Identifier
Identifier
Values
[1] => 1
[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] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 1
[1,5,2,3,4] => 1
[1,5,2,4,3] => 1
[1,5,3,2,4] => 1
[1,5,3,4,2] => 2
[1,5,4,2,3] => 1
[1,5,4,3,2] => 1
[2,1,3,4,5] => 1
[2,1,3,5,4] => 1
[2,1,4,3,5] => 1
[2,1,4,5,3] => 1
[2,1,5,3,4] => 1
[2,1,5,4,3] => 1
[2,3,1,4,5] => 1
[2,3,1,5,4] => 1
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 1
[2,3,5,4,1] => 1
[2,4,1,3,5] => 1
[2,4,1,5,3] => 1
[2,4,3,1,5] => 1
[2,4,3,5,1] => 1
[2,4,5,1,3] => 2
[2,4,5,3,1] => 1
[2,5,1,3,4] => 1
[2,5,1,4,3] => 1
[2,5,3,1,4] => 1
[2,5,3,4,1] => 2
[2,5,4,1,3] => 1
[2,5,4,3,1] => 1
[3,1,2,4,5] => 1
[3,1,2,5,4] => 1
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 1
[3,1,5,4,2] => 1
[3,2,1,4,5] => 1
[3,2,1,5,4] => 1
[3,2,4,1,5] => 1
[3,2,4,5,1] => 1
[3,2,5,1,4] => 1
[3,2,5,4,1] => 1
[3,4,1,2,5] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 3
[3,4,5,2,1] => 1
[3,5,1,2,4] => 2
[3,5,1,4,2] => 2
[3,5,2,1,4] => 1
[3,5,2,4,1] => 2
[3,5,4,1,2] => 2
[3,5,4,2,1] => 1
[4,1,2,3,5] => 1
[4,1,2,5,3] => 1
[4,1,3,2,5] => 1
[4,1,3,5,2] => 1
[4,1,5,2,3] => 2
[4,1,5,3,2] => 1
[4,2,1,3,5] => 1
[4,2,1,5,3] => 1
[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] => 1
[4,3,1,5,2] => 1
[4,3,2,1,5] => 1
[4,3,2,5,1] => 1
[4,3,5,1,2] => 2
[4,3,5,2,1] => 1
[4,5,1,2,3] => 3
[4,5,1,3,2] => 2
[4,5,2,1,3] => 2
[4,5,2,3,1] => 3
[4,5,3,1,2] => 2
[4,5,3,2,1] => 1
[5,1,2,3,4] => 1
[5,1,2,4,3] => 1
[5,1,3,2,4] => 1
[5,1,3,4,2] => 2
[5,1,4,2,3] => 1
[5,1,4,3,2] => 1
[5,2,1,3,4] => 1
[5,2,1,4,3] => 1
[5,2,3,1,4] => 2
[5,2,3,4,1] => 4
[5,2,4,1,3] => 2
[5,2,4,3,1] => 2
[5,3,1,2,4] => 1
[5,3,1,4,2] => 2
[5,3,2,1,4] => 1
[5,3,2,4,1] => 2
[5,3,4,1,2] => 3
[5,3,4,2,1] => 2
[5,4,1,2,3] => 1
[5,4,1,3,2] => 1
[5,4,2,1,3] => 1
[5,4,2,3,1] => 2
[5,4,3,1,2] => 1
[5,4,3,2,1] => 1
[1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => 1
[1,2,3,5,4,6] => 1
[1,2,3,5,6,4] => 1
[1,2,3,6,4,5] => 1
[1,2,3,6,5,4] => 1
[1,2,4,3,5,6] => 1
[1,2,4,3,6,5] => 1
[1,2,4,5,3,6] => 1
[1,2,4,5,6,3] => 1
[1,2,4,6,3,5] => 1
[1,2,4,6,5,3] => 1
[1,2,5,3,4,6] => 1
[1,2,5,3,6,4] => 1
[1,2,5,4,3,6] => 1
[1,2,5,4,6,3] => 1
[1,2,5,6,3,4] => 2
[1,2,5,6,4,3] => 1
[1,2,6,3,4,5] => 1
[1,2,6,3,5,4] => 1
[1,2,6,4,3,5] => 1
[1,2,6,4,5,3] => 2
[1,2,6,5,3,4] => 1
[1,2,6,5,4,3] => 1
[1,3,2,4,5,6] => 1
[1,3,2,4,6,5] => 1
[1,3,2,5,4,6] => 1
[1,3,2,5,6,4] => 1
[1,3,2,6,4,5] => 1
[1,3,2,6,5,4] => 1
[1,3,4,2,5,6] => 1
[1,3,4,2,6,5] => 1
[1,3,4,5,2,6] => 1
[1,3,4,5,6,2] => 1
[1,3,4,6,2,5] => 1
[1,3,4,6,5,2] => 1
[1,3,5,2,4,6] => 1
[1,3,5,2,6,4] => 1
[1,3,5,4,2,6] => 1
[1,3,5,4,6,2] => 1
[1,3,5,6,2,4] => 2
[1,3,5,6,4,2] => 1
[1,3,6,2,4,5] => 1
[1,3,6,2,5,4] => 1
[1,3,6,4,2,5] => 1
[1,3,6,4,5,2] => 2
[1,3,6,5,2,4] => 1
[1,3,6,5,4,2] => 1
[1,4,2,3,5,6] => 1
[1,4,2,3,6,5] => 1
[1,4,2,5,3,6] => 1
[1,4,2,5,6,3] => 1
[1,4,2,6,3,5] => 1
[1,4,2,6,5,3] => 1
[1,4,3,2,5,6] => 1
[1,4,3,2,6,5] => 1
[1,4,3,5,2,6] => 1
[1,4,3,5,6,2] => 1
[1,4,3,6,2,5] => 1
[1,4,3,6,5,2] => 1
[1,4,5,2,3,6] => 2
[1,4,5,2,6,3] => 2
[1,4,5,3,2,6] => 1
[1,4,5,3,6,2] => 1
[1,4,5,6,2,3] => 3
[1,4,5,6,3,2] => 1
[1,4,6,2,3,5] => 2
[1,4,6,2,5,3] => 2
[1,4,6,3,2,5] => 1
[1,4,6,3,5,2] => 2
[1,4,6,5,2,3] => 2
[1,4,6,5,3,2] => 1
[1,5,2,3,4,6] => 1
[1,5,2,3,6,4] => 1
[1,5,2,4,3,6] => 1
[1,5,2,4,6,3] => 1
[1,5,2,6,3,4] => 2
[1,5,2,6,4,3] => 1
[1,5,3,2,4,6] => 1
[1,5,3,2,6,4] => 1
[1,5,3,4,2,6] => 2
[1,5,3,4,6,2] => 2
[1,5,3,6,2,4] => 2
[1,5,3,6,4,2] => 2
[1,5,4,2,3,6] => 1
[1,5,4,2,6,3] => 1
[1,5,4,3,2,6] => 1
[1,5,4,3,6,2] => 1
[1,5,4,6,2,3] => 2
[1,5,4,6,3,2] => 1
[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] => 3
[1,5,6,4,2,3] => 2
[1,5,6,4,3,2] => 1
[1,6,2,3,4,5] => 1
[1,6,2,3,5,4] => 1
[1,6,2,4,3,5] => 1
[1,6,2,4,5,3] => 2
[1,6,2,5,3,4] => 1
[1,6,2,5,4,3] => 1
[1,6,3,2,4,5] => 1
[1,6,3,2,5,4] => 1
[1,6,3,4,2,5] => 2
[1,6,3,4,5,2] => 4
[1,6,3,5,2,4] => 2
[1,6,3,5,4,2] => 2
[1,6,4,2,3,5] => 1
[1,6,4,2,5,3] => 2
[1,6,4,3,2,5] => 1
[1,6,4,3,5,2] => 2
[1,6,4,5,2,3] => 3
[1,6,4,5,3,2] => 2
[1,6,5,2,3,4] => 1
[1,6,5,2,4,3] => 1
[1,6,5,3,2,4] => 1
[1,6,5,3,4,2] => 2
[1,6,5,4,2,3] => 1
[1,6,5,4,3,2] => 1
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 1
[2,1,3,5,4,6] => 1
[2,1,3,5,6,4] => 1
[2,1,3,6,4,5] => 1
[2,1,3,6,5,4] => 1
[2,1,4,3,5,6] => 1
[2,1,4,3,6,5] => 1
[2,1,4,5,3,6] => 1
[2,1,4,5,6,3] => 1
[2,1,4,6,3,5] => 1
[2,1,4,6,5,3] => 1
[2,1,5,3,4,6] => 1
[2,1,5,3,6,4] => 1
[2,1,5,4,3,6] => 1
[2,1,5,4,6,3] => 1
[2,1,5,6,3,4] => 2
[2,1,5,6,4,3] => 1
[2,1,6,3,4,5] => 1
[2,1,6,3,5,4] => 1
[2,1,6,4,3,5] => 1
[2,1,6,4,5,3] => 2
[2,1,6,5,3,4] => 1
[2,1,6,5,4,3] => 1
[2,3,1,4,5,6] => 1
[2,3,1,4,6,5] => 1
[2,3,1,5,4,6] => 1
[2,3,1,5,6,4] => 1
[2,3,1,6,4,5] => 1
[2,3,1,6,5,4] => 1
[2,3,4,1,5,6] => 1
[2,3,4,1,6,5] => 1
[2,3,4,5,1,6] => 1
[2,3,4,5,6,1] => 1
[2,3,4,6,1,5] => 1
[2,3,4,6,5,1] => 1
[2,3,5,1,4,6] => 1
[2,3,5,1,6,4] => 1
[2,3,5,4,1,6] => 1
[2,3,5,4,6,1] => 1
[2,3,5,6,1,4] => 2
[2,3,5,6,4,1] => 1
[2,3,6,1,4,5] => 1
[2,3,6,1,5,4] => 1
[2,3,6,4,1,5] => 1
[2,3,6,4,5,1] => 2
[2,3,6,5,1,4] => 1
[2,3,6,5,4,1] => 1
[2,4,1,3,5,6] => 1
[2,4,1,3,6,5] => 1
[2,4,1,5,3,6] => 1
[2,4,1,5,6,3] => 1
[2,4,1,6,3,5] => 1
[2,4,1,6,5,3] => 1
[2,4,3,1,5,6] => 1
[2,4,3,1,6,5] => 1
[2,4,3,5,1,6] => 1
[2,4,3,5,6,1] => 1
[2,4,3,6,1,5] => 1
[2,4,3,6,5,1] => 1
[2,4,5,1,3,6] => 2
[2,4,5,1,6,3] => 2
[2,4,5,3,1,6] => 1
[2,4,5,3,6,1] => 1
[2,4,5,6,1,3] => 3
[2,4,5,6,3,1] => 1
[2,4,6,1,3,5] => 2
[2,4,6,1,5,3] => 2
[2,4,6,3,1,5] => 1
[2,4,6,3,5,1] => 2
[2,4,6,5,1,3] => 2
[2,4,6,5,3,1] => 1
[2,5,1,3,4,6] => 1
[2,5,1,3,6,4] => 1
[2,5,1,4,3,6] => 1
[2,5,1,4,6,3] => 1
[2,5,1,6,3,4] => 2
[2,5,1,6,4,3] => 1
[2,5,3,1,4,6] => 1
[2,5,3,1,6,4] => 1
[2,5,3,4,1,6] => 2
[2,5,3,4,6,1] => 2
[2,5,3,6,1,4] => 2
[2,5,3,6,4,1] => 2
[2,5,4,1,3,6] => 1
[2,5,4,1,6,3] => 1
[2,5,4,3,1,6] => 1
[2,5,4,3,6,1] => 1
[2,5,4,6,1,3] => 2
[2,5,4,6,3,1] => 1
[2,5,6,1,3,4] => 3
[2,5,6,1,4,3] => 2
[2,5,6,3,1,4] => 2
[2,5,6,3,4,1] => 3
[2,5,6,4,1,3] => 2
[2,5,6,4,3,1] => 1
[2,6,1,3,4,5] => 1
[2,6,1,3,5,4] => 1
[2,6,1,4,3,5] => 1
[2,6,1,4,5,3] => 2
[2,6,1,5,3,4] => 1
[2,6,1,5,4,3] => 1
[2,6,3,1,4,5] => 1
[2,6,3,1,5,4] => 1
[2,6,3,4,1,5] => 2
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 2
[2,6,3,5,4,1] => 2
[2,6,4,1,3,5] => 1
[2,6,4,1,5,3] => 2
[2,6,4,3,1,5] => 1
[2,6,4,3,5,1] => 2
[2,6,4,5,1,3] => 3
[2,6,4,5,3,1] => 2
[2,6,5,1,3,4] => 1
[2,6,5,1,4,3] => 1
[2,6,5,3,1,4] => 1
[2,6,5,3,4,1] => 2
[2,6,5,4,1,3] => 1
[2,6,5,4,3,1] => 1
[3,1,2,4,5,6] => 1
[3,1,2,4,6,5] => 1
[3,1,2,5,4,6] => 1
[3,1,2,5,6,4] => 1
[3,1,2,6,4,5] => 1
[3,1,2,6,5,4] => 1
[3,1,4,2,5,6] => 1
[3,1,4,2,6,5] => 1
[3,1,4,5,2,6] => 1
[3,1,4,5,6,2] => 1
[3,1,4,6,2,5] => 1
[3,1,4,6,5,2] => 1
[3,1,5,2,4,6] => 1
[3,1,5,2,6,4] => 1
[3,1,5,4,2,6] => 1
[3,1,5,4,6,2] => 1
[3,1,5,6,2,4] => 2
[3,1,5,6,4,2] => 1
[3,1,6,2,4,5] => 1
[3,1,6,2,5,4] => 1
[3,1,6,4,2,5] => 1
[3,1,6,4,5,2] => 2
[3,1,6,5,2,4] => 1
[3,1,6,5,4,2] => 1
[3,2,1,4,5,6] => 1
[3,2,1,4,6,5] => 1
[3,2,1,5,4,6] => 1
[3,2,1,5,6,4] => 1
[3,2,1,6,4,5] => 1
[3,2,1,6,5,4] => 1
[3,2,4,1,5,6] => 1
[3,2,4,1,6,5] => 1
[3,2,4,5,1,6] => 1
[3,2,4,5,6,1] => 1
[3,2,4,6,1,5] => 1
[3,2,4,6,5,1] => 1
[3,2,5,1,4,6] => 1
[3,2,5,1,6,4] => 1
[3,2,5,4,1,6] => 1
[3,2,5,4,6,1] => 1
[3,2,5,6,1,4] => 2
[3,2,5,6,4,1] => 1
[3,2,6,1,4,5] => 1
[3,2,6,1,5,4] => 1
[3,2,6,4,1,5] => 1
[3,2,6,4,5,1] => 2
[3,2,6,5,1,4] => 1
[3,2,6,5,4,1] => 1
[3,4,1,2,5,6] => 2
[3,4,1,2,6,5] => 2
[3,4,1,5,2,6] => 2
[3,4,1,5,6,2] => 2
[3,4,1,6,2,5] => 2
[3,4,1,6,5,2] => 2
[3,4,2,1,5,6] => 1
[3,4,2,1,6,5] => 1
[3,4,2,5,1,6] => 1
[3,4,2,5,6,1] => 1
[3,4,2,6,1,5] => 1
[3,4,2,6,5,1] => 1
[3,4,5,1,2,6] => 3
[3,4,5,1,6,2] => 3
[3,4,5,2,1,6] => 1
[3,4,5,2,6,1] => 1
[3,4,5,6,1,2] => 5
[3,4,5,6,2,1] => 1
[3,4,6,1,2,5] => 3
[3,4,6,1,5,2] => 3
[3,4,6,2,1,5] => 1
[3,4,6,2,5,1] => 2
[3,4,6,5,1,2] => 3
[3,4,6,5,2,1] => 1
[3,5,1,2,4,6] => 2
[3,5,1,2,6,4] => 2
[3,5,1,4,2,6] => 2
[3,5,1,4,6,2] => 2
[3,5,1,6,2,4] => 4
[3,5,1,6,4,2] => 2
[3,5,2,1,4,6] => 1
[3,5,2,1,6,4] => 1
[3,5,2,4,1,6] => 2
[3,5,2,4,6,1] => 2
[3,5,2,6,1,4] => 2
[3,5,2,6,4,1] => 2
[3,5,4,1,2,6] => 2
[3,5,4,1,6,2] => 2
[3,5,4,2,1,6] => 1
[3,5,4,2,6,1] => 1
[3,5,4,6,1,2] => 4
[3,5,4,6,2,1] => 1
[3,5,6,1,2,4] => 5
[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] => 1
[3,6,1,2,4,5] => 2
[3,6,1,2,5,4] => 2
[3,6,1,4,2,5] => 2
[3,6,1,4,5,2] => 4
[3,6,1,5,2,4] => 2
[3,6,1,5,4,2] => 2
[3,6,2,1,4,5] => 1
[3,6,2,1,5,4] => 1
[3,6,2,4,1,5] => 2
[3,6,2,4,5,1] => 4
[3,6,2,5,1,4] => 2
[3,6,2,5,4,1] => 2
[3,6,4,1,2,5] => 2
[3,6,4,1,5,2] => 4
[3,6,4,2,1,5] => 1
[3,6,4,2,5,1] => 2
[3,6,4,5,1,2] => 6
[3,6,4,5,2,1] => 2
[3,6,5,1,2,4] => 2
[3,6,5,1,4,2] => 2
[3,6,5,2,1,4] => 1
[3,6,5,2,4,1] => 2
[3,6,5,4,1,2] => 2
[3,6,5,4,2,1] => 1
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 1
[4,1,2,5,3,6] => 1
[4,1,2,5,6,3] => 1
[4,1,2,6,3,5] => 1
[4,1,2,6,5,3] => 1
[4,1,3,2,5,6] => 1
[4,1,3,2,6,5] => 1
[4,1,3,5,2,6] => 1
[4,1,3,5,6,2] => 1
[4,1,3,6,2,5] => 1
[4,1,3,6,5,2] => 1
[4,1,5,2,3,6] => 2
[4,1,5,2,6,3] => 2
[4,1,5,3,2,6] => 1
[4,1,5,3,6,2] => 1
[4,1,5,6,2,3] => 3
[4,1,5,6,3,2] => 1
[4,1,6,2,3,5] => 2
[4,1,6,2,5,3] => 2
[4,1,6,3,2,5] => 1
[4,1,6,3,5,2] => 2
[4,1,6,5,2,3] => 2
[4,1,6,5,3,2] => 1
[4,2,1,3,5,6] => 1
[4,2,1,3,6,5] => 1
[4,2,1,5,3,6] => 1
[4,2,1,5,6,3] => 1
[4,2,1,6,3,5] => 1
[4,2,1,6,5,3] => 1
[4,2,3,1,5,6] => 2
[4,2,3,1,6,5] => 2
[4,2,3,5,1,6] => 2
[4,2,3,5,6,1] => 2
[4,2,3,6,1,5] => 2
[4,2,3,6,5,1] => 2
[4,2,5,1,3,6] => 2
[4,2,5,1,6,3] => 2
[4,2,5,3,1,6] => 2
[4,2,5,3,6,1] => 2
[4,2,5,6,1,3] => 3
[4,2,5,6,3,1] => 2
[4,2,6,1,3,5] => 2
[4,2,6,1,5,3] => 2
[4,2,6,3,1,5] => 2
[4,2,6,3,5,1] => 4
[4,2,6,5,1,3] => 2
[4,2,6,5,3,1] => 2
[4,3,1,2,5,6] => 1
[4,3,1,2,6,5] => 1
[4,3,1,5,2,6] => 1
[4,3,1,5,6,2] => 1
[4,3,1,6,2,5] => 1
[4,3,1,6,5,2] => 1
[4,3,2,1,5,6] => 1
[4,3,2,1,6,5] => 1
[4,3,2,5,1,6] => 1
[4,3,2,5,6,1] => 1
[4,3,2,6,1,5] => 1
[4,3,2,6,5,1] => 1
[4,3,5,1,2,6] => 2
[4,3,5,1,6,2] => 2
[4,3,5,2,1,6] => 1
[4,3,5,2,6,1] => 1
[4,3,5,6,1,2] => 3
[4,3,5,6,2,1] => 1
[4,3,6,1,2,5] => 2
[4,3,6,1,5,2] => 2
[4,3,6,2,1,5] => 1
[4,3,6,2,5,1] => 2
[4,3,6,5,1,2] => 2
[4,3,6,5,2,1] => 1
[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] => 2
[4,5,1,6,2,3] => 5
[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] => 3
[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] => 2
[4,5,3,2,1,6] => 1
[4,5,3,2,6,1] => 1
[4,5,3,6,1,2] => 3
[4,5,3,6,2,1] => 1
[4,5,6,1,2,3] => 10
[4,5,6,1,3,2] => 4
[4,5,6,2,1,3] => 4
[4,5,6,2,3,1] => 6
[4,5,6,3,1,2] => 3
[4,5,6,3,2,1] => 1
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 3
[4,6,1,3,2,5] => 2
[4,6,1,3,5,2] => 4
[4,6,1,5,2,3] => 3
[4,6,1,5,3,2] => 2
[4,6,2,1,3,5] => 2
[4,6,2,1,5,3] => 2
[4,6,2,3,1,5] => 3
[4,6,2,3,5,1] => 6
[4,6,2,5,1,3] => 3
[4,6,2,5,3,1] => 3
[4,6,3,1,2,5] => 2
[4,6,3,1,5,2] => 3
[4,6,3,2,1,5] => 1
[4,6,3,2,5,1] => 2
[4,6,3,5,1,2] => 4
[4,6,3,5,2,1] => 2
[4,6,5,1,2,3] => 4
[4,6,5,1,3,2] => 3
[4,6,5,2,1,3] => 2
[4,6,5,2,3,1] => 4
[4,6,5,3,1,2] => 2
[4,6,5,3,2,1] => 1
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 1
[5,1,2,4,3,6] => 1
[5,1,2,4,6,3] => 1
[5,1,2,6,3,4] => 2
[5,1,2,6,4,3] => 1
[5,1,3,2,4,6] => 1
[5,1,3,2,6,4] => 1
[5,1,3,4,2,6] => 2
[5,1,3,4,6,2] => 2
[5,1,3,6,2,4] => 2
[5,1,3,6,4,2] => 2
[5,1,4,2,3,6] => 1
[5,1,4,2,6,3] => 1
[5,1,4,3,2,6] => 1
[5,1,4,3,6,2] => 1
[5,1,4,6,2,3] => 2
[5,1,4,6,3,2] => 1
[5,1,6,2,3,4] => 3
[5,1,6,2,4,3] => 2
[5,1,6,3,2,4] => 2
[5,1,6,3,4,2] => 3
[5,1,6,4,2,3] => 2
[5,1,6,4,3,2] => 1
[5,2,1,3,4,6] => 1
[5,2,1,3,6,4] => 1
[5,2,1,4,3,6] => 1
[5,2,1,4,6,3] => 1
[5,2,1,6,3,4] => 2
[5,2,1,6,4,3] => 1
[5,2,3,1,4,6] => 2
[5,2,3,1,6,4] => 2
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 4
[5,2,3,6,1,4] => 4
[5,2,3,6,4,1] => 4
[5,2,4,1,3,6] => 2
[5,2,4,1,6,3] => 2
[5,2,4,3,1,6] => 2
[5,2,4,3,6,1] => 2
[5,2,4,6,1,3] => 4
[5,2,4,6,3,1] => 2
[5,2,6,1,3,4] => 3
[5,2,6,1,4,3] => 2
[5,2,6,3,1,4] => 4
[5,2,6,3,4,1] => 6
[5,2,6,4,1,3] => 3
[5,2,6,4,3,1] => 2
[5,3,1,2,4,6] => 1
[5,3,1,2,6,4] => 1
[5,3,1,4,2,6] => 2
[5,3,1,4,6,2] => 2
[5,3,1,6,2,4] => 2
[5,3,1,6,4,2] => 2
[5,3,2,1,4,6] => 1
[5,3,2,1,6,4] => 1
[5,3,2,4,1,6] => 2
[5,3,2,4,6,1] => 2
[5,3,2,6,1,4] => 2
[5,3,2,6,4,1] => 2
[5,3,4,1,2,6] => 3
[5,3,4,1,6,2] => 3
[5,3,4,2,1,6] => 2
[5,3,4,2,6,1] => 2
[5,3,4,6,1,2] => 6
[5,3,4,6,2,1] => 2
[5,3,6,1,2,4] => 3
[5,3,6,1,4,2] => 3
[5,3,6,2,1,4] => 2
[5,3,6,2,4,1] => 3
[5,3,6,4,1,2] => 4
[5,3,6,4,2,1] => 2
[5,4,1,2,3,6] => 1
[5,4,1,2,6,3] => 1
[5,4,1,3,2,6] => 1
[5,4,1,3,6,2] => 1
[5,4,1,6,2,3] => 2
[5,4,1,6,3,2] => 1
[5,4,2,1,3,6] => 1
[5,4,2,1,6,3] => 1
[5,4,2,3,1,6] => 2
[5,4,2,3,6,1] => 2
[5,4,2,6,1,3] => 2
[5,4,2,6,3,1] => 2
[5,4,3,1,2,6] => 1
[5,4,3,1,6,2] => 1
[5,4,3,2,1,6] => 1
[5,4,3,2,6,1] => 1
[5,4,3,6,1,2] => 2
[5,4,3,6,2,1] => 1
[5,4,6,1,2,3] => 4
[5,4,6,1,3,2] => 2
[5,4,6,2,1,3] => 3
[5,4,6,2,3,1] => 4
[5,4,6,3,1,2] => 2
[5,4,6,3,2,1] => 1
[5,6,1,2,3,4] => 5
[5,6,1,2,4,3] => 3
[5,6,1,3,2,4] => 4
[5,6,1,3,4,2] => 6
[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] => 2
[5,6,2,3,1,4] => 6
[5,6,2,3,4,1] => 9
[5,6,2,4,1,3] => 4
[5,6,2,4,3,1] => 3
[5,6,3,1,2,4] => 3
[5,6,3,1,4,2] => 4
[5,6,3,2,1,4] => 2
[5,6,3,2,4,1] => 3
[5,6,3,4,1,2] => 8
[5,6,3,4,2,1] => 4
[5,6,4,1,2,3] => 3
[5,6,4,1,3,2] => 2
[5,6,4,2,1,3] => 2
[5,6,4,2,3,1] => 3
[5,6,4,3,1,2] => 2
[5,6,4,3,2,1] => 1
[6,1,2,3,4,5] => 1
[6,1,2,3,5,4] => 1
[6,1,2,4,3,5] => 1
[6,1,2,4,5,3] => 2
[6,1,2,5,3,4] => 1
[6,1,2,5,4,3] => 1
[6,1,3,2,4,5] => 1
[6,1,3,2,5,4] => 1
[6,1,3,4,2,5] => 2
[6,1,3,4,5,2] => 4
[6,1,3,5,2,4] => 2
[6,1,3,5,4,2] => 2
[6,1,4,2,3,5] => 1
[6,1,4,2,5,3] => 2
[6,1,4,3,2,5] => 1
[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] => 1
[6,1,5,2,4,3] => 1
[6,1,5,3,2,4] => 1
[6,1,5,3,4,2] => 2
[6,1,5,4,2,3] => 1
[6,1,5,4,3,2] => 1
[6,2,1,3,4,5] => 1
[6,2,1,3,5,4] => 1
[6,2,1,4,3,5] => 1
[6,2,1,4,5,3] => 2
[6,2,1,5,3,4] => 1
[6,2,1,5,4,3] => 1
[6,2,3,1,4,5] => 2
[6,2,3,1,5,4] => 2
[6,2,3,4,1,5] => 4
[6,2,3,4,5,1] => 8
[6,2,3,5,1,4] => 4
[6,2,3,5,4,1] => 4
[6,2,4,1,3,5] => 2
[6,2,4,1,5,3] => 4
[6,2,4,3,1,5] => 2
[6,2,4,3,5,1] => 4
[6,2,4,5,1,3] => 6
[6,2,4,5,3,1] => 4
[6,2,5,1,3,4] => 2
[6,2,5,1,4,3] => 2
[6,2,5,3,1,4] => 2
[6,2,5,3,4,1] => 4
[6,2,5,4,1,3] => 2
[6,2,5,4,3,1] => 2
[6,3,1,2,4,5] => 1
[6,3,1,2,5,4] => 1
[6,3,1,4,2,5] => 2
[6,3,1,4,5,2] => 4
[6,3,1,5,2,4] => 2
[6,3,1,5,4,2] => 2
[6,3,2,1,4,5] => 1
[6,3,2,1,5,4] => 1
[6,3,2,4,1,5] => 2
[6,3,2,4,5,1] => 4
[6,3,2,5,1,4] => 2
[6,3,2,5,4,1] => 3
[6,3,4,1,2,5] => 3
[6,3,4,1,5,2] => 6
[6,3,4,2,1,5] => 2
[6,3,4,2,5,1] => 4
[6,3,4,5,1,2] => 9
[6,3,4,5,2,1] => 4
[6,3,5,1,2,4] => 3
[6,3,5,1,4,2] => 3
[6,3,5,2,1,4] => 2
[6,3,5,2,4,1] => 4
[6,3,5,4,1,2] => 3
[6,3,5,4,2,1] => 2
[6,4,1,2,3,5] => 1
[6,4,1,2,5,3] => 2
[6,4,1,3,2,5] => 1
[6,4,1,3,5,2] => 2
[6,4,1,5,2,3] => 3
[6,4,1,5,3,2] => 2
[6,4,2,1,3,5] => 1
[6,4,2,1,5,3] => 2
[6,4,2,3,1,5] => 2
[6,4,2,3,5,1] => 4
[6,4,2,5,1,3] => 3
[6,4,2,5,3,1] => 4
[6,4,3,1,2,5] => 1
[6,4,3,1,5,2] => 2
[6,4,3,2,1,5] => 1
[6,4,3,2,5,1] => 2
[6,4,3,5,1,2] => 3
[6,4,3,5,2,1] => 2
[6,4,5,1,2,3] => 6
[6,4,5,1,3,2] => 4
[6,4,5,2,1,3] => 4
[6,4,5,2,3,1] => 8
[6,4,5,3,1,2] => 3
[6,4,5,3,2,1] => 2
[6,5,1,2,3,4] => 1
[6,5,1,2,4,3] => 1
[6,5,1,3,2,4] => 1
[6,5,1,3,4,2] => 2
[6,5,1,4,2,3] => 1
[6,5,1,4,3,2] => 1
[6,5,2,1,3,4] => 1
[6,5,2,1,4,3] => 1
[6,5,2,3,1,4] => 2
[6,5,2,3,4,1] => 4
[6,5,2,4,1,3] => 2
[6,5,2,4,3,1] => 2
[6,5,3,1,2,4] => 1
[6,5,3,1,4,2] => 2
[6,5,3,2,1,4] => 1
[6,5,3,2,4,1] => 2
[6,5,3,4,1,2] => 4
[6,5,3,4,2,1] => 3
[6,5,4,1,2,3] => 1
[6,5,4,1,3,2] => 1
[6,5,4,2,1,3] => 1
[6,5,4,2,3,1] => 2
[6,5,4,3,1,2] => 1
[6,5,4,3,2,1] => 1
[1,2,3,4,5,6,7] => 1
[1,2,3,4,5,7,6] => 1
[1,2,3,4,6,5,7] => 1
[1,2,3,6,4,7,5] => 1
[1,2,6,3,4,7,5] => 1
[1,3,2,4,5,6,7] => 1
[1,3,7,2,4,5,6] => 1
[1,3,7,6,5,4,2] => 1
[1,4,6,7,2,3,5] => 5
[1,4,6,7,5,3,2] => 1
[1,4,7,2,3,5,6] => 2
[1,4,7,6,5,3,2] => 1
[1,5,2,3,4,7,6] => 1
[1,5,2,7,3,4,6] => 2
[1,5,4,7,6,3,2] => 1
[1,5,6,2,7,3,4] => 5
[1,5,6,4,7,3,2] => 1
[1,5,6,7,2,3,4] => 10
[1,5,6,7,4,3,2] => 1
[1,5,7,2,3,4,6] => 3
[1,5,7,6,4,3,2] => 1
[1,6,2,3,4,7,5] => 1
[1,6,2,3,7,4,5] => 2
[1,6,2,7,3,4,5] => 3
[1,6,5,4,3,7,2] => 1
[1,6,5,4,7,3,2] => 1
[1,6,5,7,4,3,2] => 1
[1,6,7,2,3,4,5] => 5
[1,6,7,5,4,3,2] => 1
[1,7,2,3,4,5,6] => 1
[1,7,3,5,6,4,2] => 4
[1,7,3,6,5,4,2] => 2
[1,7,4,3,6,5,2] => 3
[1,7,4,5,3,6,2] => 4
[1,7,4,5,6,3,2] => 4
[1,7,4,6,5,3,2] => 2
[1,7,5,4,3,6,2] => 2
[1,7,5,4,6,3,2] => 2
[1,7,6,4,5,3,2] => 3
[1,7,6,5,4,3,2] => 1
[2,1,3,4,5,6,7] => 1
[2,1,3,4,7,5,6] => 1
[2,1,3,7,4,5,6] => 1
[2,1,6,3,4,5,7] => 1
[2,1,7,3,4,5,6] => 1
[2,1,7,6,5,4,3] => 1
[2,3,1,4,5,6,7] => 1
[2,3,4,1,5,6,7] => 1
[2,3,4,1,5,7,6] => 1
[2,3,4,5,1,6,7] => 1
[2,3,4,5,1,7,6] => 1
[2,3,4,5,6,1,7] => 1
[2,3,4,5,6,7,1] => 1
[2,3,4,5,7,1,6] => 1
[2,3,4,6,1,7,5] => 1
[2,3,4,6,7,1,5] => 2
[2,3,4,7,1,5,6] => 1
[2,3,5,6,7,1,4] => 3
[2,4,1,3,5,6,7] => 1
[2,5,1,3,4,6,7] => 1
[2,5,7,1,3,4,6] => 3
[2,6,1,3,4,5,7] => 1
[2,6,1,7,3,4,5] => 3
[2,6,5,4,3,1,7] => 1
[2,6,7,1,3,4,5] => 5
[2,7,1,3,4,5,6] => 1
[3,1,2,4,5,6,7] => 1
[3,1,2,4,5,7,6] => 1
[3,2,1,4,5,6,7] => 1
[3,2,4,5,6,7,1] => 1
[3,4,1,2,5,6,7] => 2
[3,4,2,5,6,7,1] => 1
[3,4,5,2,6,7,1] => 1
[3,4,5,6,2,7,1] => 1
[3,4,5,6,7,1,2] => 8
[3,5,6,1,2,4,7] => 5
[3,5,6,4,2,1,7] => 1
[3,6,1,2,4,5,7] => 2
[3,6,5,4,2,1,7] => 1
[4,1,2,3,5,6,7] => 1
[4,1,2,3,5,7,6] => 1
[4,1,6,2,3,5,7] => 2
[4,2,3,5,6,7,1] => 2
[4,2,5,1,3,6,7] => 2
[4,3,2,1,5,6,7] => 1
[4,3,2,5,6,7,1] => 1
[4,3,5,2,6,7,1] => 1
[4,3,6,5,2,1,7] => 1
[4,5,1,6,2,3,7] => 5
[4,5,2,3,6,7,1] => 3
[4,5,3,6,2,1,7] => 1
[4,5,6,1,2,3,7] => 10
[4,5,6,3,2,1,7] => 1
[4,6,1,2,3,5,7] => 3
[4,6,1,2,3,7,5] => 3
[4,6,5,3,2,1,7] => 1
[5,1,2,3,4,6,7] => 1
[5,1,2,3,4,7,6] => 1
[5,1,2,3,6,4,7] => 1
[5,1,2,3,6,7,4] => 1
[5,1,2,3,7,4,6] => 1
[5,1,2,6,3,4,7] => 2
[5,1,6,2,3,4,7] => 3
[5,1,6,2,3,7,4] => 3
[5,2,3,4,6,7,1] => 4
[5,2,6,7,1,3,4] => 10
[5,3,2,4,6,7,1] => 2
[5,3,2,6,1,4,7] => 2
[5,3,6,7,1,2,4] => 10
[5,4,3,2,1,6,7] => 1
[5,4,3,2,1,7,6] => 1
[5,4,3,2,6,1,7] => 1
[5,4,3,6,2,1,7] => 1
[5,4,6,3,2,1,7] => 1
[5,4,6,7,1,2,3] => 12
[5,6,1,2,3,4,7] => 5
[5,6,1,2,3,7,4] => 5
[5,6,3,4,1,2,7] => 8
[5,6,4,3,2,1,7] => 1
[5,6,7,1,2,3,4] => 26
[6,1,2,3,4,5,7] => 1
[6,1,2,3,4,7,5] => 1
[6,1,2,3,7,4,5] => 2
[6,1,2,7,3,4,5] => 3
[6,2,3,4,5,7,1] => 8
[6,2,4,5,3,1,7] => 4
[6,2,5,4,3,1,7] => 2
[6,3,2,5,4,1,7] => 3
[6,3,4,2,5,1,7] => 4
[6,3,4,5,2,1,7] => 4
[6,3,4,7,1,2,5] => 9
[6,3,5,4,2,1,7] => 2
[6,4,3,2,5,1,7] => 2
[6,4,3,2,7,1,5] => 2
[6,4,3,5,2,1,7] => 2
[6,4,5,7,1,2,3] => 20
[6,4,7,2,5,1,3] => 8
[6,4,7,3,5,1,2] => 8
[6,5,3,4,2,1,7] => 3
[6,5,4,3,2,1,7] => 1
[6,5,7,3,4,1,2] => 10
[6,7,1,2,3,4,5] => 8
[6,7,3,4,1,2,5] => 14
[6,7,4,5,1,2,3] => 17
[7,1,2,3,4,5,6] => 1
[7,2,1,3,4,5,6] => 1
[7,2,3,1,4,5,6] => 2
[7,2,3,4,1,5,6] => 4
[7,2,3,4,5,1,6] => 8
[7,3,1,2,4,5,6] => 1
[7,3,2,1,4,5,6] => 1
[7,3,2,4,1,5,6] => 2
[7,3,4,1,2,5,6] => 3
[7,4,1,2,3,5,6] => 1
[7,4,2,1,3,5,6] => 1
[7,4,5,6,1,2,3] => 31
[7,5,1,2,3,4,6] => 1
[7,5,6,1,2,3,4] => 11
[7,5,6,3,4,1,2] => 17
[7,6,1,2,3,4,5] => 1
[7,6,4,5,1,2,3] => 7
[7,6,5,1,2,3,4] => 1
[7,6,5,3,4,1,2] => 4
[7,6,5,4,1,2,3] => 1
[7,6,5,4,3,1,2] => 1
[7,6,5,4,3,2,1] => 1
[8,7,6,5,4,3,2,1] => 1
[7,6,8,5,4,3,2,1] => 1
[7,8,5,6,4,3,2,1] => 4
[7,8,6,4,5,3,2,1] => 5
[8,6,7,4,5,3,2,1] => 12
[7,6,5,4,8,3,2,1] => 1
[6,5,7,4,8,3,2,1] => 1
[7,8,6,5,3,4,2,1] => 4
[8,6,7,5,3,4,2,1] => 10
[8,7,5,6,3,4,2,1] => 18
[6,7,8,3,4,5,2,1] => 49
[7,8,6,5,4,2,3,1] => 3
[8,6,7,5,4,2,3,1] => 8
[8,7,5,6,4,2,3,1] => 10
[8,7,6,4,5,2,3,1] => 12
[8,6,5,7,3,2,4,1] => 10
[6,7,8,5,2,3,4,1] => 31
[8,5,6,7,2,3,4,1] => 198
[7,8,3,2,4,5,6,1] => 27
[7,6,5,4,3,2,8,1] => 1
[6,5,7,4,3,2,8,1] => 1
[6,5,4,3,7,2,8,1] => 1
[5,4,6,3,7,2,8,1] => 1
[3,4,5,2,6,7,8,1] => 1
[5,2,3,4,6,7,8,1] => 4
[4,3,2,5,6,7,8,1] => 1
[3,4,2,5,6,7,8,1] => 1
[4,2,3,5,6,7,8,1] => 2
[3,2,4,5,6,7,8,1] => 1
[2,3,4,5,6,7,8,1] => 1
[8,7,6,5,4,3,1,2] => 1
[7,8,6,5,4,3,1,2] => 2
[8,6,7,5,4,3,1,2] => 3
[8,7,5,6,4,3,1,2] => 4
[8,7,6,4,5,3,1,2] => 5
[8,7,6,5,3,4,1,2] => 4
[8,7,5,6,3,4,1,2] => 28
[7,8,5,6,3,4,1,2] => 56
[5,6,7,8,3,4,1,2] => 42
[7,8,3,4,5,6,1,2] => 239
[3,4,5,6,7,8,1,2] => 13
[8,7,6,5,4,2,1,3] => 1
[7,6,8,5,4,2,1,3] => 3
[8,7,6,5,4,1,2,3] => 1
[6,7,8,5,4,1,2,3] => 10
[8,5,6,7,4,1,2,3] => 31
[8,7,6,4,5,1,2,3] => 8
[8,6,7,4,5,1,2,3] => 48
[8,7,4,5,6,1,2,3] => 49
[7,8,4,5,6,1,2,3] => 166
[6,7,4,5,8,1,2,3] => 62
[7,6,5,8,3,2,1,4] => 4
[7,5,6,8,2,3,1,4] => 72
[6,7,5,8,3,1,2,4] => 13
[6,5,7,8,2,1,3,4] => 24
[8,7,6,5,1,2,3,4] => 1
[8,7,5,6,1,2,3,4] => 14
[7,8,5,6,1,2,3,4] => 42
[8,5,6,7,1,2,3,4] => 114
[5,6,7,8,1,2,3,4] => 145
[8,7,6,4,3,2,1,5] => 1
[8,7,6,4,2,1,3,5] => 1
[8,7,6,1,2,3,4,5] => 1
[8,6,7,1,2,3,4,5] => 20
[6,7,8,1,2,3,4,5] => 73
[7,8,5,4,2,1,3,6] => 3
[7,8,5,4,1,2,3,6] => 5
[8,7,1,2,3,4,5,6] => 1
[7,8,1,2,3,4,5,6] => 13
[8,6,5,4,3,2,1,7] => 1
[8,6,5,4,3,1,2,7] => 1
[8,5,6,3,4,1,2,7] => 17
[8,6,5,4,2,1,3,7] => 1
[8,6,4,3,2,1,5,7] => 1
[8,6,4,2,1,3,5,7] => 1
[8,2,3,4,1,5,6,7] => 4
[8,4,1,2,3,5,6,7] => 1
[8,3,2,1,4,5,6,7] => 1
[8,2,3,1,4,5,6,7] => 2
[8,3,1,2,4,5,6,7] => 1
[8,2,1,3,4,5,6,7] => 1
[8,1,2,3,4,5,6,7] => 1
[7,6,5,4,3,2,1,8] => 1
[6,7,5,4,3,2,1,8] => 1
[6,5,7,4,3,2,1,8] => 1
[5,6,7,4,3,2,1,8] => 1
[7,6,4,5,3,2,1,8] => 3
[6,5,4,7,3,2,1,8] => 1
[7,5,6,3,4,2,1,8] => 10
[7,6,4,3,5,2,1,8] => 3
[7,6,3,4,5,2,1,8] => 9
[7,5,4,3,6,2,1,8] => 2
[6,7,5,4,2,3,1,8] => 3
[6,5,7,3,2,4,1,8] => 4
[5,6,7,2,3,4,1,8] => 31
[2,3,4,5,6,7,1,8] => 1
[5,6,7,1,2,3,4,8] => 26
[6,7,1,2,3,4,5,8] => 8
[7,1,2,3,4,5,6,8] => 1
[6,5,4,3,2,1,7,8] => 1
[2,3,4,5,6,1,7,8] => 1
[5,6,3,4,1,2,7,8] => 8
[4,5,6,1,2,3,7,8] => 10
[6,1,2,3,4,5,7,8] => 1
[5,4,3,2,1,6,7,8] => 1
[2,3,4,5,1,6,7,8] => 1
[5,1,2,3,4,6,7,8] => 1
[4,3,2,1,5,6,7,8] => 1
[3,4,1,2,5,6,7,8] => 2
[3,2,1,4,5,6,7,8] => 1
[2,1,3,4,5,6,7,8] => 1
[1,2,3,4,5,6,7,8] => 1
[6,5,8,7,4,3,2,1] => 1
[3,5,8,7,6,4,2,1] => 1
[4,3,6,5,8,7,2,1] => 1
[2,6,8,7,5,4,3,1] => 1
[2,3,5,6,8,7,4,1] => 1
[3,2,6,5,4,8,7,1] => 1
[4,3,6,5,7,2,8,1] => 1
[3,2,6,5,7,4,8,1] => 1
[4,3,5,2,7,6,8,1] => 1
[3,2,5,4,7,6,8,1] => 1
[1,8,7,6,5,4,3,2] => 1
[1,7,8,6,5,4,3,2] => 1
[1,6,8,7,5,4,3,2] => 1
[1,7,6,8,5,4,3,2] => 1
[1,6,7,8,5,4,3,2] => 1
[1,5,8,7,6,4,3,2] => 1
[1,7,6,5,8,4,3,2] => 1
[1,3,5,7,8,6,4,2] => 1
[1,4,3,8,7,6,5,2] => 1
[1,6,5,4,3,8,7,2] => 1
[1,4,3,6,5,8,7,2] => 1
[2,1,8,7,6,5,4,3] => 1
[2,1,6,5,7,4,8,3] => 1
[2,1,5,4,7,6,8,3] => 1
[1,2,6,7,8,5,4,3] => 1
[3,2,1,8,7,6,5,4] => 1
[3,2,1,6,5,8,7,4] => 1
[1,2,3,5,6,7,8,4] => 1
[4,3,2,1,8,7,6,5] => 1
[3,4,2,1,7,8,6,5] => 1
[2,4,3,1,6,8,7,5] => 1
[2,4,3,1,7,6,8,5] => 1
[3,2,4,1,6,8,7,5] => 1
[3,2,4,1,7,6,8,5] => 1
[2,3,4,1,6,7,8,5] => 1
[2,1,4,3,8,7,6,5] => 1
[2,1,4,3,6,8,7,5] => 1
[2,1,4,3,7,6,8,5] => 1
[5,4,3,2,1,8,7,6] => 1
[3,2,5,4,1,8,7,6] => 1
[3,2,1,4,5,8,7,6] => 1
[2,3,1,4,5,7,8,6] => 1
[6,5,4,3,2,1,8,7] => 1
[4,3,5,2,6,1,8,7] => 1
[3,2,5,4,6,1,8,7] => 1
[2,3,4,5,6,1,8,7] => 1
[2,1,6,5,4,3,8,7] => 1
[2,1,4,6,5,3,8,7] => 1
[2,1,5,4,6,3,8,7] => 1
[4,3,2,1,6,5,8,7] => 1
[2,4,3,1,6,5,8,7] => 1
[3,2,4,1,6,5,8,7] => 1
[2,1,4,3,6,5,8,7] => 1
[2,1,3,4,5,6,8,7] => 1
[1,2,3,4,5,6,8,7] => 1
[5,7,6,4,3,2,1,8] => 1
[4,7,6,5,3,2,1,8] => 1
[3,2,7,6,5,4,1,8] => 1
[5,4,3,2,7,6,1,8] => 1
[3,2,5,4,7,6,1,8] => 1
[1,4,3,2,7,6,5,8] => 1
[1,3,4,2,6,7,5,8] => 1
[1,3,2,5,4,7,6,8] => 1
[1,2,3,5,4,7,6,8] => 1
[1,3,2,4,5,7,6,8] => 1
[1,2,3,4,5,7,6,8] => 1
[1,2,4,3,6,5,7,8] => 1
[1,3,2,5,4,6,7,8] => 1
[1,2,3,5,4,6,7,8] => 1
[1,3,2,4,5,6,7,8] => 1
[1,8,4,7,6,5,3,2] => 2
[1,8,5,4,7,6,3,2] => 3
[1,8,7,4,5,6,3,2] => 9
[1,8,7,4,6,5,3,2] => 3
[1,8,6,5,4,7,3,2] => 2
[1,8,7,5,4,6,3,2] => 3
[1,8,7,5,6,4,3,2] => 3
[2,1,4,5,8,6,7,3] => 2
[2,1,8,5,4,7,6,3] => 3
[2,3,4,8,5,6,7,1] => 4
[2,3,6,4,5,1,8,7] => 2
[2,3,7,4,5,6,8,1] => 4
[2,8,4,3,5,7,6,1] => 5
[4,2,3,1,8,6,7,5] => 4
[8,2,7,4,6,5,3,1] => 6
[4,3,2,7,5,6,8,1] => 2
[4,3,2,8,5,7,6,1] => 2
[6,3,2,5,4,1,8,7] => 3
[8,3,2,5,4,7,6,1] => 9
[8,3,2,7,6,5,4,1] => 3
[8,3,7,4,6,5,2,1] => 6
[7,3,6,5,4,2,1,8] => 2
[7,4,3,6,5,2,1,8] => 3
[7,6,3,5,4,2,1,8] => 3
[8,5,4,3,2,7,6,1] => 3
[8,7,4,3,6,5,2,1] => 6
[2,4,6,8,1,3,5,7] => 5
[2,4,6,1,7,3,8,5] => 4
[2,1,4,3,7,5,8,6] => 1
[2,4,1,3,7,5,8,6] => 1
[2,1,4,3,7,8,5,6] => 2
[2,4,7,8,1,3,5,6] => 8
[2,1,4,3,8,5,6,7] => 1
[2,4,5,6,7,1,8,3] => 5
[2,1,5,3,6,4,8,7] => 1
[2,1,5,6,3,4,8,7] => 2
[2,5,6,8,1,3,4,7] => 10
[2,1,5,3,7,4,8,6] => 1
[2,5,1,7,3,4,8,6] => 2
[2,1,5,7,3,8,4,6] => 4
[2,5,7,8,1,3,4,6] => 16
[2,3,5,6,1,7,8,4] => 2
[2,1,6,3,7,4,8,5] => 2
[2,6,7,1,3,4,8,5] => 5
[2,1,6,7,8,3,4,5] => 10
[2,6,7,8,1,3,4,5] => 26
[2,1,6,3,4,7,8,5] => 1
[2,1,6,3,4,5,8,7] => 1
[2,6,1,3,4,5,7,8] => 1
[2,7,8,1,3,4,5,6] => 8
[2,7,1,3,4,5,6,8] => 1
[2,3,4,5,7,8,1,6] => 2
[2,1,8,3,4,5,6,7] => 1
[2,8,1,3,4,5,6,7] => 1
[2,1,3,8,4,5,6,7] => 1
[2,3,4,5,6,8,1,7] => 1
[3,1,4,2,6,5,8,7] => 1
[3,4,1,2,6,5,8,7] => 2
[3,1,4,2,6,8,5,7] => 1
[3,4,6,8,1,2,5,7] => 8
[3,1,4,2,7,5,8,6] => 1
[3,4,1,2,7,8,5,6] => 4
[3,4,7,8,1,2,5,6] => 13
[3,1,5,2,6,4,8,7] => 1
[3,5,1,6,2,4,8,7] => 4
[3,1,5,6,2,8,4,7] => 2
[3,5,6,8,1,2,4,7] => 16
[3,1,5,2,7,4,8,6] => 1
[3,5,1,7,2,8,4,6] => 8
[3,5,7,8,1,2,4,6] => 26
[1,3,2,5,4,8,6,7] => 1
[1,3,5,8,2,4,6,7] => 2
[1,3,5,6,2,7,4,8] => 2
[3,1,6,2,7,4,8,5] => 2
[3,6,7,1,2,8,4,5] => 13
[3,6,1,7,8,2,4,5] => 20
[3,6,7,8,1,2,4,5] => 43
[1,3,2,6,4,7,5,8] => 1
[3,1,2,6,7,4,8,5] => 2
[1,3,2,6,4,8,5,7] => 1
[1,3,2,6,4,5,7,8] => 1
[1,3,2,7,4,8,5,6] => 2
[1,3,2,7,4,5,6,8] => 1
[1,3,2,8,4,5,6,7] => 1
[3,1,2,4,5,8,6,7] => 1
[1,3,2,4,5,8,6,7] => 1
[4,1,5,2,6,3,8,7] => 2
[4,5,6,1,2,3,8,7] => 10
[4,1,5,6,8,2,3,7] => 5
[4,5,6,8,1,2,3,7] => 26
[4,1,5,2,7,3,8,6] => 2
[4,1,5,2,7,8,3,6] => 4
[4,5,7,1,2,8,3,6] => 20
[4,5,1,7,8,2,3,6] => 13
[4,5,7,8,1,2,3,6] => 43
[1,4,2,5,3,7,6,8] => 1
[1,4,2,5,3,8,6,7] => 1
[1,4,2,5,3,6,7,8] => 1
[1,4,5,6,7,2,3,8] => 5
[4,1,6,2,7,3,8,5] => 4
[4,6,7,1,8,2,3,5] => 51
[4,6,7,8,1,2,3,5] => 73
[1,4,2,6,3,7,5,8] => 1
[1,4,2,6,7,8,3,5] => 3
[1,4,2,6,3,8,5,7] => 1
[4,1,2,3,6,5,8,7] => 1
[1,4,2,6,3,5,7,8] => 1
[1,4,2,7,3,8,5,6] => 2
[4,7,1,2,3,5,6,8] => 3
[1,4,2,7,3,5,6,8] => 1
[1,4,2,3,7,5,6,8] => 1
[1,4,2,3,5,7,6,8] => 1
[1,4,2,8,3,5,6,7] => 1
[4,1,2,3,8,5,6,7] => 1
[1,4,2,3,5,8,6,7] => 1
[1,4,2,3,5,6,7,8] => 1
[5,1,6,2,7,3,8,4] => 5
[1,5,2,6,3,7,4,8] => 2
[1,5,2,6,3,8,4,7] => 2
[1,5,2,6,3,4,7,8] => 2
[1,2,5,6,3,4,7,8] => 2
[1,5,2,7,3,8,4,6] => 4
[5,7,1,2,3,4,6,8] => 5
[1,5,2,7,3,4,6,8] => 2
[1,5,2,3,4,7,6,8] => 1
[5,1,2,8,3,4,6,7] => 2
[1,5,2,8,3,4,6,7] => 2
[1,5,8,2,3,4,6,7] => 3
[1,5,2,3,4,8,6,7] => 1
[1,2,3,5,4,8,6,7] => 1
[5,1,2,3,4,6,8,7] => 1
[1,5,2,3,4,6,7,8] => 1
[1,6,2,7,3,8,4,5] => 5
[1,6,7,8,2,3,4,5] => 26
[6,7,1,2,3,4,8,5] => 8
[6,1,7,2,3,4,5,8] => 5
[6,1,2,7,3,4,5,8] => 3
[1,6,2,7,3,4,5,8] => 3
[1,6,7,2,3,4,5,8] => 5
[1,6,2,3,4,7,5,8] => 1
[1,2,3,6,4,7,5,8] => 1
[6,1,8,2,3,4,5,7] => 5
[1,6,2,8,3,4,5,7] => 3
[1,6,8,2,3,4,5,7] => 5
[1,6,2,3,4,8,5,7] => 1
[1,2,3,6,4,8,5,7] => 1
[6,1,2,3,4,5,8,7] => 1
[1,6,2,3,4,5,7,8] => 1
[1,2,3,6,4,5,7,8] => 1
[1,7,2,8,3,4,5,6] => 5
[1,7,8,2,3,4,5,6] => 8
[1,7,2,3,8,4,5,6] => 3
[7,1,2,3,4,8,5,6] => 2
[1,7,2,3,4,8,5,6] => 2
[1,2,3,7,4,8,5,6] => 2
[7,1,2,3,4,5,8,6] => 1
[1,7,2,3,4,5,8,6] => 1
[1,2,7,3,4,5,8,6] => 1
[1,7,2,3,4,5,6,8] => 1
[1,2,3,7,4,5,6,8] => 1
[1,8,2,3,4,5,6,7] => 1
[1,2,3,8,4,5,6,7] => 1
[1,2,3,4,5,8,6,7] => 1
[2,1,5,8,3,7,6,4] => 2
[2,1,6,8,7,3,5,4] => 3
[2,1,7,8,6,5,3,4] => 2
[2,1,7,6,8,4,3,5] => 3
[2,1,7,5,4,8,3,6] => 2
[2,1,8,6,7,4,5,3] => 8
[3,4,1,2,8,7,6,5] => 2
[3,5,1,8,2,7,6,4] => 4
[3,6,1,8,7,2,5,4] => 6
[3,6,1,5,4,2,8,7] => 2
[3,7,1,8,6,5,2,4] => 4
[3,7,1,6,8,4,2,5] => 6
[3,7,1,5,4,8,2,6] => 4
[3,8,1,5,4,7,6,2] => 6
[3,8,1,6,7,4,5,2] => 16
[3,8,1,7,6,5,4,2] => 2
[4,3,2,1,7,8,5,6] => 2
[4,5,8,1,2,7,6,3] => 10
[4,6,8,1,7,2,5,3] => 14
[4,6,5,1,3,2,8,7] => 3
[4,7,8,1,6,5,2,3] => 10
[4,7,6,1,8,3,2,5] => 9
[4,7,5,1,3,8,2,6] => 6
[4,8,5,1,3,7,6,2] => 9
[4,8,6,1,7,3,5,2] => 24
[4,8,7,1,6,5,3,2] => 3
[5,3,2,8,1,7,6,4] => 2
[5,3,2,7,1,8,4,6] => 4
[5,3,2,6,1,4,8,7] => 2
[5,4,8,2,1,7,6,3] => 3
[5,4,7,2,1,8,3,6] => 6
[5,4,6,2,1,3,8,7] => 3
[5,6,8,7,1,2,4,3] => 24
[5,6,4,3,1,2,8,7] => 2
[5,7,8,6,1,4,2,3] => 13
[5,7,6,8,1,3,2,4] => 26
[5,7,4,3,1,8,2,6] => 4
[5,8,4,3,1,7,6,2] => 4
[5,8,6,7,1,3,4,2] => 72
[5,8,7,6,1,4,3,2] => 4
[6,3,2,8,7,1,5,4] => 3
[6,3,2,7,8,1,4,5] => 10
[6,4,8,2,7,1,5,3] => 8
[6,4,7,2,8,1,3,5] => 14
[6,4,5,2,3,1,8,7] => 8
[6,5,8,7,2,1,4,3] => 6
[6,7,4,3,8,1,2,5] => 10
[6,8,4,3,7,1,5,2] => 10
[6,8,5,7,3,1,4,2] => 18
[6,8,7,5,4,1,3,2] => 3
[7,3,2,8,6,5,1,4] => 4
[7,3,2,6,8,4,1,5] => 9
[7,3,2,5,4,8,1,6] => 6
[7,4,8,2,6,5,1,3] => 10
[7,4,6,2,8,3,1,5] => 24
[7,4,5,2,3,8,1,6] => 16
[7,5,8,6,2,4,1,3] => 18
[7,5,4,3,2,8,1,6] => 2
[7,6,4,3,8,2,1,5] => 3
[7,8,4,3,6,5,1,2] => 18
[8,3,2,6,7,4,5,1] => 24
[8,4,5,2,3,7,6,1] => 24
[8,4,6,2,7,3,5,1] => 63
[8,4,7,2,6,5,3,1] => 8
[8,5,7,6,2,4,3,1] => 10
[8,6,4,3,7,2,5,1] => 8
[4,2,5,1,3,6,7,8] => 2
[5,2,6,7,1,3,4,8] => 10
[3,2,7,8,1,4,5,6] => 5
[5,3,2,6,1,4,7,8] => 2
[7,4,3,8,1,2,5,6] => 5
[4,7,3,6,8,1,2,5] => 16
[6,7,3,4,8,1,2,5] => 44
[5,4,2,7,8,1,3,6] => 5
[6,4,7,2,5,1,3,8] => 8
[6,4,3,2,7,1,5,8] => 2
[5,4,3,2,8,1,6,7] => 1
[7,5,3,8,2,6,1,4] => 8
[8,5,4,7,2,6,1,3] => 10
[8,5,4,7,3,6,1,2] => 10
[2,1,4,3,6,5,10,9,8,7] => 1
[2,1,4,3,6,5,8,7,10,9] => 1
[2,1,4,3,8,7,6,5,10,9] => 1
[2,1,4,3,10,7,6,9,8,5] => 3
[2,1,4,3,10,9,8,7,6,5] => 1
[2,1,6,5,4,3,10,9,8,7] => 1
[2,1,6,5,4,3,8,7,10,9] => 1
[2,1,8,5,4,7,6,3,10,9] => 3
[2,1,8,7,6,5,4,3,10,9] => 1
[2,1,10,5,4,7,6,9,8,3] => 9
[2,1,10,5,4,9,8,7,6,3] => 3
[2,1,10,7,6,5,4,9,8,3] => 3
[2,1,10,9,8,7,6,5,4,3] => 1
[2,1,10,9,6,5,8,7,4,3] => 6
[4,3,2,1,6,5,10,9,8,7] => 1
[4,3,2,1,6,5,8,7,10,9] => 1
[4,3,2,1,8,7,6,5,10,9] => 1
[4,3,2,1,10,7,6,9,8,5] => 3
[4,3,2,1,10,9,8,7,6,5] => 1
[6,3,2,5,4,1,10,9,8,7] => 3
[6,3,2,5,4,1,8,7,10,9] => 3
[6,5,4,3,2,1,10,9,8,7] => 1
[6,5,4,3,2,1,8,7,10,9] => 1
[6,7,8,9,10,1,2,3,4,5] => 6307
[7,3,2,8,9,10,1,4,5,6] => 147
[8,3,2,5,4,7,6,1,10,9] => 9
[8,3,2,7,6,5,4,1,10,9] => 3
[8,5,4,3,2,7,6,1,10,9] => 3
[8,5,4,3,2,9,10,1,6,7] => 10
[8,7,4,3,6,5,2,1,10,9] => 6
[8,7,6,5,4,3,2,1,10,9] => 1
[8,9,5,6,3,4,10,1,2,7] => 320
[9,7,5,10,3,8,2,6,1,4] => 56
[9,7,6,5,4,3,2,10,1,8] => 2
[10,3,2,5,4,7,6,9,8,1] => 27
[10,3,2,5,4,9,8,7,6,1] => 9
[10,3,2,7,6,5,4,9,8,1] => 9
[10,3,2,9,8,7,6,5,4,1] => 3
[10,3,2,9,6,5,8,7,4,1] => 18
[10,5,4,3,2,7,6,9,8,1] => 9
[10,5,4,3,2,9,8,7,6,1] => 5
[10,7,4,3,6,5,2,9,8,1] => 18
[10,7,6,5,4,3,2,9,8,1] => 3
[10,9,4,3,8,7,6,5,2,1] => 6
[10,9,4,3,6,5,8,7,2,1] => 36
[10,9,6,5,4,3,8,7,2,1] => 6
[10,9,8,5,4,7,6,3,2,1] => 10
[10,9,8,7,6,5,4,3,2,1] => 1
[2,4,6,8,10,1,3,5,7,9] => 26
[5,6,7,8,10,1,2,3,4,9] => 673
[4,7,8,9,10,1,2,3,5,6] => 1872
[5,7,8,9,10,1,2,3,4,6] => 3206
[4,5,6,8,10,1,2,3,7,9] => 120
[3,7,8,9,10,1,2,4,5,6] => 1114
[3,6,7,8,10,1,2,4,5,9] => 238
[3,5,8,9,10,1,2,4,6,7] => 334
[3,6,8,9,10,1,2,4,5,7] => 652
[4,6,7,8,10,1,2,3,5,9] => 394
[4,5,8,9,10,1,2,3,6,7] => 566
[5,6,8,9,10,1,2,3,4,7] => 1872
[4,6,8,9,10,1,2,3,5,7] => 1084
[3,4,6,8,10,1,2,5,7,9] => 42
[2,7,8,9,10,1,3,4,5,6] => 673
[2,6,7,8,10,1,3,4,5,9] => 145
[2,5,8,9,10,1,3,4,6,7] => 202
[2,6,8,9,10,1,3,4,5,7] => 394
[2,5,6,8,10,1,3,4,7,9] => 43
[2,4,8,9,10,1,3,5,6,7] => 120
[2,4,7,8,10,1,3,5,6,9] => 43
[2,4,6,9,10,1,3,5,7,8] => 42
[2,4,7,9,10,1,3,5,6,8] => 71
[2,5,7,8,10,1,3,4,6,9] => 73
[4,5,6,9,10,1,2,3,7,8] => 193
[5,6,7,9,10,1,2,3,4,8] => 1114
[2,5,7,9,10,1,3,4,6,8] => 120
[3,5,6,8,10,1,2,4,7,9] => 71
[3,4,8,9,10,1,2,5,6,7] => 193
[3,4,7,8,10,1,2,5,6,9] => 69
[3,4,6,9,10,1,2,5,7,8] => 68
[3,4,7,9,10,1,2,5,6,8] => 114
[4,5,7,8,10,1,2,3,6,9] => 202
[2,5,6,9,10,1,3,4,7,8] => 69
[2,6,7,9,10,1,3,4,5,8] => 238
[4,5,7,9,10,1,2,3,6,8] => 334
[3,5,7,8,10,1,2,4,6,9] => 120
[3,5,6,9,10,1,2,4,7,8] => 114
[3,6,7,9,10,1,2,4,5,8] => 390
[4,6,7,9,10,1,2,3,5,8] => 652
[3,5,7,9,10,1,2,4,6,8] => 197
[10,8,9,5,6,7,1,2,3,4] => 2902
[9,6,10,2,7,8,1,3,4,5] => 735
[10,7,8,3,4,9,1,2,5,6] => 662
[9,5,10,4,6,7,1,2,3,8] => 348
[8,4,9,3,5,10,1,2,6,7] => 120
[8,4,9,2,5,10,1,3,6,7] => 120
[10,8,9,4,5,6,1,2,3,7] => 1166
[9,6,10,5,7,8,1,2,3,4] => 928
[10,7,8,5,6,9,1,2,3,4] => 1334
[9,8,10,5,6,7,1,2,3,4] => 1664
[8,7,9,5,6,10,1,2,3,4] => 484
[8,6,9,2,7,10,1,3,4,5] => 268
[10,6,7,4,5,8,1,2,3,9] => 262
[9,5,10,2,6,7,1,3,4,8] => 301
[10,6,7,3,4,8,1,2,5,9] => 181
[9,8,10,4,5,6,1,2,3,7] => 806
[8,7,9,3,4,10,1,2,5,6] => 260
[8,6,9,5,7,10,1,2,3,4] => 340
[7,6,8,4,5,9,1,2,3,10] => 96
[7,5,8,2,6,9,1,3,4,10] => 62
[7,6,8,3,4,9,1,2,5,10] => 68
[7,5,8,4,6,9,1,2,3,10] => 72
[7,4,8,3,5,9,1,2,6,10] => 44
[7,4,8,2,5,9,1,3,6,10] => 44
[6,7,8,9,10,12,1,2,3,4,5,11] => 51312
[2,8,9,10,11,12,1,3,4,5,6,7] => 51312
[10,9,8,7,6,5,4,3,2,1,12,11] => 1
[2,1,12,11,10,9,8,7,6,5,4,3] => 1
click to show generating function       
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
References
[1] Kazhdan, D., Lusztig, G. Representations of Coxeter groups and Hecke algebras MathSciNet:0560412
[2] Björner, A., Brenti, F. Combinatorics of Coxeter groups MathSciNet:2133266
[3] Humphreys, J. E. Reflection groups and Coxeter groups MathSciNet:1066460
[4] Number of permutations w in S_n whose Kazhdan-Lusztig polynomial evaluated at q=1 is less than or equal to 2; P_id,w(1) <= 2. OEIS:A224255
[5] Number of permutations w in S_n whose Kazhdan-Lusztig polynomial P_id,w(1) <= 3. OEIS:A224316
[6] Woo, A. Permutations with Kazhdan-Lusztig polynomial $P_{id,w}(q)=1+q^h$ MathSciNet:2515773
Code
# takes a long time!
def statistic(x):
    if x.size() == 1:
        return 1
    G = WeylGroup(['A',x.size()-1])
    R.=LaurentPolynomialRing(QQ)
    KL=KazhdanLusztigPolynomial(G,q)
    w = G.from_reduced_word(x.reduced_word())
    return KL.P(1,w).substitute({q:1})

# alternative, with coxeter3 installed:
def statistic_alt(x):
    if x.size() == 1:
        return 1
    G = CoxeterGroup(['A',x.size()-1], implementation="coxeter3")
    R. = PolynomialRing(QQ)
    P = G.kazhdan_lusztig_polynomial([],x.reduced_word())
    return P.substitute({q:1})
Created
Feb 14, 2013 at 00:43 by Sara Billey
Updated
Dec 28, 2017 at 20:18 by Martin Rubey