Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 3
[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] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 3
[3,4,2,1] => 2
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 2
[1,2,3,4,5] => 4
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 2
[1,3,4,2,5] => 3
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 3
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 3
[1,4,5,3,2] => 2
[1,5,2,3,4] => 3
[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] => 1
[2,1,3,4,5] => 4
[2,1,3,5,4] => 3
[2,1,4,3,5] => 3
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 2
[2,3,1,4,5] => 3
[2,3,1,5,4] => 2
[2,3,4,1,5] => 3
[2,3,4,5,1] => 4
[2,3,5,1,4] => 3
[2,3,5,4,1] => 3
[2,4,1,3,5] => 3
[2,4,1,5,3] => 2
[2,4,3,1,5] => 2
[2,4,3,5,1] => 3
[2,4,5,1,3] => 3
[2,4,5,3,1] => 3
[2,5,1,3,4] => 3
[2,5,1,4,3] => 2
[2,5,3,1,4] => 2
[2,5,3,4,1] => 3
[2,5,4,1,3] => 2
[2,5,4,3,1] => 2
[3,1,2,4,5] => 4
[3,1,2,5,4] => 3
[3,1,4,2,5] => 3
[3,1,4,5,2] => 4
[3,1,5,2,4] => 3
[3,1,5,4,2] => 3
[3,2,1,4,5] => 3
[3,2,1,5,4] => 2
[3,2,4,1,5] => 3
[3,2,4,5,1] => 4
[3,2,5,1,4] => 3
[3,2,5,4,1] => 3
[3,4,1,2,5] => 3
[3,4,1,5,2] => 3
[3,4,2,1,5] => 2
[3,4,2,5,1] => 3
[3,4,5,1,2] => 4
[3,4,5,2,1] => 3
[3,5,1,2,4] => 3
[3,5,1,4,2] => 3
[3,5,2,1,4] => 2
[3,5,2,4,1] => 3
[3,5,4,1,2] => 3
[3,5,4,2,1] => 2
[4,1,2,3,5] => 4
[4,1,2,5,3] => 4
[4,1,3,2,5] => 3
[4,1,3,5,2] => 4
[4,1,5,2,3] => 4
[4,1,5,3,2] => 3
[4,2,1,3,5] => 3
[4,2,1,5,3] => 3
[4,2,3,1,5] => 3
[4,2,3,5,1] => 4
[4,2,5,1,3] => 4
[4,2,5,3,1] => 3
[4,3,1,2,5] => 3
[4,3,1,5,2] => 3
[4,3,2,1,5] => 2
[4,3,2,5,1] => 3
[4,3,5,1,2] => 4
[4,3,5,2,1] => 3
[4,5,1,2,3] => 4
[4,5,1,3,2] => 3
[4,5,2,1,3] => 3
[4,5,2,3,1] => 3
[4,5,3,1,2] => 3
[4,5,3,2,1] => 2
[5,1,2,3,4] => 5
[5,1,2,4,3] => 4
[5,1,3,2,4] => 4
[5,1,3,4,2] => 4
[5,1,4,2,3] => 4
[5,1,4,3,2] => 3
[5,2,1,3,4] => 4
[5,2,1,4,3] => 3
[5,2,3,1,4] => 4
[5,2,3,4,1] => 4
[5,2,4,1,3] => 4
[5,2,4,3,1] => 3
[5,3,1,2,4] => 4
[5,3,1,4,2] => 3
[5,3,2,1,4] => 3
[5,3,2,4,1] => 3
[5,3,4,1,2] => 4
[5,3,4,2,1] => 3
[5,4,1,2,3] => 4
[5,4,1,3,2] => 3
[5,4,2,1,3] => 3
[5,4,2,3,1] => 3
[5,4,3,1,2] => 3
[5,4,3,2,1] => 2
[1,2,3,4,5,6] => 5
[1,2,3,4,6,5] => 4
[1,2,3,5,4,6] => 4
[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] => 4
[1,2,4,3,6,5] => 3
[1,2,4,5,3,6] => 4
[1,2,4,5,6,3] => 4
[1,2,4,6,3,5] => 4
[1,2,4,6,5,3] => 3
[1,2,5,3,4,6] => 4
[1,2,5,3,6,4] => 3
[1,2,5,4,3,6] => 3
[1,2,5,4,6,3] => 3
[1,2,5,6,3,4] => 4
[1,2,5,6,4,3] => 3
[1,2,6,3,4,5] => 4
[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] => 2
[1,3,2,4,5,6] => 4
[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] => 4
[1,3,4,2,6,5] => 3
[1,3,4,5,2,6] => 4
[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] => 4
[1,3,5,2,6,4] => 3
[1,3,5,4,2,6] => 3
[1,3,5,4,6,2] => 3
[1,3,5,6,2,4] => 4
[1,3,5,6,4,2] => 3
[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] => 3
[1,3,6,5,2,4] => 3
[1,3,6,5,4,2] => 2
[1,4,2,3,5,6] => 4
[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] => 2
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 2
[1,4,3,5,2,6] => 3
[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] => 4
[1,4,5,2,6,3] => 3
[1,4,5,3,2,6] => 3
[1,4,5,3,6,2] => 3
[1,4,5,6,2,3] => 4
[1,4,5,6,3,2] => 3
[1,4,6,2,3,5] => 4
[1,4,6,2,5,3] => 3
[1,4,6,3,2,5] => 3
[1,4,6,3,5,2] => 3
[1,4,6,5,2,3] => 3
[1,4,6,5,3,2] => 2
[1,5,2,3,4,6] => 4
[1,5,2,3,6,4] => 3
[1,5,2,4,3,6] => 3
[1,5,2,4,6,3] => 3
[1,5,2,6,3,4] => 3
[1,5,2,6,4,3] => 2
[1,5,3,2,4,6] => 3
[1,5,3,2,6,4] => 2
[1,5,3,4,2,6] => 3
[1,5,3,4,6,2] => 3
[1,5,3,6,2,4] => 3
[1,5,3,6,4,2] => 2
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 2
[1,5,4,3,2,6] => 2
[1,5,4,3,6,2] => 2
[1,5,4,6,2,3] => 3
[1,5,4,6,3,2] => 2
[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] => 3
[1,5,6,4,2,3] => 3
[1,5,6,4,3,2] => 2
[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] => 3
[1,6,2,5,3,4] => 3
[1,6,2,5,4,3] => 2
[1,6,3,2,4,5] => 3
[1,6,3,2,5,4] => 2
[1,6,3,4,2,5] => 3
[1,6,3,4,5,2] => 3
[1,6,3,5,2,4] => 3
[1,6,3,5,4,2] => 2
[1,6,4,2,3,5] => 3
[1,6,4,2,5,3] => 2
[1,6,4,3,2,5] => 2
[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] => 3
[1,6,5,2,4,3] => 2
[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] => 1
[2,1,3,4,5,6] => 5
[2,1,3,4,6,5] => 4
[2,1,3,5,4,6] => 4
[2,1,3,5,6,4] => 4
[2,1,3,6,4,5] => 4
[2,1,3,6,5,4] => 3
[2,1,4,3,5,6] => 4
[2,1,4,3,6,5] => 3
[2,1,4,5,3,6] => 4
[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] => 4
[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] => 4
[2,1,5,6,4,3] => 3
[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] => 3
[2,1,6,5,3,4] => 3
[2,1,6,5,4,3] => 2
[2,3,1,4,5,6] => 4
[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] => 4
[2,3,4,1,6,5] => 3
[2,3,4,5,1,6] => 4
[2,3,4,5,6,1] => 5
[2,3,4,6,1,5] => 4
[2,3,4,6,5,1] => 4
[2,3,5,1,4,6] => 4
[2,3,5,1,6,4] => 3
[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] => 4
[2,3,6,1,4,5] => 4
[2,3,6,1,5,4] => 3
[2,3,6,4,1,5] => 3
[2,3,6,4,5,1] => 4
[2,3,6,5,1,4] => 3
[2,3,6,5,4,1] => 3
[2,4,1,3,5,6] => 4
[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] => 2
[2,4,3,1,5,6] => 3
[2,4,3,1,6,5] => 2
[2,4,3,5,1,6] => 3
[2,4,3,5,6,1] => 4
[2,4,3,6,1,5] => 3
[2,4,3,6,5,1] => 3
[2,4,5,1,3,6] => 4
[2,4,5,1,6,3] => 3
[2,4,5,3,1,6] => 3
[2,4,5,3,6,1] => 4
[2,4,5,6,1,3] => 4
[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] => 3
[2,4,6,3,5,1] => 4
[2,4,6,5,1,3] => 3
[2,4,6,5,3,1] => 3
[2,5,1,3,4,6] => 4
[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] => 2
[2,5,3,1,4,6] => 3
[2,5,3,1,6,4] => 2
[2,5,3,4,1,6] => 3
[2,5,3,4,6,1] => 4
[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] => 2
[2,5,4,3,1,6] => 2
[2,5,4,3,6,1] => 3
[2,5,4,6,1,3] => 3
[2,5,4,6,3,1] => 3
[2,5,6,1,3,4] => 4
[2,5,6,1,4,3] => 3
[2,5,6,3,1,4] => 3
[2,5,6,3,4,1] => 4
[2,5,6,4,1,3] => 3
[2,5,6,4,3,1] => 3
[2,6,1,3,4,5] => 4
[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] => 2
[2,6,3,1,4,5] => 3
[2,6,3,1,5,4] => 2
[2,6,3,4,1,5] => 3
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 3
[2,6,3,5,4,1] => 3
[2,6,4,1,3,5] => 3
[2,6,4,1,5,3] => 2
[2,6,4,3,1,5] => 2
[2,6,4,3,5,1] => 3
[2,6,4,5,1,3] => 3
[2,6,4,5,3,1] => 3
[2,6,5,1,3,4] => 3
[2,6,5,1,4,3] => 2
[2,6,5,3,1,4] => 2
[2,6,5,3,4,1] => 3
[2,6,5,4,1,3] => 2
[2,6,5,4,3,1] => 2
[3,1,2,4,5,6] => 5
[3,1,2,4,6,5] => 4
[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] => 3
[3,1,4,2,5,6] => 4
[3,1,4,2,6,5] => 3
[3,1,4,5,2,6] => 4
[3,1,4,5,6,2] => 5
[3,1,4,6,2,5] => 4
[3,1,4,6,5,2] => 4
[3,1,5,2,4,6] => 4
[3,1,5,2,6,4] => 3
[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] => 4
[3,1,6,2,4,5] => 4
[3,1,6,2,5,4] => 3
[3,1,6,4,2,5] => 3
[3,1,6,4,5,2] => 4
[3,1,6,5,2,4] => 3
[3,1,6,5,4,2] => 3
[3,2,1,4,5,6] => 4
[3,2,1,4,6,5] => 3
[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] => 2
[3,2,4,1,5,6] => 4
[3,2,4,1,6,5] => 3
[3,2,4,5,1,6] => 4
[3,2,4,5,6,1] => 5
[3,2,4,6,1,5] => 4
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 4
[3,2,5,1,6,4] => 3
[3,2,5,4,1,6] => 3
[3,2,5,4,6,1] => 4
[3,2,5,6,1,4] => 4
[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] => 4
[3,2,6,5,1,4] => 3
[3,2,6,5,4,1] => 3
[3,4,1,2,5,6] => 4
[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] => 3
[3,4,1,6,5,2] => 3
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 2
[3,4,2,5,1,6] => 3
[3,4,2,5,6,1] => 4
[3,4,2,6,1,5] => 3
[3,4,2,6,5,1] => 3
[3,4,5,1,2,6] => 4
[3,4,5,1,6,2] => 4
[3,4,5,2,1,6] => 3
[3,4,5,2,6,1] => 4
[3,4,5,6,1,2] => 5
[3,4,5,6,2,1] => 4
[3,4,6,1,2,5] => 4
[3,4,6,1,5,2] => 4
[3,4,6,2,1,5] => 3
[3,4,6,2,5,1] => 4
[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] => 3
[3,5,1,4,2,6] => 3
[3,5,1,4,6,2] => 4
[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] => 2
[3,5,2,4,1,6] => 3
[3,5,2,4,6,1] => 4
[3,5,2,6,1,4] => 3
[3,5,2,6,4,1] => 3
[3,5,4,1,2,6] => 3
[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] => 4
[3,5,4,6,2,1] => 3
[3,5,6,1,2,4] => 4
[3,5,6,1,4,2] => 4
[3,5,6,2,1,4] => 3
[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] => 4
[3,6,1,5,2,4] => 3
[3,6,1,5,4,2] => 3
[3,6,2,1,4,5] => 3
[3,6,2,1,5,4] => 2
[3,6,2,4,1,5] => 3
[3,6,2,4,5,1] => 4
[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] => 4
[3,6,4,5,2,1] => 3
[3,6,5,1,2,4] => 3
[3,6,5,1,4,2] => 3
[3,6,5,2,1,4] => 2
[3,6,5,2,4,1] => 3
[3,6,5,4,1,2] => 3
[3,6,5,4,2,1] => 2
[4,1,2,3,5,6] => 5
[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] => 4
[4,1,2,6,5,3] => 4
[4,1,3,2,5,6] => 4
[4,1,3,2,6,5] => 3
[4,1,3,5,2,6] => 4
[4,1,3,5,6,2] => 5
[4,1,3,6,2,5] => 4
[4,1,3,6,5,2] => 4
[4,1,5,2,3,6] => 4
[4,1,5,2,6,3] => 4
[4,1,5,3,2,6] => 3
[4,1,5,3,6,2] => 4
[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] => 4
[4,1,6,3,2,5] => 3
[4,1,6,3,5,2] => 4
[4,1,6,5,2,3] => 4
[4,1,6,5,3,2] => 3
[4,2,1,3,5,6] => 4
[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] => 3
[4,2,1,6,5,3] => 3
[4,2,3,1,5,6] => 4
[4,2,3,1,6,5] => 3
[4,2,3,5,1,6] => 4
[4,2,3,5,6,1] => 5
[4,2,3,6,1,5] => 4
[4,2,3,6,5,1] => 4
[4,2,5,1,3,6] => 4
[4,2,5,1,6,3] => 4
[4,2,5,3,1,6] => 3
[4,2,5,3,6,1] => 4
[4,2,5,6,1,3] => 5
[4,2,5,6,3,1] => 4
[4,2,6,1,3,5] => 4
[4,2,6,1,5,3] => 4
[4,2,6,3,1,5] => 3
[4,2,6,3,5,1] => 4
[4,2,6,5,1,3] => 4
[4,2,6,5,3,1] => 3
[4,3,1,2,5,6] => 4
[4,3,1,2,6,5] => 3
[4,3,1,5,2,6] => 3
[4,3,1,5,6,2] => 4
[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] => 2
[4,3,2,5,1,6] => 3
[4,3,2,5,6,1] => 4
[4,3,2,6,1,5] => 3
[4,3,2,6,5,1] => 3
[4,3,5,1,2,6] => 4
[4,3,5,1,6,2] => 4
[4,3,5,2,1,6] => 3
[4,3,5,2,6,1] => 4
[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] => 4
[4,3,6,2,1,5] => 3
[4,3,6,2,5,1] => 4
[4,3,6,5,1,2] => 4
[4,3,6,5,2,1] => 3
[4,5,1,2,3,6] => 4
[4,5,1,2,6,3] => 4
[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] => 3
[4,5,2,1,3,6] => 3
[4,5,2,1,6,3] => 3
[4,5,2,3,1,6] => 3
[4,5,2,3,6,1] => 4
[4,5,2,6,1,3] => 4
[4,5,2,6,3,1] => 3
[4,5,3,1,2,6] => 3
[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] => 4
[4,5,3,6,2,1] => 3
[4,5,6,1,2,3] => 5
[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] => 4
[4,5,6,3,2,1] => 3
[4,6,1,2,3,5] => 4
[4,6,1,2,5,3] => 4
[4,6,1,3,2,5] => 3
[4,6,1,3,5,2] => 4
[4,6,1,5,2,3] => 4
[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] => 4
[4,6,2,5,1,3] => 4
[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] => 4
[4,6,3,5,2,1] => 3
[4,6,5,1,2,3] => 4
[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] => 2
[5,1,2,3,4,6] => 5
[5,1,2,3,6,4] => 5
[5,1,2,4,3,6] => 4
[5,1,2,4,6,3] => 5
[5,1,2,6,3,4] => 5
[5,1,2,6,4,3] => 4
[5,1,3,2,4,6] => 4
[5,1,3,2,6,4] => 4
[5,1,3,4,2,6] => 4
[5,1,3,4,6,2] => 5
[5,1,3,6,2,4] => 5
[5,1,3,6,4,2] => 4
[5,1,4,2,3,6] => 4
[5,1,4,2,6,3] => 4
[5,1,4,3,2,6] => 3
[5,1,4,3,6,2] => 4
[5,1,4,6,2,3] => 5
[5,1,4,6,3,2] => 4
[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] => 4
[5,1,6,4,2,3] => 4
[5,1,6,4,3,2] => 3
[5,2,1,3,4,6] => 4
[5,2,1,3,6,4] => 4
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 4
[5,2,1,6,3,4] => 4
[5,2,1,6,4,3] => 3
[5,2,3,1,4,6] => 4
[5,2,3,1,6,4] => 4
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 5
[5,2,3,6,1,4] => 5
[5,2,3,6,4,1] => 4
[5,2,4,1,3,6] => 4
[5,2,4,1,6,3] => 4
[5,2,4,3,1,6] => 3
[5,2,4,3,6,1] => 4
[5,2,4,6,1,3] => 5
[5,2,4,6,3,1] => 4
[5,2,6,1,3,4] => 5
[5,2,6,1,4,3] => 4
[5,2,6,3,1,4] => 4
[5,2,6,3,4,1] => 4
[5,2,6,4,1,3] => 4
[5,2,6,4,3,1] => 3
[5,3,1,2,4,6] => 4
[5,3,1,2,6,4] => 4
[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] => 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] => 4
[5,3,2,6,1,4] => 4
[5,3,2,6,4,1] => 3
[5,3,4,1,2,6] => 4
[5,3,4,1,6,2] => 4
[5,3,4,2,1,6] => 3
[5,3,4,2,6,1] => 4
[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] => 4
[5,3,6,2,1,4] => 4
[5,3,6,2,4,1] => 4
[5,3,6,4,1,2] => 4
[5,3,6,4,2,1] => 3
[5,4,1,2,3,6] => 4
[5,4,1,2,6,3] => 4
[5,4,1,3,2,6] => 3
[5,4,1,3,6,2] => 4
[5,4,1,6,2,3] => 4
[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] => 3
[5,4,2,3,6,1] => 4
[5,4,2,6,1,3] => 4
[5,4,2,6,3,1] => 3
[5,4,3,1,2,6] => 3
[5,4,3,1,6,2] => 3
[5,4,3,2,1,6] => 2
[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] => 5
[5,4,6,1,3,2] => 4
[5,4,6,2,1,3] => 4
[5,4,6,2,3,1] => 4
[5,4,6,3,1,2] => 4
[5,4,6,3,2,1] => 3
[5,6,1,2,3,4] => 5
[5,6,1,2,4,3] => 4
[5,6,1,3,2,4] => 4
[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] => 4
[5,6,2,1,4,3] => 3
[5,6,2,3,1,4] => 4
[5,6,2,3,4,1] => 4
[5,6,2,4,1,3] => 4
[5,6,2,4,3,1] => 3
[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] => 3
[5,6,3,4,1,2] => 4
[5,6,3,4,2,1] => 3
[5,6,4,1,2,3] => 4
[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] => 2
[6,1,2,3,4,5] => 6
[6,1,2,3,5,4] => 5
[6,1,2,4,3,5] => 5
[6,1,2,4,5,3] => 5
[6,1,2,5,3,4] => 5
[6,1,2,5,4,3] => 4
[6,1,3,2,4,5] => 5
[6,1,3,2,5,4] => 4
[6,1,3,4,2,5] => 5
[6,1,3,4,5,2] => 5
[6,1,3,5,2,4] => 5
[6,1,3,5,4,2] => 4
[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] => 4
[6,1,4,5,2,3] => 5
[6,1,4,5,3,2] => 4
[6,1,5,2,3,4] => 5
[6,1,5,2,4,3] => 4
[6,1,5,3,2,4] => 4
[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] => 5
[6,2,1,3,5,4] => 4
[6,2,1,4,3,5] => 4
[6,2,1,4,5,3] => 4
[6,2,1,5,3,4] => 4
[6,2,1,5,4,3] => 3
[6,2,3,1,4,5] => 5
[6,2,3,1,5,4] => 4
[6,2,3,4,1,5] => 5
[6,2,3,4,5,1] => 5
[6,2,3,5,1,4] => 5
[6,2,3,5,4,1] => 4
[6,2,4,1,3,5] => 5
[6,2,4,1,5,3] => 4
[6,2,4,3,1,5] => 4
[6,2,4,3,5,1] => 4
[6,2,4,5,1,3] => 5
[6,2,4,5,3,1] => 4
[6,2,5,1,3,4] => 5
[6,2,5,1,4,3] => 4
[6,2,5,3,1,4] => 4
[6,2,5,3,4,1] => 4
[6,2,5,4,1,3] => 4
[6,2,5,4,3,1] => 3
[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] => 4
[6,3,1,5,2,4] => 4
[6,3,1,5,4,2] => 3
[6,3,2,1,4,5] => 4
[6,3,2,1,5,4] => 3
[6,3,2,4,1,5] => 4
[6,3,2,4,5,1] => 4
[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] => 4
[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] => 4
[6,3,5,2,1,4] => 4
[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] => 5
[6,4,1,2,5,3] => 4
[6,4,1,3,2,5] => 4
[6,4,1,3,5,2] => 4
[6,4,1,5,2,3] => 4
[6,4,1,5,3,2] => 3
[6,4,2,1,3,5] => 4
[6,4,2,1,5,3] => 3
[6,4,2,3,1,5] => 4
[6,4,2,3,5,1] => 4
[6,4,2,5,1,3] => 4
[6,4,2,5,3,1] => 3
[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] => 3
[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] => 4
[6,4,5,3,1,2] => 4
[6,4,5,3,2,1] => 3
[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] => 4
[6,5,1,4,2,3] => 4
[6,5,1,4,3,2] => 3
[6,5,2,1,3,4] => 4
[6,5,2,1,4,3] => 3
[6,5,2,3,1,4] => 4
[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] => 4
[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] => 4
[6,5,3,4,2,1] => 3
[6,5,4,1,2,3] => 4
[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] => 2
click to show generating function       
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree as described on the permutations page, and the statistic is given by the number of leaves of this tree.
See also St000308The height of the tree associated to a permutation. for the height of this tree.
References
[1] Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows. OEIS:A123125
[2] http://oeis.org/wiki/User:Peter_Luschny/PermutationTrees
Code
def statistic(pi):
    if pi == []: return 0
    w, i, branch, next = 0, len(pi), [0], pi[0] 
    while true:
        while next < branch[len(branch)-1]:
            branch.pop()
        current = 0
        w += 1 
        while next > current:
            i -= 1 
            if i == 0: return w
            branch.append(next)
            current, next = next, pi[i]
Created
Dec 11, 2015 at 11:05 by Peter Luschny
Updated
Dec 11, 2015 at 23:39 by Christian Stump