Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 3
[2,3,1] => 2
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 4
[2,3,1,4] => 3
[2,3,4,1] => 2
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 4
[3,2,1,4] => 3
[3,2,4,1] => 4
[3,4,1,2] => 3
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 4
[4,2,1,3] => 3
[4,2,3,1] => 4
[4,3,1,2] => 3
[4,3,2,1] => 2
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 4
[1,3,4,2,5] => 3
[1,3,4,5,2] => 2
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 3
[1,4,2,5,3] => 4
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 3
[1,4,5,3,2] => 2
[1,5,2,3,4] => 3
[1,5,2,4,3] => 4
[1,5,3,2,4] => 3
[1,5,3,4,2] => 4
[1,5,4,2,3] => 3
[1,5,4,3,2] => 2
[2,1,3,4,5] => 3
[2,1,3,5,4] => 4
[2,1,4,3,5] => 5
[2,1,4,5,3] => 4
[2,1,5,3,4] => 5
[2,1,5,4,3] => 4
[2,3,1,4,5] => 3
[2,3,1,5,4] => 4
[2,3,4,1,5] => 3
[2,3,4,5,1] => 2
[2,3,5,1,4] => 3
[2,3,5,4,1] => 2
[2,4,1,3,5] => 3
[2,4,1,5,3] => 4
[2,4,3,1,5] => 3
[2,4,3,5,1] => 4
[2,4,5,1,3] => 3
[2,4,5,3,1] => 2
[2,5,1,3,4] => 3
[2,5,1,4,3] => 4
[2,5,3,1,4] => 3
[2,5,3,4,1] => 4
[2,5,4,1,3] => 3
[2,5,4,3,1] => 2
[3,1,2,4,5] => 3
[3,1,2,5,4] => 4
[3,1,4,2,5] => 5
[3,1,4,5,2] => 4
[3,1,5,2,4] => 5
[3,1,5,4,2] => 4
[3,2,1,4,5] => 3
[3,2,1,5,4] => 4
[3,2,4,1,5] => 5
[3,2,4,5,1] => 4
[3,2,5,1,4] => 5
[3,2,5,4,1] => 4
[3,4,1,2,5] => 3
[3,4,1,5,2] => 4
[3,4,2,1,5] => 3
[3,4,2,5,1] => 4
[3,4,5,1,2] => 3
[3,4,5,2,1] => 2
[3,5,1,2,4] => 3
[3,5,1,4,2] => 4
[3,5,2,1,4] => 3
[3,5,2,4,1] => 4
[3,5,4,1,2] => 3
[3,5,4,2,1] => 2
[4,1,2,3,5] => 3
[4,1,2,5,3] => 4
[4,1,3,2,5] => 5
[4,1,3,5,2] => 4
[4,1,5,2,3] => 5
[4,1,5,3,2] => 4
[4,2,1,3,5] => 3
[4,2,1,5,3] => 4
[4,2,3,1,5] => 5
[4,2,3,5,1] => 4
[4,2,5,1,3] => 5
[4,2,5,3,1] => 4
[4,3,1,2,5] => 3
[4,3,1,5,2] => 4
[4,3,2,1,5] => 3
[4,3,2,5,1] => 4
[4,3,5,1,2] => 5
[4,3,5,2,1] => 4
[4,5,1,2,3] => 3
[4,5,1,3,2] => 4
[4,5,2,1,3] => 3
[4,5,2,3,1] => 4
[4,5,3,1,2] => 3
[4,5,3,2,1] => 2
[5,1,2,3,4] => 3
[5,1,2,4,3] => 4
[5,1,3,2,4] => 5
[5,1,3,4,2] => 4
[5,1,4,2,3] => 5
[5,1,4,3,2] => 4
[5,2,1,3,4] => 3
[5,2,1,4,3] => 4
[5,2,3,1,4] => 5
[5,2,3,4,1] => 4
[5,2,4,1,3] => 5
[5,2,4,3,1] => 4
[5,3,1,2,4] => 3
[5,3,1,4,2] => 4
[5,3,2,1,4] => 3
[5,3,2,4,1] => 4
[5,3,4,1,2] => 5
[5,3,4,2,1] => 4
[5,4,1,2,3] => 3
[5,4,1,3,2] => 4
[5,4,2,1,3] => 3
[5,4,2,3,1] => 4
[5,4,3,1,2] => 3
[5,4,3,2,1] => 2
[1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => 2
[1,2,3,5,4,6] => 3
[1,2,3,5,6,4] => 2
[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] => 4
[1,2,4,5,3,6] => 3
[1,2,4,5,6,3] => 2
[1,2,4,6,3,5] => 3
[1,2,4,6,5,3] => 2
[1,2,5,3,4,6] => 3
[1,2,5,3,6,4] => 4
[1,2,5,4,3,6] => 3
[1,2,5,4,6,3] => 4
[1,2,5,6,3,4] => 3
[1,2,5,6,4,3] => 2
[1,2,6,3,4,5] => 3
[1,2,6,3,5,4] => 4
[1,2,6,4,3,5] => 3
[1,2,6,4,5,3] => 4
[1,2,6,5,3,4] => 3
[1,2,6,5,4,3] => 2
[1,3,2,4,5,6] => 3
[1,3,2,4,6,5] => 4
[1,3,2,5,4,6] => 5
[1,3,2,5,6,4] => 4
[1,3,2,6,4,5] => 5
[1,3,2,6,5,4] => 4
[1,3,4,2,5,6] => 3
[1,3,4,2,6,5] => 4
[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] => 2
[1,3,5,2,4,6] => 3
[1,3,5,2,6,4] => 4
[1,3,5,4,2,6] => 3
[1,3,5,4,6,2] => 4
[1,3,5,6,2,4] => 3
[1,3,5,6,4,2] => 2
[1,3,6,2,4,5] => 3
[1,3,6,2,5,4] => 4
[1,3,6,4,2,5] => 3
[1,3,6,4,5,2] => 4
[1,3,6,5,2,4] => 3
[1,3,6,5,4,2] => 2
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 4
[1,4,2,5,3,6] => 5
[1,4,2,5,6,3] => 4
[1,4,2,6,3,5] => 5
[1,4,2,6,5,3] => 4
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 4
[1,4,3,5,2,6] => 5
[1,4,3,5,6,2] => 4
[1,4,3,6,2,5] => 5
[1,4,3,6,5,2] => 4
[1,4,5,2,3,6] => 3
[1,4,5,2,6,3] => 4
[1,4,5,3,2,6] => 3
[1,4,5,3,6,2] => 4
[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] => 4
[1,4,6,3,2,5] => 3
[1,4,6,3,5,2] => 4
[1,4,6,5,2,3] => 3
[1,4,6,5,3,2] => 2
[1,5,2,3,4,6] => 3
[1,5,2,3,6,4] => 4
[1,5,2,4,3,6] => 5
[1,5,2,4,6,3] => 4
[1,5,2,6,3,4] => 5
[1,5,2,6,4,3] => 4
[1,5,3,2,4,6] => 3
[1,5,3,2,6,4] => 4
[1,5,3,4,2,6] => 5
[1,5,3,4,6,2] => 4
[1,5,3,6,2,4] => 5
[1,5,3,6,4,2] => 4
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 4
[1,5,4,3,2,6] => 3
[1,5,4,3,6,2] => 4
[1,5,4,6,2,3] => 5
[1,5,4,6,3,2] => 4
[1,5,6,2,3,4] => 3
[1,5,6,2,4,3] => 4
[1,5,6,3,2,4] => 3
[1,5,6,3,4,2] => 4
[1,5,6,4,2,3] => 3
[1,5,6,4,3,2] => 2
[1,6,2,3,4,5] => 3
[1,6,2,3,5,4] => 4
[1,6,2,4,3,5] => 5
[1,6,2,4,5,3] => 4
[1,6,2,5,3,4] => 5
[1,6,2,5,4,3] => 4
[1,6,3,2,4,5] => 3
[1,6,3,2,5,4] => 4
[1,6,3,4,2,5] => 5
[1,6,3,4,5,2] => 4
[1,6,3,5,2,4] => 5
[1,6,3,5,4,2] => 4
[1,6,4,2,3,5] => 3
[1,6,4,2,5,3] => 4
[1,6,4,3,2,5] => 3
[1,6,4,3,5,2] => 4
[1,6,4,5,2,3] => 5
[1,6,4,5,3,2] => 4
[1,6,5,2,3,4] => 3
[1,6,5,2,4,3] => 4
[1,6,5,3,2,4] => 3
[1,6,5,3,4,2] => 4
[1,6,5,4,2,3] => 3
[1,6,5,4,3,2] => 2
[2,1,3,4,5,6] => 3
[2,1,3,4,6,5] => 4
[2,1,3,5,4,6] => 5
[2,1,3,5,6,4] => 4
[2,1,3,6,4,5] => 5
[2,1,3,6,5,4] => 4
[2,1,4,3,5,6] => 5
[2,1,4,3,6,5] => 6
[2,1,4,5,3,6] => 5
[2,1,4,5,6,3] => 4
[2,1,4,6,3,5] => 5
[2,1,4,6,5,3] => 4
[2,1,5,3,4,6] => 5
[2,1,5,3,6,4] => 6
[2,1,5,4,3,6] => 5
[2,1,5,4,6,3] => 6
[2,1,5,6,3,4] => 5
[2,1,5,6,4,3] => 4
[2,1,6,3,4,5] => 5
[2,1,6,3,5,4] => 6
[2,1,6,4,3,5] => 5
[2,1,6,4,5,3] => 6
[2,1,6,5,3,4] => 5
[2,1,6,5,4,3] => 4
[2,3,1,4,5,6] => 3
[2,3,1,4,6,5] => 4
[2,3,1,5,4,6] => 5
[2,3,1,5,6,4] => 4
[2,3,1,6,4,5] => 5
[2,3,1,6,5,4] => 4
[2,3,4,1,5,6] => 3
[2,3,4,1,6,5] => 4
[2,3,4,5,1,6] => 3
[2,3,4,5,6,1] => 2
[2,3,4,6,1,5] => 3
[2,3,4,6,5,1] => 2
[2,3,5,1,4,6] => 3
[2,3,5,1,6,4] => 4
[2,3,5,4,1,6] => 3
[2,3,5,4,6,1] => 4
[2,3,5,6,1,4] => 3
[2,3,5,6,4,1] => 2
[2,3,6,1,4,5] => 3
[2,3,6,1,5,4] => 4
[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] => 2
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 4
[2,4,1,5,3,6] => 5
[2,4,1,5,6,3] => 4
[2,4,1,6,3,5] => 5
[2,4,1,6,5,3] => 4
[2,4,3,1,5,6] => 3
[2,4,3,1,6,5] => 4
[2,4,3,5,1,6] => 5
[2,4,3,5,6,1] => 4
[2,4,3,6,1,5] => 5
[2,4,3,6,5,1] => 4
[2,4,5,1,3,6] => 3
[2,4,5,1,6,3] => 4
[2,4,5,3,1,6] => 3
[2,4,5,3,6,1] => 4
[2,4,5,6,1,3] => 3
[2,4,5,6,3,1] => 2
[2,4,6,1,3,5] => 3
[2,4,6,1,5,3] => 4
[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] => 2
[2,5,1,3,4,6] => 3
[2,5,1,3,6,4] => 4
[2,5,1,4,3,6] => 5
[2,5,1,4,6,3] => 4
[2,5,1,6,3,4] => 5
[2,5,1,6,4,3] => 4
[2,5,3,1,4,6] => 3
[2,5,3,1,6,4] => 4
[2,5,3,4,1,6] => 5
[2,5,3,4,6,1] => 4
[2,5,3,6,1,4] => 5
[2,5,3,6,4,1] => 4
[2,5,4,1,3,6] => 3
[2,5,4,1,6,3] => 4
[2,5,4,3,1,6] => 3
[2,5,4,3,6,1] => 4
[2,5,4,6,1,3] => 5
[2,5,4,6,3,1] => 4
[2,5,6,1,3,4] => 3
[2,5,6,1,4,3] => 4
[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] => 2
[2,6,1,3,4,5] => 3
[2,6,1,3,5,4] => 4
[2,6,1,4,3,5] => 5
[2,6,1,4,5,3] => 4
[2,6,1,5,3,4] => 5
[2,6,1,5,4,3] => 4
[2,6,3,1,4,5] => 3
[2,6,3,1,5,4] => 4
[2,6,3,4,1,5] => 5
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 5
[2,6,3,5,4,1] => 4
[2,6,4,1,3,5] => 3
[2,6,4,1,5,3] => 4
[2,6,4,3,1,5] => 3
[2,6,4,3,5,1] => 4
[2,6,4,5,1,3] => 5
[2,6,4,5,3,1] => 4
[2,6,5,1,3,4] => 3
[2,6,5,1,4,3] => 4
[2,6,5,3,1,4] => 3
[2,6,5,3,4,1] => 4
[2,6,5,4,1,3] => 3
[2,6,5,4,3,1] => 2
[3,1,2,4,5,6] => 3
[3,1,2,4,6,5] => 4
[3,1,2,5,4,6] => 5
[3,1,2,5,6,4] => 4
[3,1,2,6,4,5] => 5
[3,1,2,6,5,4] => 4
[3,1,4,2,5,6] => 5
[3,1,4,2,6,5] => 6
[3,1,4,5,2,6] => 5
[3,1,4,5,6,2] => 4
[3,1,4,6,2,5] => 5
[3,1,4,6,5,2] => 4
[3,1,5,2,4,6] => 5
[3,1,5,2,6,4] => 6
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 6
[3,1,5,6,2,4] => 5
[3,1,5,6,4,2] => 4
[3,1,6,2,4,5] => 5
[3,1,6,2,5,4] => 6
[3,1,6,4,2,5] => 5
[3,1,6,4,5,2] => 6
[3,1,6,5,2,4] => 5
[3,1,6,5,4,2] => 4
[3,2,1,4,5,6] => 3
[3,2,1,4,6,5] => 4
[3,2,1,5,4,6] => 5
[3,2,1,5,6,4] => 4
[3,2,1,6,4,5] => 5
[3,2,1,6,5,4] => 4
[3,2,4,1,5,6] => 5
[3,2,4,1,6,5] => 6
[3,2,4,5,1,6] => 5
[3,2,4,5,6,1] => 4
[3,2,4,6,1,5] => 5
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 6
[3,2,5,4,1,6] => 5
[3,2,5,4,6,1] => 6
[3,2,5,6,1,4] => 5
[3,2,5,6,4,1] => 4
[3,2,6,1,4,5] => 5
[3,2,6,1,5,4] => 6
[3,2,6,4,1,5] => 5
[3,2,6,4,5,1] => 6
[3,2,6,5,1,4] => 5
[3,2,6,5,4,1] => 4
[3,4,1,2,5,6] => 3
[3,4,1,2,6,5] => 4
[3,4,1,5,2,6] => 5
[3,4,1,5,6,2] => 4
[3,4,1,6,2,5] => 5
[3,4,1,6,5,2] => 4
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 4
[3,4,2,5,1,6] => 5
[3,4,2,5,6,1] => 4
[3,4,2,6,1,5] => 5
[3,4,2,6,5,1] => 4
[3,4,5,1,2,6] => 3
[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] => 3
[3,4,5,6,2,1] => 2
[3,4,6,1,2,5] => 3
[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] => 3
[3,4,6,5,2,1] => 2
[3,5,1,2,4,6] => 3
[3,5,1,2,6,4] => 4
[3,5,1,4,2,6] => 5
[3,5,1,4,6,2] => 4
[3,5,1,6,2,4] => 5
[3,5,1,6,4,2] => 4
[3,5,2,1,4,6] => 3
[3,5,2,1,6,4] => 4
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 4
[3,5,2,6,1,4] => 5
[3,5,2,6,4,1] => 4
[3,5,4,1,2,6] => 3
[3,5,4,1,6,2] => 4
[3,5,4,2,1,6] => 3
[3,5,4,2,6,1] => 4
[3,5,4,6,1,2] => 5
[3,5,4,6,2,1] => 4
[3,5,6,1,2,4] => 3
[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] => 3
[3,5,6,4,2,1] => 2
[3,6,1,2,4,5] => 3
[3,6,1,2,5,4] => 4
[3,6,1,4,2,5] => 5
[3,6,1,4,5,2] => 4
[3,6,1,5,2,4] => 5
[3,6,1,5,4,2] => 4
[3,6,2,1,4,5] => 3
[3,6,2,1,5,4] => 4
[3,6,2,4,1,5] => 5
[3,6,2,4,5,1] => 4
[3,6,2,5,1,4] => 5
[3,6,2,5,4,1] => 4
[3,6,4,1,2,5] => 3
[3,6,4,1,5,2] => 4
[3,6,4,2,1,5] => 3
[3,6,4,2,5,1] => 4
[3,6,4,5,1,2] => 5
[3,6,4,5,2,1] => 4
[3,6,5,1,2,4] => 3
[3,6,5,1,4,2] => 4
[3,6,5,2,1,4] => 3
[3,6,5,2,4,1] => 4
[3,6,5,4,1,2] => 3
[3,6,5,4,2,1] => 2
[4,1,2,3,5,6] => 3
[4,1,2,3,6,5] => 4
[4,1,2,5,3,6] => 5
[4,1,2,5,6,3] => 4
[4,1,2,6,3,5] => 5
[4,1,2,6,5,3] => 4
[4,1,3,2,5,6] => 5
[4,1,3,2,6,5] => 6
[4,1,3,5,2,6] => 5
[4,1,3,5,6,2] => 4
[4,1,3,6,2,5] => 5
[4,1,3,6,5,2] => 4
[4,1,5,2,3,6] => 5
[4,1,5,2,6,3] => 6
[4,1,5,3,2,6] => 5
[4,1,5,3,6,2] => 6
[4,1,5,6,2,3] => 5
[4,1,5,6,3,2] => 4
[4,1,6,2,3,5] => 5
[4,1,6,2,5,3] => 6
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 6
[4,1,6,5,2,3] => 5
[4,1,6,5,3,2] => 4
[4,2,1,3,5,6] => 3
[4,2,1,3,6,5] => 4
[4,2,1,5,3,6] => 5
[4,2,1,5,6,3] => 4
[4,2,1,6,3,5] => 5
[4,2,1,6,5,3] => 4
[4,2,3,1,5,6] => 5
[4,2,3,1,6,5] => 6
[4,2,3,5,1,6] => 5
[4,2,3,5,6,1] => 4
[4,2,3,6,1,5] => 5
[4,2,3,6,5,1] => 4
[4,2,5,1,3,6] => 5
[4,2,5,1,6,3] => 6
[4,2,5,3,1,6] => 5
[4,2,5,3,6,1] => 6
[4,2,5,6,1,3] => 5
[4,2,5,6,3,1] => 4
[4,2,6,1,3,5] => 5
[4,2,6,1,5,3] => 6
[4,2,6,3,1,5] => 5
[4,2,6,3,5,1] => 6
[4,2,6,5,1,3] => 5
[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] => 5
[4,3,1,5,6,2] => 4
[4,3,1,6,2,5] => 5
[4,3,1,6,5,2] => 4
[4,3,2,1,5,6] => 3
[4,3,2,1,6,5] => 4
[4,3,2,5,1,6] => 5
[4,3,2,5,6,1] => 4
[4,3,2,6,1,5] => 5
[4,3,2,6,5,1] => 4
[4,3,5,1,2,6] => 5
[4,3,5,1,6,2] => 6
[4,3,5,2,1,6] => 5
[4,3,5,2,6,1] => 6
[4,3,5,6,1,2] => 5
[4,3,5,6,2,1] => 4
[4,3,6,1,2,5] => 5
[4,3,6,1,5,2] => 6
[4,3,6,2,1,5] => 5
[4,3,6,2,5,1] => 6
[4,3,6,5,1,2] => 5
[4,3,6,5,2,1] => 4
[4,5,1,2,3,6] => 3
[4,5,1,2,6,3] => 4
[4,5,1,3,2,6] => 5
[4,5,1,3,6,2] => 4
[4,5,1,6,2,3] => 5
[4,5,1,6,3,2] => 4
[4,5,2,1,3,6] => 3
[4,5,2,1,6,3] => 4
[4,5,2,3,1,6] => 5
[4,5,2,3,6,1] => 4
[4,5,2,6,1,3] => 5
[4,5,2,6,3,1] => 4
[4,5,3,1,2,6] => 3
[4,5,3,1,6,2] => 4
[4,5,3,2,1,6] => 3
[4,5,3,2,6,1] => 4
[4,5,3,6,1,2] => 5
[4,5,3,6,2,1] => 4
[4,5,6,1,2,3] => 3
[4,5,6,1,3,2] => 4
[4,5,6,2,1,3] => 3
[4,5,6,2,3,1] => 4
[4,5,6,3,1,2] => 3
[4,5,6,3,2,1] => 2
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 4
[4,6,1,3,2,5] => 5
[4,6,1,3,5,2] => 4
[4,6,1,5,2,3] => 5
[4,6,1,5,3,2] => 4
[4,6,2,1,3,5] => 3
[4,6,2,1,5,3] => 4
[4,6,2,3,1,5] => 5
[4,6,2,3,5,1] => 4
[4,6,2,5,1,3] => 5
[4,6,2,5,3,1] => 4
[4,6,3,1,2,5] => 3
[4,6,3,1,5,2] => 4
[4,6,3,2,1,5] => 3
[4,6,3,2,5,1] => 4
[4,6,3,5,1,2] => 5
[4,6,3,5,2,1] => 4
[4,6,5,1,2,3] => 3
[4,6,5,1,3,2] => 4
[4,6,5,2,1,3] => 3
[4,6,5,2,3,1] => 4
[4,6,5,3,1,2] => 3
[4,6,5,3,2,1] => 2
[5,1,2,3,4,6] => 3
[5,1,2,3,6,4] => 4
[5,1,2,4,3,6] => 5
[5,1,2,4,6,3] => 4
[5,1,2,6,3,4] => 5
[5,1,2,6,4,3] => 4
[5,1,3,2,4,6] => 5
[5,1,3,2,6,4] => 6
[5,1,3,4,2,6] => 5
[5,1,3,4,6,2] => 4
[5,1,3,6,2,4] => 5
[5,1,3,6,4,2] => 4
[5,1,4,2,3,6] => 5
[5,1,4,2,6,3] => 6
[5,1,4,3,2,6] => 5
[5,1,4,3,6,2] => 6
[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] => 6
[5,1,6,3,2,4] => 5
[5,1,6,3,4,2] => 6
[5,1,6,4,2,3] => 5
[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] => 5
[5,2,1,4,6,3] => 4
[5,2,1,6,3,4] => 5
[5,2,1,6,4,3] => 4
[5,2,3,1,4,6] => 5
[5,2,3,1,6,4] => 6
[5,2,3,4,1,6] => 5
[5,2,3,4,6,1] => 4
[5,2,3,6,1,4] => 5
[5,2,3,6,4,1] => 4
[5,2,4,1,3,6] => 5
[5,2,4,1,6,3] => 6
[5,2,4,3,1,6] => 5
[5,2,4,3,6,1] => 6
[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] => 6
[5,2,6,3,1,4] => 5
[5,2,6,3,4,1] => 6
[5,2,6,4,1,3] => 5
[5,2,6,4,3,1] => 4
[5,3,1,2,4,6] => 3
[5,3,1,2,6,4] => 4
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 4
[5,3,1,6,2,4] => 5
[5,3,1,6,4,2] => 4
[5,3,2,1,4,6] => 3
[5,3,2,1,6,4] => 4
[5,3,2,4,1,6] => 5
[5,3,2,4,6,1] => 4
[5,3,2,6,1,4] => 5
[5,3,2,6,4,1] => 4
[5,3,4,1,2,6] => 5
[5,3,4,1,6,2] => 6
[5,3,4,2,1,6] => 5
[5,3,4,2,6,1] => 6
[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] => 6
[5,3,6,2,1,4] => 5
[5,3,6,2,4,1] => 6
[5,3,6,4,1,2] => 5
[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] => 5
[5,4,1,3,6,2] => 4
[5,4,1,6,2,3] => 5
[5,4,1,6,3,2] => 4
[5,4,2,1,3,6] => 3
[5,4,2,1,6,3] => 4
[5,4,2,3,1,6] => 5
[5,4,2,3,6,1] => 4
[5,4,2,6,1,3] => 5
[5,4,2,6,3,1] => 4
[5,4,3,1,2,6] => 3
[5,4,3,1,6,2] => 4
[5,4,3,2,1,6] => 3
[5,4,3,2,6,1] => 4
[5,4,3,6,1,2] => 5
[5,4,3,6,2,1] => 4
[5,4,6,1,2,3] => 5
[5,4,6,1,3,2] => 6
[5,4,6,2,1,3] => 5
[5,4,6,2,3,1] => 6
[5,4,6,3,1,2] => 5
[5,4,6,3,2,1] => 4
[5,6,1,2,3,4] => 3
[5,6,1,2,4,3] => 4
[5,6,1,3,2,4] => 5
[5,6,1,3,4,2] => 4
[5,6,1,4,2,3] => 5
[5,6,1,4,3,2] => 4
[5,6,2,1,3,4] => 3
[5,6,2,1,4,3] => 4
[5,6,2,3,1,4] => 5
[5,6,2,3,4,1] => 4
[5,6,2,4,1,3] => 5
[5,6,2,4,3,1] => 4
[5,6,3,1,2,4] => 3
[5,6,3,1,4,2] => 4
[5,6,3,2,1,4] => 3
[5,6,3,2,4,1] => 4
[5,6,3,4,1,2] => 5
[5,6,3,4,2,1] => 4
[5,6,4,1,2,3] => 3
[5,6,4,1,3,2] => 4
[5,6,4,2,1,3] => 3
[5,6,4,2,3,1] => 4
[5,6,4,3,1,2] => 3
[5,6,4,3,2,1] => 2
[6,1,2,3,4,5] => 3
[6,1,2,3,5,4] => 4
[6,1,2,4,3,5] => 5
[6,1,2,4,5,3] => 4
[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] => 6
[6,1,3,4,2,5] => 5
[6,1,3,4,5,2] => 4
[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] => 6
[6,1,4,3,2,5] => 5
[6,1,4,3,5,2] => 6
[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] => 6
[6,1,5,3,2,4] => 5
[6,1,5,3,4,2] => 6
[6,1,5,4,2,3] => 5
[6,1,5,4,3,2] => 4
[6,2,1,3,4,5] => 3
[6,2,1,3,5,4] => 4
[6,2,1,4,3,5] => 5
[6,2,1,4,5,3] => 4
[6,2,1,5,3,4] => 5
[6,2,1,5,4,3] => 4
[6,2,3,1,4,5] => 5
[6,2,3,1,5,4] => 6
[6,2,3,4,1,5] => 5
[6,2,3,4,5,1] => 4
[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] => 6
[6,2,4,3,1,5] => 5
[6,2,4,3,5,1] => 6
[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] => 6
[6,2,5,3,1,4] => 5
[6,2,5,3,4,1] => 6
[6,2,5,4,1,3] => 5
[6,2,5,4,3,1] => 4
[6,3,1,2,4,5] => 3
[6,3,1,2,5,4] => 4
[6,3,1,4,2,5] => 5
[6,3,1,4,5,2] => 4
[6,3,1,5,2,4] => 5
[6,3,1,5,4,2] => 4
[6,3,2,1,4,5] => 3
[6,3,2,1,5,4] => 4
[6,3,2,4,1,5] => 5
[6,3,2,4,5,1] => 4
[6,3,2,5,1,4] => 5
[6,3,2,5,4,1] => 4
[6,3,4,1,2,5] => 5
[6,3,4,1,5,2] => 6
[6,3,4,2,1,5] => 5
[6,3,4,2,5,1] => 6
[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] => 6
[6,3,5,2,1,4] => 5
[6,3,5,2,4,1] => 6
[6,3,5,4,1,2] => 5
[6,3,5,4,2,1] => 4
[6,4,1,2,3,5] => 3
[6,4,1,2,5,3] => 4
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 4
[6,4,1,5,2,3] => 5
[6,4,1,5,3,2] => 4
[6,4,2,1,3,5] => 3
[6,4,2,1,5,3] => 4
[6,4,2,3,1,5] => 5
[6,4,2,3,5,1] => 4
[6,4,2,5,1,3] => 5
[6,4,2,5,3,1] => 4
[6,4,3,1,2,5] => 3
[6,4,3,1,5,2] => 4
[6,4,3,2,1,5] => 3
[6,4,3,2,5,1] => 4
[6,4,3,5,1,2] => 5
[6,4,3,5,2,1] => 4
[6,4,5,1,2,3] => 5
[6,4,5,1,3,2] => 6
[6,4,5,2,1,3] => 5
[6,4,5,2,3,1] => 6
[6,4,5,3,1,2] => 5
[6,4,5,3,2,1] => 4
[6,5,1,2,3,4] => 3
[6,5,1,2,4,3] => 4
[6,5,1,3,2,4] => 5
[6,5,1,3,4,2] => 4
[6,5,1,4,2,3] => 5
[6,5,1,4,3,2] => 4
[6,5,2,1,3,4] => 3
[6,5,2,1,4,3] => 4
[6,5,2,3,1,4] => 5
[6,5,2,3,4,1] => 4
[6,5,2,4,1,3] => 5
[6,5,2,4,3,1] => 4
[6,5,3,1,2,4] => 3
[6,5,3,1,4,2] => 4
[6,5,3,2,1,4] => 3
[6,5,3,2,4,1] => 4
[6,5,3,4,1,2] => 5
[6,5,3,4,2,1] => 4
[6,5,4,1,2,3] => 3
[6,5,4,1,3,2] => 4
[6,5,4,2,1,3] => 3
[6,5,4,2,3,1] => 4
[6,5,4,3,1,2] => 3
[6,5,4,3,2,1] => 2
click to show generating function       
Description
The number of up-down runs of a permutation.
An up-down run of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$.
For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and
$643$.
References
[1] Zhuang, Y. Counting permutations by runs MathSciNet:3499494
[2] Zhuang, Y. Eulerian polynomials and descent statistics arXiv:1610.07218
Code
def statistic(x):
   if len(x)>1 and x[0]>x[1]:
        return x.number_of_peaks() + x.complement().number_of_peaks() + 2
   elif len(x)>0:
        return x.number_of_peaks() + x.complement().number_of_peaks() + 1
   else:
        return 0
Created
Oct 27, 2016 at 21:11 by Yan Zhuang
Updated
Oct 27, 2016 at 21:11 by Yan Zhuang