Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[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] => 1
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 1
[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] => 1
[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] => 1
[2,1,3,4,5] => 2
[2,1,3,5,4] => 2
[2,1,4,3,5] => 3
[2,1,4,5,3] => 2
[2,1,5,3,4] => 3
[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] => 1
[2,3,5,1,4] => 2
[2,3,5,4,1] => 1
[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] => 1
[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] => 1
[3,1,2,4,5] => 2
[3,1,2,5,4] => 2
[3,1,4,2,5] => 3
[3,1,4,5,2] => 2
[3,1,5,2,4] => 3
[3,1,5,4,2] => 2
[3,2,1,4,5] => 2
[3,2,1,5,4] => 2
[3,2,4,1,5] => 3
[3,2,4,5,1] => 2
[3,2,5,1,4] => 3
[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] => 1
[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] => 1
[4,1,2,3,5] => 2
[4,1,2,5,3] => 2
[4,1,3,2,5] => 3
[4,1,3,5,2] => 2
[4,1,5,2,3] => 3
[4,1,5,3,2] => 2
[4,2,1,3,5] => 2
[4,2,1,5,3] => 2
[4,2,3,1,5] => 3
[4,2,3,5,1] => 2
[4,2,5,1,3] => 3
[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] => 3
[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] => 1
[5,1,2,3,4] => 2
[5,1,2,4,3] => 2
[5,1,3,2,4] => 3
[5,1,3,4,2] => 2
[5,1,4,2,3] => 3
[5,1,4,3,2] => 2
[5,2,1,3,4] => 2
[5,2,1,4,3] => 2
[5,2,3,1,4] => 3
[5,2,3,4,1] => 2
[5,2,4,1,3] => 3
[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] => 3
[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] => 1
[1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => 1
[1,2,3,5,4,6] => 2
[1,2,3,5,6,4] => 1
[1,2,3,6,4,5] => 2
[1,2,3,6,5,4] => 1
[1,2,4,3,5,6] => 2
[1,2,4,3,6,5] => 2
[1,2,4,5,3,6] => 2
[1,2,4,5,6,3] => 1
[1,2,4,6,3,5] => 2
[1,2,4,6,5,3] => 1
[1,2,5,3,4,6] => 2
[1,2,5,3,6,4] => 2
[1,2,5,4,3,6] => 2
[1,2,5,4,6,3] => 2
[1,2,5,6,3,4] => 2
[1,2,5,6,4,3] => 1
[1,2,6,3,4,5] => 2
[1,2,6,3,5,4] => 2
[1,2,6,4,3,5] => 2
[1,2,6,4,5,3] => 2
[1,2,6,5,3,4] => 2
[1,2,6,5,4,3] => 1
[1,3,2,4,5,6] => 2
[1,3,2,4,6,5] => 2
[1,3,2,5,4,6] => 3
[1,3,2,5,6,4] => 2
[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] => 2
[1,3,4,5,2,6] => 2
[1,3,4,5,6,2] => 1
[1,3,4,6,2,5] => 2
[1,3,4,6,5,2] => 1
[1,3,5,2,4,6] => 2
[1,3,5,2,6,4] => 2
[1,3,5,4,2,6] => 2
[1,3,5,4,6,2] => 2
[1,3,5,6,2,4] => 2
[1,3,5,6,4,2] => 1
[1,3,6,2,4,5] => 2
[1,3,6,2,5,4] => 2
[1,3,6,4,2,5] => 2
[1,3,6,4,5,2] => 2
[1,3,6,5,2,4] => 2
[1,3,6,5,4,2] => 1
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 2
[1,4,2,5,3,6] => 3
[1,4,2,5,6,3] => 2
[1,4,2,6,3,5] => 3
[1,4,2,6,5,3] => 2
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 2
[1,4,3,5,2,6] => 3
[1,4,3,5,6,2] => 2
[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] => 2
[1,4,5,3,2,6] => 2
[1,4,5,3,6,2] => 2
[1,4,5,6,2,3] => 2
[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] => 2
[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] => 2
[1,5,2,3,6,4] => 2
[1,5,2,4,3,6] => 3
[1,5,2,4,6,3] => 2
[1,5,2,6,3,4] => 3
[1,5,2,6,4,3] => 2
[1,5,3,2,4,6] => 2
[1,5,3,2,6,4] => 2
[1,5,3,4,2,6] => 3
[1,5,3,4,6,2] => 2
[1,5,3,6,2,4] => 3
[1,5,3,6,4,2] => 2
[1,5,4,2,3,6] => 2
[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] => 2
[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] => 1
[1,6,2,3,4,5] => 2
[1,6,2,3,5,4] => 2
[1,6,2,4,3,5] => 3
[1,6,2,4,5,3] => 2
[1,6,2,5,3,4] => 3
[1,6,2,5,4,3] => 2
[1,6,3,2,4,5] => 2
[1,6,3,2,5,4] => 2
[1,6,3,4,2,5] => 3
[1,6,3,4,5,2] => 2
[1,6,3,5,2,4] => 3
[1,6,3,5,4,2] => 2
[1,6,4,2,3,5] => 2
[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] => 2
[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] => 2
[2,1,3,4,6,5] => 2
[2,1,3,5,4,6] => 3
[2,1,3,5,6,4] => 2
[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] => 2
[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] => 2
[2,1,6,3,4,5] => 3
[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] => 2
[2,3,1,4,6,5] => 2
[2,3,1,5,4,6] => 3
[2,3,1,5,6,4] => 2
[2,3,1,6,4,5] => 3
[2,3,1,6,5,4] => 2
[2,3,4,1,5,6] => 2
[2,3,4,1,6,5] => 2
[2,3,4,5,1,6] => 2
[2,3,4,5,6,1] => 1
[2,3,4,6,1,5] => 2
[2,3,4,6,5,1] => 1
[2,3,5,1,4,6] => 2
[2,3,5,1,6,4] => 2
[2,3,5,4,1,6] => 2
[2,3,5,4,6,1] => 2
[2,3,5,6,1,4] => 2
[2,3,5,6,4,1] => 1
[2,3,6,1,4,5] => 2
[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] => 1
[2,4,1,3,5,6] => 2
[2,4,1,3,6,5] => 2
[2,4,1,5,3,6] => 3
[2,4,1,5,6,3] => 2
[2,4,1,6,3,5] => 3
[2,4,1,6,5,3] => 2
[2,4,3,1,5,6] => 2
[2,4,3,1,6,5] => 2
[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] => 2
[2,4,5,1,6,3] => 2
[2,4,5,3,1,6] => 2
[2,4,5,3,6,1] => 2
[2,4,5,6,1,3] => 2
[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] => 2
[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] => 2
[2,5,1,3,6,4] => 2
[2,5,1,4,3,6] => 3
[2,5,1,4,6,3] => 2
[2,5,1,6,3,4] => 3
[2,5,1,6,4,3] => 2
[2,5,3,1,4,6] => 2
[2,5,3,1,6,4] => 2
[2,5,3,4,1,6] => 3
[2,5,3,4,6,1] => 2
[2,5,3,6,1,4] => 3
[2,5,3,6,4,1] => 2
[2,5,4,1,3,6] => 2
[2,5,4,1,6,3] => 2
[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] => 2
[2,5,6,1,3,4] => 2
[2,5,6,1,4,3] => 2
[2,5,6,3,1,4] => 2
[2,5,6,3,4,1] => 2
[2,5,6,4,1,3] => 2
[2,5,6,4,3,1] => 1
[2,6,1,3,4,5] => 2
[2,6,1,3,5,4] => 2
[2,6,1,4,3,5] => 3
[2,6,1,4,5,3] => 2
[2,6,1,5,3,4] => 3
[2,6,1,5,4,3] => 2
[2,6,3,1,4,5] => 2
[2,6,3,1,5,4] => 2
[2,6,3,4,1,5] => 3
[2,6,3,4,5,1] => 2
[2,6,3,5,1,4] => 3
[2,6,3,5,4,1] => 2
[2,6,4,1,3,5] => 2
[2,6,4,1,5,3] => 2
[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] => 2
[2,6,5,1,3,4] => 2
[2,6,5,1,4,3] => 2
[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] => 1
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 2
[3,1,2,5,4,6] => 3
[3,1,2,5,6,4] => 2
[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] => 2
[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] => 2
[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] => 2
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 2
[3,2,1,5,4,6] => 3
[3,2,1,5,6,4] => 2
[3,2,1,6,4,5] => 3
[3,2,1,6,5,4] => 2
[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] => 3
[3,2,5,6,1,4] => 3
[3,2,5,6,4,1] => 2
[3,2,6,1,4,5] => 3
[3,2,6,1,5,4] => 3
[3,2,6,4,1,5] => 3
[3,2,6,4,5,1] => 3
[3,2,6,5,1,4] => 3
[3,2,6,5,4,1] => 2
[3,4,1,2,5,6] => 2
[3,4,1,2,6,5] => 2
[3,4,1,5,2,6] => 3
[3,4,1,5,6,2] => 2
[3,4,1,6,2,5] => 3
[3,4,1,6,5,2] => 2
[3,4,2,1,5,6] => 2
[3,4,2,1,6,5] => 2
[3,4,2,5,1,6] => 3
[3,4,2,5,6,1] => 2
[3,4,2,6,1,5] => 3
[3,4,2,6,5,1] => 2
[3,4,5,1,2,6] => 2
[3,4,5,1,6,2] => 2
[3,4,5,2,1,6] => 2
[3,4,5,2,6,1] => 2
[3,4,5,6,1,2] => 2
[3,4,5,6,2,1] => 1
[3,4,6,1,2,5] => 2
[3,4,6,1,5,2] => 2
[3,4,6,2,1,5] => 2
[3,4,6,2,5,1] => 2
[3,4,6,5,1,2] => 2
[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] => 3
[3,5,1,4,6,2] => 2
[3,5,1,6,2,4] => 3
[3,5,1,6,4,2] => 2
[3,5,2,1,4,6] => 2
[3,5,2,1,6,4] => 2
[3,5,2,4,1,6] => 3
[3,5,2,4,6,1] => 2
[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] => 2
[3,5,4,2,1,6] => 2
[3,5,4,2,6,1] => 2
[3,5,4,6,1,2] => 3
[3,5,4,6,2,1] => 2
[3,5,6,1,2,4] => 2
[3,5,6,1,4,2] => 2
[3,5,6,2,1,4] => 2
[3,5,6,2,4,1] => 2
[3,5,6,4,1,2] => 2
[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] => 3
[3,6,1,4,5,2] => 2
[3,6,1,5,2,4] => 3
[3,6,1,5,4,2] => 2
[3,6,2,1,4,5] => 2
[3,6,2,1,5,4] => 2
[3,6,2,4,1,5] => 3
[3,6,2,4,5,1] => 2
[3,6,2,5,1,4] => 3
[3,6,2,5,4,1] => 2
[3,6,4,1,2,5] => 2
[3,6,4,1,5,2] => 2
[3,6,4,2,1,5] => 2
[3,6,4,2,5,1] => 2
[3,6,4,5,1,2] => 3
[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] => 2
[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] => 2
[4,1,2,3,6,5] => 2
[4,1,2,5,3,6] => 3
[4,1,2,5,6,3] => 2
[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] => 2
[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] => 3
[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] => 3
[4,1,6,5,3,2] => 2
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 2
[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] => 2
[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] => 3
[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] => 2
[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] => 2
[4,3,1,2,5,6] => 2
[4,3,1,2,6,5] => 2
[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] => 2
[4,3,2,1,5,6] => 2
[4,3,2,1,6,5] => 2
[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] => 2
[4,3,5,1,2,6] => 3
[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] => 2
[4,3,6,1,2,5] => 3
[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] => 2
[4,5,1,2,3,6] => 2
[4,5,1,2,6,3] => 2
[4,5,1,3,2,6] => 3
[4,5,1,3,6,2] => 2
[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] => 3
[4,5,2,3,6,1] => 2
[4,5,2,6,1,3] => 3
[4,5,2,6,3,1] => 2
[4,5,3,1,2,6] => 2
[4,5,3,1,6,2] => 2
[4,5,3,2,1,6] => 2
[4,5,3,2,6,1] => 2
[4,5,3,6,1,2] => 3
[4,5,3,6,2,1] => 2
[4,5,6,1,2,3] => 2
[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] => 1
[4,6,1,2,3,5] => 2
[4,6,1,2,5,3] => 2
[4,6,1,3,2,5] => 3
[4,6,1,3,5,2] => 2
[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] => 2
[4,6,2,5,1,3] => 3
[4,6,2,5,3,1] => 2
[4,6,3,1,2,5] => 2
[4,6,3,1,5,2] => 2
[4,6,3,2,1,5] => 2
[4,6,3,2,5,1] => 2
[4,6,3,5,1,2] => 3
[4,6,3,5,2,1] => 2
[4,6,5,1,2,3] => 2
[4,6,5,1,3,2] => 2
[4,6,5,2,1,3] => 2
[4,6,5,2,3,1] => 2
[4,6,5,3,1,2] => 2
[4,6,5,3,2,1] => 1
[5,1,2,3,4,6] => 2
[5,1,2,3,6,4] => 2
[5,1,2,4,3,6] => 3
[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] => 3
[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] => 3
[5,1,4,2,6,3] => 3
[5,1,4,3,2,6] => 3
[5,1,4,3,6,2] => 3
[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] => 2
[5,2,1,3,6,4] => 2
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 2
[5,2,1,6,3,4] => 3
[5,2,1,6,4,3] => 2
[5,2,3,1,4,6] => 3
[5,2,3,1,6,4] => 3
[5,2,3,4,1,6] => 3
[5,2,3,4,6,1] => 2
[5,2,3,6,1,4] => 3
[5,2,3,6,4,1] => 2
[5,2,4,1,3,6] => 3
[5,2,4,1,6,3] => 3
[5,2,4,3,1,6] => 3
[5,2,4,3,6,1] => 3
[5,2,4,6,1,3] => 3
[5,2,4,6,3,1] => 2
[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] => 2
[5,3,1,2,4,6] => 2
[5,3,1,2,6,4] => 2
[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] => 2
[5,3,2,1,4,6] => 2
[5,3,2,1,6,4] => 2
[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] => 2
[5,3,4,1,2,6] => 3
[5,3,4,1,6,2] => 3
[5,3,4,2,1,6] => 3
[5,3,4,2,6,1] => 3
[5,3,4,6,1,2] => 3
[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] => 3
[5,3,6,2,4,1] => 3
[5,3,6,4,1,2] => 3
[5,3,6,4,2,1] => 2
[5,4,1,2,3,6] => 2
[5,4,1,2,6,3] => 2
[5,4,1,3,2,6] => 3
[5,4,1,3,6,2] => 2
[5,4,1,6,2,3] => 3
[5,4,1,6,3,2] => 2
[5,4,2,1,3,6] => 2
[5,4,2,1,6,3] => 2
[5,4,2,3,1,6] => 3
[5,4,2,3,6,1] => 2
[5,4,2,6,1,3] => 3
[5,4,2,6,3,1] => 2
[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] => 3
[5,4,3,6,2,1] => 2
[5,4,6,1,2,3] => 3
[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] => 2
[5,6,1,2,3,4] => 2
[5,6,1,2,4,3] => 2
[5,6,1,3,2,4] => 3
[5,6,1,3,4,2] => 2
[5,6,1,4,2,3] => 3
[5,6,1,4,3,2] => 2
[5,6,2,1,3,4] => 2
[5,6,2,1,4,3] => 2
[5,6,2,3,1,4] => 3
[5,6,2,3,4,1] => 2
[5,6,2,4,1,3] => 3
[5,6,2,4,3,1] => 2
[5,6,3,1,2,4] => 2
[5,6,3,1,4,2] => 2
[5,6,3,2,1,4] => 2
[5,6,3,2,4,1] => 2
[5,6,3,4,1,2] => 3
[5,6,3,4,2,1] => 2
[5,6,4,1,2,3] => 2
[5,6,4,1,3,2] => 2
[5,6,4,2,1,3] => 2
[5,6,4,2,3,1] => 2
[5,6,4,3,1,2] => 2
[5,6,4,3,2,1] => 1
[6,1,2,3,4,5] => 2
[6,1,2,3,5,4] => 2
[6,1,2,4,3,5] => 3
[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] => 3
[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] => 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] => 3
[6,1,4,3,5,2] => 3
[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] => 3
[6,2,1,4,5,3] => 2
[6,2,1,5,3,4] => 3
[6,2,1,5,4,3] => 2
[6,2,3,1,4,5] => 3
[6,2,3,1,5,4] => 3
[6,2,3,4,1,5] => 3
[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] => 3
[6,2,4,3,5,1] => 3
[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] => 3
[6,2,5,4,1,3] => 3
[6,2,5,4,3,1] => 2
[6,3,1,2,4,5] => 2
[6,3,1,2,5,4] => 2
[6,3,1,4,2,5] => 3
[6,3,1,4,5,2] => 2
[6,3,1,5,2,4] => 3
[6,3,1,5,4,2] => 2
[6,3,2,1,4,5] => 2
[6,3,2,1,5,4] => 2
[6,3,2,4,1,5] => 3
[6,3,2,4,5,1] => 2
[6,3,2,5,1,4] => 3
[6,3,2,5,4,1] => 2
[6,3,4,1,2,5] => 3
[6,3,4,1,5,2] => 3
[6,3,4,2,1,5] => 3
[6,3,4,2,5,1] => 3
[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] => 2
[6,4,1,2,3,5] => 2
[6,4,1,2,5,3] => 2
[6,4,1,3,2,5] => 3
[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] => 2
[6,4,2,1,5,3] => 2
[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] => 2
[6,4,3,1,2,5] => 2
[6,4,3,1,5,2] => 2
[6,4,3,2,1,5] => 2
[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] => 3
[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] => 2
[6,5,1,2,3,4] => 2
[6,5,1,2,4,3] => 2
[6,5,1,3,2,4] => 3
[6,5,1,3,4,2] => 2
[6,5,1,4,2,3] => 3
[6,5,1,4,3,2] => 2
[6,5,2,1,3,4] => 2
[6,5,2,1,4,3] => 2
[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] => 2
[6,5,3,1,2,4] => 2
[6,5,3,1,4,2] => 2
[6,5,3,2,1,4] => 2
[6,5,3,2,4,1] => 2
[6,5,3,4,1,2] => 3
[6,5,3,4,2,1] => 2
[6,5,4,1,2,3] => 2
[6,5,4,1,3,2] => 2
[6,5,4,2,1,3] => 2
[6,5,4,2,3,1] => 2
[6,5,4,3,1,2] => 2
[6,5,4,3,2,1] => 1
click to show generating function       
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
References
[1] Claesson, A., Kitaev, S. Classification of bijections between 321- and 132-avoiding permutations MathSciNet:2465405
Code
def statistic(pi):
    pi = [0] + list(pi) + [0]
    n = len(pi)
    return sum( 1 for i in [1 .. n-2] if pi[i-1] < pi[i] > pi[i+1] )

Created
Jun 13, 2013 at 16:10 by Chris Berg
Updated
Jun 05, 2017 at 10:07 by Christian Stump