Identifier
Identifier
Values
[1] => 0
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 2
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 2
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
[1,5,2,3,4] => 2
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 2
[1,5,4,2,3] => 2
[1,5,4,3,2] => 2
[2,1,3,4,5] => 2
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 2
[2,3,1,4,5] => 2
[2,3,1,5,4] => 2
[2,3,4,1,5] => 2
[2,3,4,5,1] => 2
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[2,4,1,3,5] => 2
[2,4,1,5,3] => 2
[2,4,3,1,5] => 2
[2,4,3,5,1] => 2
[2,4,5,1,3] => 2
[2,4,5,3,1] => 2
[2,5,1,3,4] => 2
[2,5,1,4,3] => 2
[2,5,3,1,4] => 2
[2,5,3,4,1] => 2
[2,5,4,1,3] => 2
[2,5,4,3,1] => 2
[3,1,2,4,5] => 2
[3,1,2,5,4] => 2
[3,1,4,2,5] => 2
[3,1,4,5,2] => 2
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 2
[3,2,1,5,4] => 2
[3,2,4,1,5] => 2
[3,2,4,5,1] => 2
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 2
[3,4,2,5,1] => 2
[3,4,5,1,2] => 2
[3,4,5,2,1] => 2
[3,5,1,2,4] => 2
[3,5,1,4,2] => 2
[3,5,2,1,4] => 2
[3,5,2,4,1] => 2
[3,5,4,1,2] => 2
[3,5,4,2,1] => 2
[4,1,2,3,5] => 2
[4,1,2,5,3] => 2
[4,1,3,2,5] => 2
[4,1,3,5,2] => 2
[4,1,5,2,3] => 2
[4,1,5,3,2] => 2
[4,2,1,3,5] => 2
[4,2,1,5,3] => 2
[4,2,3,1,5] => 2
[4,2,3,5,1] => 2
[4,2,5,1,3] => 2
[4,2,5,3,1] => 2
[4,3,1,2,5] => 2
[4,3,1,5,2] => 2
[4,3,2,1,5] => 2
[4,3,2,5,1] => 2
[4,3,5,1,2] => 2
[4,3,5,2,1] => 2
[4,5,1,2,3] => 2
[4,5,1,3,2] => 2
[4,5,2,1,3] => 2
[4,5,2,3,1] => 2
[4,5,3,1,2] => 2
[4,5,3,2,1] => 2
[5,1,2,3,4] => 2
[5,1,2,4,3] => 2
[5,1,3,2,4] => 2
[5,1,3,4,2] => 2
[5,1,4,2,3] => 2
[5,1,4,3,2] => 2
[5,2,1,3,4] => 2
[5,2,1,4,3] => 2
[5,2,3,1,4] => 2
[5,2,3,4,1] => 2
[5,2,4,1,3] => 2
[5,2,4,3,1] => 2
[5,3,1,2,4] => 2
[5,3,1,4,2] => 2
[5,3,2,1,4] => 2
[5,3,2,4,1] => 2
[5,3,4,1,2] => 2
[5,3,4,2,1] => 2
[5,4,1,2,3] => 2
[5,4,1,3,2] => 2
[5,4,2,1,3] => 2
[5,4,2,3,1] => 2
[5,4,3,1,2] => 2
[5,4,3,2,1] => 2
[1,2,3,4,5,6] => 3
[1,2,3,4,6,5] => 3
[1,2,3,5,4,6] => 3
[1,2,3,5,6,4] => 3
[1,2,3,6,4,5] => 3
[1,2,3,6,5,4] => 2
[1,2,4,3,5,6] => 3
[1,2,4,3,6,5] => 3
[1,2,4,5,3,6] => 3
[1,2,4,5,6,3] => 3
[1,2,4,6,3,5] => 3
[1,2,4,6,5,3] => 3
[1,2,5,3,4,6] => 3
[1,2,5,3,6,4] => 3
[1,2,5,4,3,6] => 2
[1,2,5,4,6,3] => 3
[1,2,5,6,3,4] => 3
[1,2,5,6,4,3] => 3
[1,2,6,3,4,5] => 3
[1,2,6,3,5,4] => 3
[1,2,6,4,3,5] => 3
[1,2,6,4,5,3] => 3
[1,2,6,5,3,4] => 3
[1,2,6,5,4,3] => 3
[1,3,2,4,5,6] => 3
[1,3,2,4,6,5] => 3
[1,3,2,5,4,6] => 3
[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] => 3
[1,3,4,2,6,5] => 3
[1,3,4,5,2,6] => 3
[1,3,4,5,6,2] => 2
[1,3,4,6,2,5] => 3
[1,3,4,6,5,2] => 3
[1,3,5,2,4,6] => 3
[1,3,5,2,6,4] => 3
[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] => 3
[1,3,6,2,4,5] => 3
[1,3,6,2,5,4] => 3
[1,3,6,4,2,5] => 3
[1,3,6,4,5,2] => 3
[1,3,6,5,2,4] => 3
[1,3,6,5,4,2] => 3
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 3
[1,4,2,5,3,6] => 3
[1,4,2,5,6,3] => 3
[1,4,2,6,3,5] => 3
[1,4,2,6,5,3] => 3
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 3
[1,4,3,5,2,6] => 2
[1,4,3,5,6,2] => 2
[1,4,3,6,2,5] => 3
[1,4,3,6,5,2] => 3
[1,4,5,2,3,6] => 3
[1,4,5,2,6,3] => 3
[1,4,5,3,2,6] => 2
[1,4,5,3,6,2] => 2
[1,4,5,6,2,3] => 3
[1,4,5,6,3,2] => 2
[1,4,6,2,3,5] => 3
[1,4,6,2,5,3] => 2
[1,4,6,3,2,5] => 3
[1,4,6,3,5,2] => 3
[1,4,6,5,2,3] => 2
[1,4,6,5,3,2] => 3
[1,5,2,3,4,6] => 3
[1,5,2,3,6,4] => 3
[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] => 3
[1,5,3,2,4,6] => 2
[1,5,3,2,6,4] => 3
[1,5,3,4,2,6] => 2
[1,5,3,4,6,2] => 2
[1,5,3,6,2,4] => 3
[1,5,3,6,4,2] => 3
[1,5,4,2,3,6] => 2
[1,5,4,2,6,3] => 3
[1,5,4,3,2,6] => 2
[1,5,4,3,6,2] => 2
[1,5,4,6,2,3] => 2
[1,5,4,6,3,2] => 2
[1,5,6,2,3,4] => 3
[1,5,6,2,4,3] => 2
[1,5,6,3,2,4] => 2
[1,5,6,3,4,2] => 2
[1,5,6,4,2,3] => 2
[1,5,6,4,3,2] => 2
[1,6,2,3,4,5] => 2
[1,6,2,3,5,4] => 3
[1,6,2,4,3,5] => 3
[1,6,2,4,5,3] => 3
[1,6,2,5,3,4] => 3
[1,6,2,5,4,3] => 3
[1,6,3,2,4,5] => 2
[1,6,3,2,5,4] => 3
[1,6,3,4,2,5] => 2
[1,6,3,4,5,2] => 2
[1,6,3,5,2,4] => 3
[1,6,3,5,4,2] => 3
[1,6,4,2,3,5] => 2
[1,6,4,2,5,3] => 3
[1,6,4,3,2,5] => 2
[1,6,4,3,5,2] => 2
[1,6,4,5,2,3] => 2
[1,6,4,5,3,2] => 2
[1,6,5,2,3,4] => 2
[1,6,5,2,4,3] => 3
[1,6,5,3,2,4] => 2
[1,6,5,3,4,2] => 2
[1,6,5,4,2,3] => 2
[1,6,5,4,3,2] => 2
[2,1,3,4,5,6] => 3
[2,1,3,4,6,5] => 3
[2,1,3,5,4,6] => 3
[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] => 3
[2,1,4,3,6,5] => 3
[2,1,4,5,3,6] => 3
[2,1,4,5,6,3] => 2
[2,1,4,6,3,5] => 3
[2,1,4,6,5,3] => 3
[2,1,5,3,4,6] => 3
[2,1,5,3,6,4] => 3
[2,1,5,4,3,6] => 3
[2,1,5,4,6,3] => 3
[2,1,5,6,3,4] => 3
[2,1,5,6,4,3] => 3
[2,1,6,3,4,5] => 2
[2,1,6,3,5,4] => 3
[2,1,6,4,3,5] => 3
[2,1,6,4,5,3] => 3
[2,1,6,5,3,4] => 3
[2,1,6,5,4,3] => 3
[2,3,1,4,5,6] => 3
[2,3,1,4,6,5] => 3
[2,3,1,5,4,6] => 3
[2,3,1,5,6,4] => 3
[2,3,1,6,4,5] => 3
[2,3,1,6,5,4] => 2
[2,3,4,1,5,6] => 3
[2,3,4,1,6,5] => 2
[2,3,4,5,1,6] => 2
[2,3,4,5,6,1] => 2
[2,3,4,6,1,5] => 2
[2,3,4,6,5,1] => 2
[2,3,5,1,4,6] => 3
[2,3,5,1,6,4] => 3
[2,3,5,4,1,6] => 2
[2,3,5,4,6,1] => 2
[2,3,5,6,1,4] => 3
[2,3,5,6,4,1] => 3
[2,3,6,1,4,5] => 3
[2,3,6,1,5,4] => 2
[2,3,6,4,1,5] => 2
[2,3,6,4,5,1] => 2
[2,3,6,5,1,4] => 2
[2,3,6,5,4,1] => 2
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 3
[2,4,1,5,3,6] => 3
[2,4,1,5,6,3] => 3
[2,4,1,6,3,5] => 3
[2,4,1,6,5,3] => 3
[2,4,3,1,5,6] => 3
[2,4,3,1,6,5] => 3
[2,4,3,5,1,6] => 3
[2,4,3,5,6,1] => 2
[2,4,3,6,1,5] => 3
[2,4,3,6,5,1] => 2
[2,4,5,1,3,6] => 3
[2,4,5,1,6,3] => 3
[2,4,5,3,1,6] => 2
[2,4,5,3,6,1] => 3
[2,4,5,6,1,3] => 3
[2,4,5,6,3,1] => 3
[2,4,6,1,3,5] => 3
[2,4,6,1,5,3] => 3
[2,4,6,3,1,5] => 2
[2,4,6,3,5,1] => 3
[2,4,6,5,1,3] => 3
[2,4,6,5,3,1] => 3
[2,5,1,3,4,6] => 3
[2,5,1,3,6,4] => 3
[2,5,1,4,3,6] => 3
[2,5,1,4,6,3] => 3
[2,5,1,6,3,4] => 3
[2,5,1,6,4,3] => 3
[2,5,3,1,4,6] => 3
[2,5,3,1,6,4] => 3
[2,5,3,4,1,6] => 2
[2,5,3,4,6,1] => 2
[2,5,3,6,1,4] => 3
[2,5,3,6,4,1] => 3
[2,5,4,1,3,6] => 3
[2,5,4,1,6,3] => 3
[2,5,4,3,1,6] => 2
[2,5,4,3,6,1] => 2
[2,5,4,6,1,3] => 3
[2,5,4,6,3,1] => 3
[2,5,6,1,3,4] => 3
[2,5,6,1,4,3] => 2
[2,5,6,3,1,4] => 3
[2,5,6,3,4,1] => 3
[2,5,6,4,1,3] => 3
[2,5,6,4,3,1] => 3
[2,6,1,3,4,5] => 2
[2,6,1,3,5,4] => 3
[2,6,1,4,3,5] => 3
[2,6,1,4,5,3] => 3
[2,6,1,5,3,4] => 3
[2,6,1,5,4,3] => 3
[2,6,3,1,4,5] => 2
[2,6,3,1,5,4] => 3
[2,6,3,4,1,5] => 2
[2,6,3,4,5,1] => 2
[2,6,3,5,1,4] => 2
[2,6,3,5,4,1] => 2
[2,6,4,1,3,5] => 2
[2,6,4,1,5,3] => 3
[2,6,4,3,1,5] => 2
[2,6,4,3,5,1] => 2
[2,6,4,5,1,3] => 3
[2,6,4,5,3,1] => 3
[2,6,5,1,3,4] => 2
[2,6,5,1,4,3] => 3
[2,6,5,3,1,4] => 2
[2,6,5,3,4,1] => 2
[2,6,5,4,1,3] => 2
[2,6,5,4,3,1] => 2
[3,1,2,4,5,6] => 3
[3,1,2,4,6,5] => 3
[3,1,2,5,4,6] => 3
[3,1,2,5,6,4] => 3
[3,1,2,6,4,5] => 3
[3,1,2,6,5,4] => 2
[3,1,4,2,5,6] => 3
[3,1,4,2,6,5] => 3
[3,1,4,5,2,6] => 3
[3,1,4,5,6,2] => 2
[3,1,4,6,2,5] => 3
[3,1,4,6,5,2] => 3
[3,1,5,2,4,6] => 3
[3,1,5,2,6,4] => 3
[3,1,5,4,2,6] => 3
[3,1,5,4,6,2] => 3
[3,1,5,6,2,4] => 3
[3,1,5,6,4,2] => 3
[3,1,6,2,4,5] => 3
[3,1,6,2,5,4] => 3
[3,1,6,4,2,5] => 3
[3,1,6,4,5,2] => 3
[3,1,6,5,2,4] => 3
[3,1,6,5,4,2] => 3
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 2
[3,2,1,5,4,6] => 2
[3,2,1,5,6,4] => 2
[3,2,1,6,4,5] => 2
[3,2,1,6,5,4] => 3
[3,2,4,1,5,6] => 3
[3,2,4,1,6,5] => 3
[3,2,4,5,1,6] => 3
[3,2,4,5,6,1] => 2
[3,2,4,6,1,5] => 3
[3,2,4,6,5,1] => 2
[3,2,5,1,4,6] => 3
[3,2,5,1,6,4] => 3
[3,2,5,4,1,6] => 3
[3,2,5,4,6,1] => 2
[3,2,5,6,1,4] => 2
[3,2,5,6,4,1] => 2
[3,2,6,1,4,5] => 2
[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] => 3
[3,2,6,5,4,1] => 3
[3,4,1,2,5,6] => 3
[3,4,1,2,6,5] => 3
[3,4,1,5,2,6] => 3
[3,4,1,5,6,2] => 3
[3,4,1,6,2,5] => 3
[3,4,1,6,5,2] => 2
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 3
[3,4,2,5,1,6] => 3
[3,4,2,5,6,1] => 3
[3,4,2,6,1,5] => 3
[3,4,2,6,5,1] => 2
[3,4,5,1,2,6] => 3
[3,4,5,1,6,2] => 3
[3,4,5,2,1,6] => 2
[3,4,5,2,6,1] => 3
[3,4,5,6,1,2] => 3
[3,4,5,6,2,1] => 3
[3,4,6,1,2,5] => 3
[3,4,6,1,5,2] => 3
[3,4,6,2,1,5] => 2
[3,4,6,2,5,1] => 3
[3,4,6,5,1,2] => 3
[3,4,6,5,2,1] => 3
[3,5,1,2,4,6] => 3
[3,5,1,2,6,4] => 3
[3,5,1,4,2,6] => 3
[3,5,1,4,6,2] => 3
[3,5,1,6,2,4] => 3
[3,5,1,6,4,2] => 3
[3,5,2,1,4,6] => 3
[3,5,2,1,6,4] => 3
[3,5,2,4,1,6] => 3
[3,5,2,4,6,1] => 3
[3,5,2,6,1,4] => 3
[3,5,2,6,4,1] => 2
[3,5,4,1,2,6] => 2
[3,5,4,1,6,2] => 3
[3,5,4,2,1,6] => 2
[3,5,4,2,6,1] => 3
[3,5,4,6,1,2] => 3
[3,5,4,6,2,1] => 3
[3,5,6,1,2,4] => 3
[3,5,6,1,4,2] => 3
[3,5,6,2,1,4] => 2
[3,5,6,2,4,1] => 3
[3,5,6,4,1,2] => 3
[3,5,6,4,2,1] => 3
[3,6,1,2,4,5] => 3
[3,6,1,2,5,4] => 2
[3,6,1,4,2,5] => 3
[3,6,1,4,5,2] => 3
[3,6,1,5,2,4] => 3
[3,6,1,5,4,2] => 3
[3,6,2,1,4,5] => 2
[3,6,2,1,5,4] => 3
[3,6,2,4,1,5] => 2
[3,6,2,4,5,1] => 3
[3,6,2,5,1,4] => 3
[3,6,2,5,4,1] => 3
[3,6,4,1,2,5] => 3
[3,6,4,1,5,2] => 3
[3,6,4,2,1,5] => 2
[3,6,4,2,5,1] => 3
[3,6,4,5,1,2] => 3
[3,6,4,5,2,1] => 3
[3,6,5,1,2,4] => 2
[3,6,5,1,4,2] => 3
[3,6,5,2,1,4] => 3
[3,6,5,2,4,1] => 3
[3,6,5,4,1,2] => 2
[3,6,5,4,2,1] => 3
[4,1,2,3,5,6] => 3
[4,1,2,3,6,5] => 2
[4,1,2,5,3,6] => 3
[4,1,2,5,6,3] => 3
[4,1,2,6,3,5] => 3
[4,1,2,6,5,3] => 2
[4,1,3,2,5,6] => 3
[4,1,3,2,6,5] => 3
[4,1,3,5,2,6] => 3
[4,1,3,5,6,2] => 2
[4,1,3,6,2,5] => 3
[4,1,3,6,5,2] => 3
[4,1,5,2,3,6] => 3
[4,1,5,2,6,3] => 3
[4,1,5,3,2,6] => 3
[4,1,5,3,6,2] => 2
[4,1,5,6,2,3] => 3
[4,1,5,6,3,2] => 2
[4,1,6,2,3,5] => 3
[4,1,6,2,5,3] => 3
[4,1,6,3,2,5] => 3
[4,1,6,3,5,2] => 3
[4,1,6,5,2,3] => 2
[4,1,6,5,3,2] => 3
[4,2,1,3,5,6] => 3
[4,2,1,3,6,5] => 3
[4,2,1,5,3,6] => 3
[4,2,1,5,6,3] => 2
[4,2,1,6,3,5] => 3
[4,2,1,6,5,3] => 3
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 3
[4,2,3,5,1,6] => 3
[4,2,3,5,6,1] => 2
[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] => 3
[4,2,5,6,1,3] => 3
[4,2,5,6,3,1] => 3
[4,2,6,1,3,5] => 3
[4,2,6,1,5,3] => 3
[4,2,6,3,1,5] => 3
[4,2,6,3,5,1] => 3
[4,2,6,5,1,3] => 3
[4,2,6,5,3,1] => 3
[4,3,1,2,5,6] => 3
[4,3,1,2,6,5] => 3
[4,3,1,5,2,6] => 3
[4,3,1,5,6,2] => 2
[4,3,1,6,2,5] => 3
[4,3,1,6,5,2] => 3
[4,3,2,1,5,6] => 3
[4,3,2,1,6,5] => 3
[4,3,2,5,1,6] => 3
[4,3,2,5,6,1] => 2
[4,3,2,6,1,5] => 3
[4,3,2,6,5,1] => 3
[4,3,5,1,2,6] => 2
[4,3,5,1,6,2] => 3
[4,3,5,2,1,6] => 3
[4,3,5,2,6,1] => 3
[4,3,5,6,1,2] => 3
[4,3,5,6,2,1] => 3
[4,3,6,1,2,5] => 2
[4,3,6,1,5,2] => 3
[4,3,6,2,1,5] => 3
[4,3,6,2,5,1] => 3
[4,3,6,5,1,2] => 3
[4,3,6,5,2,1] => 3
[4,5,1,2,3,6] => 3
[4,5,1,2,6,3] => 3
[4,5,1,3,2,6] => 2
[4,5,1,3,6,2] => 3
[4,5,1,6,2,3] => 3
[4,5,1,6,3,2] => 2
[4,5,2,1,3,6] => 2
[4,5,2,1,6,3] => 2
[4,5,2,3,1,6] => 2
[4,5,2,3,6,1] => 3
[4,5,2,6,1,3] => 3
[4,5,2,6,3,1] => 3
[4,5,3,1,2,6] => 2
[4,5,3,1,6,2] => 3
[4,5,3,2,1,6] => 2
[4,5,3,2,6,1] => 3
[4,5,3,6,1,2] => 3
[4,5,3,6,2,1] => 3
[4,5,6,1,2,3] => 3
[4,5,6,1,3,2] => 2
[4,5,6,2,1,3] => 2
[4,5,6,2,3,1] => 2
[4,5,6,3,1,2] => 2
[4,5,6,3,2,1] => 2
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 3
[4,6,1,3,2,5] => 3
[4,6,1,3,5,2] => 3
[4,6,1,5,2,3] => 3
[4,6,1,5,3,2] => 3
[4,6,2,1,3,5] => 3
[4,6,2,1,5,3] => 3
[4,6,2,3,1,5] => 3
[4,6,2,3,5,1] => 3
[4,6,2,5,1,3] => 3
[4,6,2,5,3,1] => 3
[4,6,3,1,2,5] => 3
[4,6,3,1,5,2] => 3
[4,6,3,2,1,5] => 2
[4,6,3,2,5,1] => 3
[4,6,3,5,1,2] => 3
[4,6,3,5,2,1] => 3
[4,6,5,1,2,3] => 2
[4,6,5,1,3,2] => 3
[4,6,5,2,1,3] => 3
[4,6,5,2,3,1] => 3
[4,6,5,3,1,2] => 3
[4,6,5,3,2,1] => 3
[5,1,2,3,4,6] => 2
[5,1,2,3,6,4] => 2
[5,1,2,4,3,6] => 2
[5,1,2,4,6,3] => 2
[5,1,2,6,3,4] => 3
[5,1,2,6,4,3] => 2
[5,1,3,2,4,6] => 3
[5,1,3,2,6,4] => 3
[5,1,3,4,2,6] => 2
[5,1,3,4,6,2] => 2
[5,1,3,6,2,4] => 3
[5,1,3,6,4,2] => 2
[5,1,4,2,3,6] => 2
[5,1,4,2,6,3] => 2
[5,1,4,3,2,6] => 2
[5,1,4,3,6,2] => 2
[5,1,4,6,2,3] => 3
[5,1,4,6,3,2] => 2
[5,1,6,2,3,4] => 3
[5,1,6,2,4,3] => 3
[5,1,6,3,2,4] => 3
[5,1,6,3,4,2] => 3
[5,1,6,4,2,3] => 3
[5,1,6,4,3,2] => 2
[5,2,1,3,4,6] => 3
[5,2,1,3,6,4] => 3
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 3
[5,2,1,6,3,4] => 2
[5,2,1,6,4,3] => 3
[5,2,3,1,4,6] => 3
[5,2,3,1,6,4] => 3
[5,2,3,4,1,6] => 2
[5,2,3,4,6,1] => 2
[5,2,3,6,1,4] => 3
[5,2,3,6,4,1] => 3
[5,2,4,1,3,6] => 3
[5,2,4,1,6,3] => 3
[5,2,4,3,1,6] => 2
[5,2,4,3,6,1] => 2
[5,2,4,6,1,3] => 3
[5,2,4,6,3,1] => 3
[5,2,6,1,3,4] => 3
[5,2,6,1,4,3] => 3
[5,2,6,3,1,4] => 3
[5,2,6,3,4,1] => 3
[5,2,6,4,1,3] => 3
[5,2,6,4,3,1] => 3
[5,3,1,2,4,6] => 3
[5,3,1,2,6,4] => 3
[5,3,1,4,2,6] => 3
[5,3,1,4,6,2] => 2
[5,3,1,6,2,4] => 3
[5,3,1,6,4,2] => 3
[5,3,2,1,4,6] => 3
[5,3,2,1,6,4] => 3
[5,3,2,4,1,6] => 3
[5,3,2,4,6,1] => 2
[5,3,2,6,1,4] => 3
[5,3,2,6,4,1] => 3
[5,3,4,1,2,6] => 2
[5,3,4,1,6,2] => 3
[5,3,4,2,1,6] => 2
[5,3,4,2,6,1] => 3
[5,3,4,6,1,2] => 3
[5,3,4,6,2,1] => 3
[5,3,6,1,2,4] => 3
[5,3,6,1,4,2] => 3
[5,3,6,2,1,4] => 3
[5,3,6,2,4,1] => 3
[5,3,6,4,1,2] => 3
[5,3,6,4,2,1] => 3
[5,4,1,2,3,6] => 2
[5,4,1,2,6,3] => 2
[5,4,1,3,2,6] => 2
[5,4,1,3,6,2] => 2
[5,4,1,6,2,3] => 2
[5,4,1,6,3,2] => 3
[5,4,2,1,3,6] => 3
[5,4,2,1,6,3] => 3
[5,4,2,3,1,6] => 2
[5,4,2,3,6,1] => 2
[5,4,2,6,1,3] => 3
[5,4,2,6,3,1] => 3
[5,4,3,1,2,6] => 2
[5,4,3,1,6,2] => 2
[5,4,3,2,1,6] => 2
[5,4,3,2,6,1] => 2
[5,4,3,6,1,2] => 2
[5,4,3,6,2,1] => 3
[5,4,6,1,2,3] => 2
[5,4,6,1,3,2] => 3
[5,4,6,2,1,3] => 3
[5,4,6,2,3,1] => 3
[5,4,6,3,1,2] => 3
[5,4,6,3,2,1] => 3
[5,6,1,2,3,4] => 3
[5,6,1,2,4,3] => 3
[5,6,1,3,2,4] => 3
[5,6,1,3,4,2] => 3
[5,6,1,4,2,3] => 3
[5,6,1,4,3,2] => 2
[5,6,2,1,3,4] => 3
[5,6,2,1,4,3] => 3
[5,6,2,3,1,4] => 3
[5,6,2,3,4,1] => 3
[5,6,2,4,1,3] => 3
[5,6,2,4,3,1] => 3
[5,6,3,1,2,4] => 3
[5,6,3,1,4,2] => 3
[5,6,3,2,1,4] => 2
[5,6,3,2,4,1] => 3
[5,6,3,4,1,2] => 3
[5,6,3,4,2,1] => 3
[5,6,4,1,2,3] => 2
[5,6,4,1,3,2] => 3
[5,6,4,2,1,3] => 3
[5,6,4,2,3,1] => 3
[5,6,4,3,1,2] => 3
[5,6,4,3,2,1] => 3
[6,1,2,3,4,5] => 2
[6,1,2,3,5,4] => 2
[6,1,2,4,3,5] => 2
[6,1,2,4,5,3] => 2
[6,1,2,5,3,4] => 3
[6,1,2,5,4,3] => 2
[6,1,3,2,4,5] => 2
[6,1,3,2,5,4] => 2
[6,1,3,4,2,5] => 2
[6,1,3,4,5,2] => 2
[6,1,3,5,2,4] => 3
[6,1,3,5,4,2] => 2
[6,1,4,2,3,5] => 3
[6,1,4,2,5,3] => 3
[6,1,4,3,2,5] => 2
[6,1,4,3,5,2] => 2
[6,1,4,5,2,3] => 3
[6,1,4,5,3,2] => 2
[6,1,5,2,3,4] => 3
[6,1,5,2,4,3] => 3
[6,1,5,3,2,4] => 3
[6,1,5,3,4,2] => 3
[6,1,5,4,2,3] => 3
[6,1,5,4,3,2] => 2
[6,2,1,3,4,5] => 2
[6,2,1,3,5,4] => 2
[6,2,1,4,3,5] => 2
[6,2,1,4,5,3] => 2
[6,2,1,5,3,4] => 2
[6,2,1,5,4,3] => 3
[6,2,3,1,4,5] => 2
[6,2,3,1,5,4] => 2
[6,2,3,4,1,5] => 2
[6,2,3,4,5,1] => 2
[6,2,3,5,1,4] => 3
[6,2,3,5,4,1] => 2
[6,2,4,1,3,5] => 3
[6,2,4,1,5,3] => 3
[6,2,4,3,1,5] => 2
[6,2,4,3,5,1] => 2
[6,2,4,5,1,3] => 3
[6,2,4,5,3,1] => 2
[6,2,5,1,3,4] => 3
[6,2,5,1,4,3] => 3
[6,2,5,3,1,4] => 3
[6,2,5,3,4,1] => 2
[6,2,5,4,1,3] => 3
[6,2,5,4,3,1] => 3
[6,3,1,2,4,5] => 3
[6,3,1,2,5,4] => 2
[6,3,1,4,2,5] => 3
[6,3,1,4,5,2] => 3
[6,3,1,5,2,4] => 2
[6,3,1,5,4,2] => 3
[6,3,2,1,4,5] => 2
[6,3,2,1,5,4] => 3
[6,3,2,4,1,5] => 2
[6,3,2,4,5,1] => 2
[6,3,2,5,1,4] => 3
[6,3,2,5,4,1] => 3
[6,3,4,1,2,5] => 3
[6,3,4,1,5,2] => 3
[6,3,4,2,1,5] => 2
[6,3,4,2,5,1] => 2
[6,3,4,5,1,2] => 3
[6,3,4,5,2,1] => 2
[6,3,5,1,2,4] => 3
[6,3,5,1,4,2] => 3
[6,3,5,2,1,4] => 3
[6,3,5,2,4,1] => 3
[6,3,5,4,1,2] => 3
[6,3,5,4,2,1] => 3
[6,4,1,2,3,5] => 3
[6,4,1,2,5,3] => 3
[6,4,1,3,2,5] => 3
[6,4,1,3,5,2] => 3
[6,4,1,5,2,3] => 3
[6,4,1,5,3,2] => 3
[6,4,2,1,3,5] => 3
[6,4,2,1,5,3] => 3
[6,4,2,3,1,5] => 3
[6,4,2,3,5,1] => 2
[6,4,2,5,1,3] => 3
[6,4,2,5,3,1] => 3
[6,4,3,1,2,5] => 3
[6,4,3,1,5,2] => 3
[6,4,3,2,1,5] => 2
[6,4,3,2,5,1] => 3
[6,4,3,5,1,2] => 3
[6,4,3,5,2,1] => 3
[6,4,5,1,2,3] => 2
[6,4,5,1,3,2] => 3
[6,4,5,2,1,3] => 3
[6,4,5,2,3,1] => 3
[6,4,5,3,1,2] => 3
[6,4,5,3,2,1] => 3
[6,5,1,2,3,4] => 3
[6,5,1,2,4,3] => 3
[6,5,1,3,2,4] => 3
[6,5,1,3,4,2] => 3
[6,5,1,4,2,3] => 3
[6,5,1,4,3,2] => 3
[6,5,2,1,3,4] => 3
[6,5,2,1,4,3] => 3
[6,5,2,3,1,4] => 3
[6,5,2,3,4,1] => 2
[6,5,2,4,1,3] => 3
[6,5,2,4,3,1] => 3
[6,5,3,1,2,4] => 3
[6,5,3,1,4,2] => 3
[6,5,3,2,1,4] => 3
[6,5,3,2,4,1] => 3
[6,5,3,4,1,2] => 3
[6,5,3,4,2,1] => 3
[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] => 3
[6,5,4,3,1,2] => 3
[6,5,4,3,2,1] => 3
click to show generating function       
Description
The length of a longest pair of twins in a permutation.
A pair of twins in a permutation is a pair of two disjoint subsequences which are order isomorphic.
References
[1] Dudek, A., Grytczuk, Jarosław, Ruciński, A. Variations on twins in permutations arXiv:2001.05589
Code
from sage.combinat.permutation import to_standard
def statistic(pi):
    n = len(pi)
    S = set(range(n))
    for k in range(n//2, 0, -1):
        for s1 in Subsets(S, k):
            for s2 in Subsets(S.difference(s1), k):
                pi1 = [pi[i] for i in s1]
                pi2 = [pi[i] for i in s2]
                if to_standard(pi1) == to_standard(pi2):
                    return k
    return 0

Created
Jan 17, 2020 at 09:24 by Martin Rubey
Updated
Jan 17, 2020 at 09:24 by Martin Rubey