3,6,4,2,5] => 3
[1,3,6,4,5,2] => 2
[1,3,6,5,2,4] => 4
[1,3,6,5,4,2] => 3
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 3
[1,4,2,5,3,6] => 3
[1,4,2,5,6,3] => 4
[1,4,2,6,3,5] => 4
[1,4,2,6,5,3] => 3
[1,4,3,2,5,6] => 1
[1,4,3,2,6,5] => 2
[1,4,3,5,2,6] => 2
[1,4,3,5,6,2] => 3
[1,4,3,6,2,5] => 3
[1,4,3,6,5,2] => 2
[1,4,5,2,3,6] => 2
[1,4,5,2,6,3] => 3
[1,4,5,3,2,6] => 3
[1,4,5,3,6,2] => 4
[1,4,5,6,2,3] => 4
[1,4,5,6,3,2] => 3
[1,4,6,2,3,5] => 3
[1,4,6,2,5,3] => 2
[1,4,6,3,2,5] => 4
[1,4,6,3,5,2] => 3
[1,4,6,5,2,3] => 3
[1,4,6,5,3,2] => 4
[1,5,2,3,4,6] => 3
[1,5,2,3,6,4] => 4
[1,5,2,4,3,6] => 2
[1,5,2,4,6,3] => 3
[1,5,2,6,3,4] => 3
[1,5,2,6,4,3] => 4
[1,5,3,2,4,6] => 2
[1,5,3,2,6,4] => 3
[1,5,3,4,2,6] => 1
[1,5,3,4,6,2] => 2
[1,5,3,6,2,4] => 2
[1,5,3,6,4,2] => 3
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 4
[1,5,4,3,2,6] => 2
[1,5,4,3,6,2] => 3
[1,5,4,6,2,3] => 3
[1,5,4,6,3,2] => 4
[1,5,6,2,3,4] => 4
[1,5,6,2,4,3] => 3
[1,5,6,3,2,4] => 3
[1,5,6,3,4,2] => 4
[1,5,6,4,2,3] => 2
[1,5,6,4,3,2] => 3
[1,6,2,3,4,5] => 4
[1,6,2,3,5,4] => 3
[1,6,2,4,3,5] => 3
[1,6,2,4,5,3] => 2
[1,6,2,5,3,4] => 4
[1,6,2,5,4,3] => 3
[1,6,3,2,4,5] => 3
[1,6,3,2,5,4] => 2
[1,6,3,4,2,5] => 2
[1,6,3,4,5,2] => 1
[1,6,3,5,2,4] => 3
[1,6,3,5,4,2] => 2
[1,6,4,2,3,5] => 4
[1,6,4,2,5,3] => 3
[1,6,4,3,2,5] => 3
[1,6,4,3,5,2] => 2
[1,6,4,5,2,3] => 4
[1,6,4,5,3,2] => 3
[1,6,5,2,3,4] => 3
[1,6,5,2,4,3] => 4
[1,6,5,3,2,4] => 4
[1,6,5,3,4,2] => 3
[1,6,5,4,2,3] => 3
[1,6,5,4,3,2] => 2
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 2
[2,1,3,5,4,6] => 2
[2,1,3,5,6,4] => 3
[2,1,3,6,4,5] => 3
[2,1,3,6,5,4] => 2
[2,1,4,3,5,6] => 2
[2,1,4,3,6,5] => 3
[2,1,4,5,3,6] => 3
[2,1,4,5,6,3] => 4
[2,1,4,6,3,5] => 4
[2,1,4,6,5,3] => 3
[2,1,5,3,4,6] => 3
[2,1,5,3,6,4] => 4
[2,1,5,4,3,6] => 2
[2,1,5,4,6,3] => 3
[2,1,5,6,3,4] => 3
[2,1,5,6,4,3] => 4
[2,1,6,3,4,5] => 4
[2,1,6,3,5,4] => 3
[2,1,6,4,3,5] => 3
[2,1,6,4,5,3] => 2
[2,1,6,5,3,4] => 4
[2,1,6,5,4,3] => 3
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 3
[2,3,1,5,4,6] => 3
[2,3,1,5,6,4] => 4
[2,3,1,6,4,5] => 4
[2,3,1,6,5,4] => 3
[2,3,4,1,5,6] => 3
[2,3,4,1,6,5] => 4
[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] => 4
[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] => 4
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 4
[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] => 3
[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] => 3
[2,4,5,1,6,3] => 4
[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] => 4
[2,4,6,1,3,5] => 4
[2,4,6,1,5,3] => 3
[2,4,6,3,1,5] => 5
[2,4,6,3,5,1] => 4
[2,4,6,5,1,3] => 4
[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] => 4
[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] => 3
[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] => 3
[2,5,4,3,6,1] => 4
[2,5,4,6,1,3] => 4
[2,5,4,6,3,1] => 5
[2,5,6,1,3,4] => 5
[2,5,6,1,4,3] => 4
[2,5,6,3,1,4] => 4
[2,5,6,3,4,1] => 5
[2,5,6,4,1,3] => 3
[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] => 4
[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] => 3
[2,6,4,1,3,5] => 5
[2,6,4,1,5,3] => 4
[2,6,4,3,1,5] => 4
[2,6,4,3,5,1] => 3
[2,6,4,5,1,3] => 5
[2,6,4,5,3,1] => 4
[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] => 3
[3,1,2,5,4,6] => 3
[3,1,2,5,6,4] => 4
[3,1,2,6,4,5] => 4
[3,1,2,6,5,4] => 3
[3,1,4,2,5,6] => 3
[3,1,4,2,6,5] => 4
[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] => 4
[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] => 4
[3,2,1,4,5,6] => 1
[3,2,1,4,6,5] => 2
[3,2,1,5,4,6] => 2
[3,2,1,5,6,4] => 3
[3,2,1,6,4,5] => 3
[3,2,1,6,5,4] => 2
[3,2,4,1,5,6] => 2
[3,2,4,1,6,5] => 3
[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] => 3
[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] => 3
[3,4,1,2,5,6] => 2
[3,4,1,2,6,5] => 3
[3,4,1,5,2,6] => 3
[3,4,1,5,6,2] => 4
[3,4,1,6,2,5] => 4
[3,4,1,6,5,2] => 3
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 4
[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] => 3
[3,4,5,2,6,1] => 4
[3,4,5,6,1,2] => 4
[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] => 4
[3,4,6,2,5,1] => 3
[3,4,6,5,1,2] => 5
[3,4,6,5,2,1] => 4
[3,5,1,2,4,6] => 3
[3,5,1,2,6,4] => 4
[3,5,1,4,2,6] => 2
[3,5,1,4,6,2] => 3
[3,5,1,6,2,4] => 3
[3,5,1,6,4,2] => 4
[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] => 4
[3,5,2,6,4,1] => 5
[3,5,4,1,2,6] => 3
[3,5,4,1,6,2] => 4
[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] => 4
[3,5,6,1,2,4] => 4
[3,5,6,1,4,2] => 5
[3,5,6,2,1,4] => 5
[3,5,6,2,4,1] => 4
[3,5,6,4,1,2] => 4
[3,5,6,4,2,1] => 3
[3,6,1,2,4,5] => 4
[3,6,1,2,5,4] => 3
[3,6,1,4,2,5] => 3
[3,6,1,4,5,2] => 2
[3,6,1,5,2,4] => 4
[3,6,1,5,4,2] => 3
[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] => 4
[3,6,4,1,2,5] => 4
[3,6,4,1,5,2] => 3
[3,6,4,2,1,5] => 5
[3,6,4,2,5,1] => 4
[3,6,4,5,1,2] => 4
[3,6,4,5,2,1] => 5
[3,6,5,1,2,4] => 5
[3,6,5,1,4,2] => 4
[3,6,5,2,1,4] => 4
[3,6,5,2,4,1] => 5
[3,6,5,4,1,2] => 3
[3,6,5,4,2,1] => 4
[4,1,2,3,5,6] => 3
[4,1,2,3,6,5] => 4
[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] => 3
[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] => 3
[4,1,5,2,6,3] => 4
[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] => 4
[4,1,6,2,3,5] => 4
[4,1,6,2,5,3] => 3
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 4
[4,1,6,5,2,3] => 4
[4,1,6,5,3,2] => 5
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 3
[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] => 2
[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] => 2
[4,2,5,1,6,3] => 3
[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] => 3
[4,2,6,1,3,5] => 3
[4,2,6,1,5,3] => 2
[4,2,6,3,1,5] => 4
[4,2,6,3,5,1] => 3
[4,2,6,5,1,3] => 3
[4,2,6,5,3,1] => 4
[4,3,1,2,5,6] => 3
[4,3,1,2,6,5] => 4
[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] => 2
[4,3,2,1,6,5] => 3
[4,3,2,5,1,6] => 3
[4,3,2,5,6,1] => 4
[4,3,2,6,1,5] => 4
[4,3,2,6,5,1] => 3
[4,3,5,1,2,6] => 3
[4,3,5,1,6,2] => 4
[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] => 4
[4,3,6,1,2,5] => 4
[4,3,6,1,5,2] => 3
[4,3,6,2,1,5] => 5
[4,3,6,2,5,1] => 4
[4,3,6,5,1,2] => 4
[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] => 3
[4,5,1,3,6,2] => 4
[4,5,1,6,2,3] => 4
[4,5,1,6,3,2] => 5
[4,5,2,1,3,6] => 3
[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] => 4
[4,5,3,1,2,6] => 2
[4,5,3,1,6,2] => 3
[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] => 3
[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] => 5
[4,5,6,3,1,2] => 5
[4,5,6,3,2,1] => 4
[4,6,1,2,3,5] => 5
[4,6,1,2,5,3] => 4
[4,6,1,3,2,5] => 4
[4,6,1,3,5,2] => 3
[4,6,1,5,2,3] => 5
[4,6,1,5,3,2] => 4
[4,6,2,1,3,5] => 4
[4,6,2,1,5,3] => 3
[4,6,2,3,1,5] => 5
[4,6,2,3,5,1] => 4
[4,6,2,5,1,3] => 4
[4,6,2,5,3,1] => 5
[4,6,3,1,2,5] => 3
[4,6,3,1,5,2] => 2
[4,6,3,2,1,5] => 4
[4,6,3,2,5,1] => 3
[4,6,3,5,1,2] => 3
[4,6,3,5,2,1] => 4
[4,6,5,1,2,3] => 4
[4,6,5,1,3,2] => 3
[4,6,5,2,1,3] => 5
[4,6,5,2,3,1] => 4
[4,6,5,3,1,2] => 4
[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] => 4
[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] => 3
[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] => 3
[5,1,4,3,6,2] => 4
[5,1,4,6,2,3] => 4
[5,1,4,6,3,2] => 5
[5,1,6,2,3,4] => 5
[5,1,6,2,4,3] => 4
[5,1,6,3,2,4] => 4
[5,1,6,3,4,2] => 5
[5,1,6,4,2,3] => 3
[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] => 3
[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] => 2
[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] => 2
[5,2,4,3,6,1] => 3
[5,2,4,6,1,3] => 3
[5,2,4,6,3,1] => 4
[5,2,6,1,3,4] => 4
[5,2,6,1,4,3] => 3
[5,2,6,3,1,4] => 3
[5,2,6,3,4,1] => 4
[5,2,6,4,1,3] => 2
[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] => 4
[5,3,1,6,4,2] => 5
[5,3,2,1,4,6] => 3
[5,3,2,1,6,4] => 4
[5,3,2,4,1,6] => 2
[5,3,2,4,6,1] => 3
[5,3,2,6,1,4] => 3
[5,3,2,6,4,1] => 4
[5,3,4,1,2,6] => 4
[5,3,4,1,6,2] => 5
[5,3,4,2,1,6] => 3
[5,3,4,2,6,1] => 4
[5,3,4,6,1,2] => 4
[5,3,4,6,2,1] => 5
[5,3,6,1,2,4] => 5
[5,3,6,1,4,2] => 4
[5,3,6,2,1,4] => 4
[5,3,6,2,4,1] => 5
[5,3,6,4,1,2] => 3
[5,3,6,4,2,1] => 4
[5,4,1,2,3,6] => 3
[5,4,1,2,6,3] => 4
[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] => 4
[5,4,2,1,3,6] => 4
[5,4,2,1,6,3] => 5
[5,4,2,3,1,6] => 3
[5,4,2,3,6,1] => 4
[5,4,2,6,1,3] => 4
[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] => 2
[5,4,3,2,6,1] => 3
[5,4,3,6,1,2] => 3
[5,4,3,6,2,1] => 4
[5,4,6,1,2,3] => 4
[5,4,6,1,3,2] => 5
[5,4,6,2,1,3] => 3
[5,4,6,2,3,1] => 4
[5,4,6,3,1,2] => 4
[5,4,6,3,2,1] => 5
[5,6,1,2,3,4] => 4
[5,6,1,2,4,3] => 5
[5,6,1,3,2,4] => 5
[5,6,1,3,4,2] => 4
[5,6,1,4,2,3] => 4
[5,6,1,4,3,2] => 3
[5,6,2,1,3,4] => 5
[5,6,2,1,4,3] => 4
[5,6,2,3,1,4] => 4
[5,6,2,3,4,1] => 5
[5,6,2,4,1,3] => 3
[5,6,2,4,3,1] => 4
[5,6,3,1,2,4] => 4
[5,6,3,1,4,2] => 3
[5,6,3,2,1,4] => 3
[5,6,3,2,4,1] => 4
[5,6,3,4,1,2] => 2
[5,6,3,4,2,1] => 3
[5,6,4,1,2,3] => 5
[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] => 4
[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] => 4
[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] => 3
[6,1,4,2,3,5] => 5
[6,1,4,2,5,3] => 4
[6,1,4,3,2,5] => 4
[6,1,4,3,5,2] => 3
[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] => 4
[6,1,5,4,2,3] => 4
[6,1,5,4,3,2] => 3
[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] => 3
[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] => 2
[6,2,4,1,3,5] => 4
[6,2,4,1,5,3] => 3
[6,2,4,3,1,5] => 3
[6,2,4,3,5,1] => 2
[6,2,4,5,1,3] => 4
[6,2,4,5,3,1] => 3
[6,2,5,1,3,4] => 3
[6,2,5,1,4,3] => 4
[6,2,5,3,1,4] => 4
[6,2,5,3,4,1] => 3
[6,2,5,4,1,3] => 3
[6,2,5,4,3,1] => 2
[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] => 4
[6,3,2,1,4,5] => 4
[6,3,2,1,5,4] => 3
[6,3,2,4,1,5] => 3
[6,3,2,4,5,1] => 2
[6,3,2,5,1,4] => 4
[6,3,2,5,4,1] => 3
[6,3,4,1,2,5] => 5
[6,3,4,1,5,2] => 4
[6,3,4,2,1,5] => 4
[6,3,4,2,5,1] => 3
[6,3,4,5,1,2] => 5
[6,3,4,5,2,1] => 4
[6,3,5,1,2,4] => 4
[6,3,5,1,4,2] => 5
[6,3,5,2,1,4] => 5
[6,3,5,2,4,1] => 4
[6,3,5,4,1,2] => 4
[6,3,5,4,2,1] => 3
[6,4,1,2,3,5] => 4
[6,4,1,2,5,3] => 3
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 4
[6,4,1,5,2,3] => 4
[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] => 4
[6,4,2,3,5,1] => 3
[6,4,2,5,1,3] => 5
[6,4,2,5,3,1] => 4
[6,4,3,1,2,5] => 4
[6,4,3,1,5,2] => 3
[6,4,3,2,1,5] => 3
[6,4,3,2,5,1] => 2
[6,4,3,5,1,2] => 4
[6,4,3,5,2,1] => 3
[6,4,5,1,2,3] => 5
[6,4,5,1,3,2] => 4
[6,4,5,2,1,3] => 4
[6,4,5,2,3,1] => 3
[6,4,5,3,1,2] => 5
[6,4,5,3,2,1] => 4
[6,5,1,2,3,4] => 5
[6,5,1,2,4,3] => 4
[6,5,1,3,2,4] => 4
[6,5,1,3,4,2] => 5
[6,5,1,4,2,3] => 3
[6,5,1,4,3,2] => 4
[6,5,2,1,3,4] => 4
[6,5,2,1,4,3] => 5
[6,5,2,3,1,4] => 5
[6,5,2,3,4,1] => 4
[6,5,2,4,1,3] => 4
[6,5,2,4,3,1] => 3
[6,5,3,1,2,4] => 3
[6,5,3,1,4,2] => 4
[6,5,3,2,1,4] => 4
[6,5,3,2,4,1] => 3
[6,5,3,4,1,2] => 3
[6,5,3,4,2,1] => 2
[6,5,4,1,2,3] => 4
[6,5,4,1,3,2] => 5
[6,5,4,2,1,3] => 5
[6,5,4,2,3,1] => 4
[6,5,4,3,1,2] => 4
[6,5,4,3,2,1] => 3
click to show generating function       
Description
The absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
References
[1] Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind (1<=k<=n; in other words, the unsigned Stirling numbers of the first kind in reverse order). OEIS:A094638
Code
def statistic(pi):
      return len(pi) - len(pi.cycle_tuples())
Created
Aug 29, 2014 at 22:28 by Martin Rubey
Updated
Jul 03, 2019 at 19:02 by Christian Stump