Identifier
Identifier
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 4
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 5
[4,1,2,3] => 6
[4,1,3,2] => 4
[4,2,1,3] => 5
[4,2,3,1] => 3
[4,3,1,2] => 5
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 4
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 4
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 4
[1,4,5,3,2] => 5
[1,5,2,3,4] => 6
[1,5,2,4,3] => 4
[1,5,3,2,4] => 5
[1,5,3,4,2] => 3
[1,5,4,2,3] => 5
[1,5,4,3,2] => 4
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 3
[2,1,5,3,4] => 4
[2,1,5,4,3] => 3
[2,3,1,4,5] => 2
[2,3,1,5,4] => 3
[2,3,4,1,5] => 3
[2,3,4,5,1] => 4
[2,3,5,1,4] => 5
[2,3,5,4,1] => 4
[2,4,1,3,5] => 4
[2,4,1,5,3] => 5
[2,4,3,1,5] => 3
[2,4,3,5,1] => 4
[2,4,5,1,3] => 5
[2,4,5,3,1] => 6
[2,5,1,3,4] => 7
[2,5,1,4,3] => 5
[2,5,3,1,4] => 6
[2,5,3,4,1] => 4
[2,5,4,1,3] => 6
[2,5,4,3,1] => 5
[3,1,2,4,5] => 3
[3,1,2,5,4] => 4
[3,1,4,2,5] => 4
[3,1,4,5,2] => 5
[3,1,5,2,4] => 6
[3,1,5,4,2] => 5
[3,2,1,4,5] => 2
[3,2,1,5,4] => 3
[3,2,4,1,5] => 3
[3,2,4,5,1] => 4
[3,2,5,1,4] => 5
[3,2,5,4,1] => 4
[3,4,1,2,5] => 4
[3,4,1,5,2] => 5
[3,4,2,1,5] => 5
[3,4,2,5,1] => 6
[3,4,5,1,2] => 7
[3,4,5,2,1] => 6
[3,5,1,2,4] => 7
[3,5,1,4,2] => 5
[3,5,2,1,4] => 8
[3,5,2,4,1] => 6
[3,5,4,1,2] => 6
[3,5,4,2,1] => 7
[4,1,2,3,5] => 6
[4,1,2,5,3] => 7
[4,1,3,2,5] => 4
[4,1,3,5,2] => 5
[4,1,5,2,3] => 6
[4,1,5,3,2] => 8
[4,2,1,3,5] => 5
[4,2,1,5,3] => 6
[4,2,3,1,5] => 3
[4,2,3,5,1] => 4
[4,2,5,1,3] => 5
[4,2,5,3,1] => 7
[4,3,1,2,5] => 5
[4,3,1,5,2] => 6
[4,3,2,1,5] => 4
[4,3,2,5,1] => 5
[4,3,5,1,2] => 6
[4,3,5,2,1] => 7
[4,5,1,2,3] => 8
[4,5,1,3,2] => 8
[4,5,2,1,3] => 7
[4,5,2,3,1] => 9
[4,5,3,1,2] => 6
[4,5,3,2,1] => 7
[5,1,2,3,4] => 10
[5,1,2,4,3] => 7
[5,1,3,2,4] => 8
[5,1,3,4,2] => 5
[5,1,4,2,3] => 8
[5,1,4,3,2] => 6
[5,2,1,3,4] => 9
[5,2,1,4,3] => 6
[5,2,3,1,4] => 7
[5,2,3,4,1] => 4
[5,2,4,1,3] => 7
[5,2,4,3,1] => 5
[5,3,1,2,4] => 9
[5,3,1,4,2] => 6
[5,3,2,1,4] => 8
[5,3,2,4,1] => 5
[5,3,4,1,2] => 7
[5,3,4,2,1] => 6
[5,4,1,2,3] => 8
[5,4,1,3,2] => 8
[5,4,2,1,3] => 9
[5,4,2,3,1] => 7
[5,4,3,1,2] => 7
[5,4,3,2,1] => 6
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 1
[1,2,3,5,4,6] => 1
[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] => 1
[1,2,4,3,6,5] => 2
[1,2,4,5,3,6] => 2
[1,2,4,5,6,3] => 3
[1,2,4,6,3,5] => 4
[1,2,4,6,5,3] => 3
[1,2,5,3,4,6] => 3
[1,2,5,3,6,4] => 4
[1,2,5,4,3,6] => 2
[1,2,5,4,6,3] => 3
[1,2,5,6,3,4] => 4
[1,2,5,6,4,3] => 5
[1,2,6,3,4,5] => 6
[1,2,6,3,5,4] => 4
[1,2,6,4,3,5] => 5
[1,2,6,4,5,3] => 3
[1,2,6,5,3,4] => 5
[1,2,6,5,4,3] => 4
[1,3,2,4,5,6] => 1
[1,3,2,4,6,5] => 2
[1,3,2,5,4,6] => 2
[1,3,2,5,6,4] => 3
[1,3,2,6,4,5] => 4
[1,3,2,6,5,4] => 3
[1,3,4,2,5,6] => 2
[1,3,4,2,6,5] => 3
[1,3,4,5,2,6] => 3
[1,3,4,5,6,2] => 4
[1,3,4,6,2,5] => 5
[1,3,4,6,5,2] => 4
[1,3,5,2,4,6] => 4
[1,3,5,2,6,4] => 5
[1,3,5,4,2,6] => 3
[1,3,5,4,6,2] => 4
[1,3,5,6,2,4] => 5
[1,3,5,6,4,2] => 6
[1,3,6,2,4,5] => 7
[1,3,6,2,5,4] => 5
[1,3,6,4,2,5] => 6
[1,3,6,4,5,2] => 4
[1,3,6,5,2,4] => 6
[1,3,6,5,4,2] => 5
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 4
[1,4,2,5,3,6] => 4
[1,4,2,5,6,3] => 5
[1,4,2,6,3,5] => 6
[1,4,2,6,5,3] => 5
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 3
[1,4,3,5,2,6] => 3
[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] => 4
[1,4,5,2,6,3] => 5
[1,4,5,3,2,6] => 5
[1,4,5,3,6,2] => 6
[1,4,5,6,2,3] => 7
[1,4,5,6,3,2] => 6
[1,4,6,2,3,5] => 7
[1,4,6,2,5,3] => 5
[1,4,6,3,2,5] => 8
[1,4,6,3,5,2] => 6
[1,4,6,5,2,3] => 6
[1,4,6,5,3,2] => 7
[1,5,2,3,4,6] => 6
[1,5,2,3,6,4] => 7
[1,5,2,4,3,6] => 4
[1,5,2,4,6,3] => 5
[1,5,2,6,3,4] => 6
[1,5,2,6,4,3] => 8
[1,5,3,2,4,6] => 5
[1,5,3,2,6,4] => 6
[1,5,3,4,2,6] => 3
[1,5,3,4,6,2] => 4
[1,5,3,6,2,4] => 5
[1,5,3,6,4,2] => 7
[1,5,4,2,3,6] => 5
[1,5,4,2,6,3] => 6
[1,5,4,3,2,6] => 4
[1,5,4,3,6,2] => 5
[1,5,4,6,2,3] => 6
[1,5,4,6,3,2] => 7
[1,5,6,2,3,4] => 8
[1,5,6,2,4,3] => 8
[1,5,6,3,2,4] => 7
[1,5,6,3,4,2] => 9
[1,5,6,4,2,3] => 6
[1,5,6,4,3,2] => 7
[1,6,2,3,4,5] => 10
[1,6,2,3,5,4] => 7
[1,6,2,4,3,5] => 8
[1,6,2,4,5,3] => 5
[1,6,2,5,3,4] => 8
[1,6,2,5,4,3] => 6
[1,6,3,2,4,5] => 9
[1,6,3,2,5,4] => 6
[1,6,3,4,2,5] => 7
[1,6,3,4,5,2] => 4
[1,6,3,5,2,4] => 7
[1,6,3,5,4,2] => 5
[1,6,4,2,3,5] => 9
[1,6,4,2,5,3] => 6
[1,6,4,3,2,5] => 8
[1,6,4,3,5,2] => 5
[1,6,4,5,2,3] => 7
[1,6,4,5,3,2] => 6
[1,6,5,2,3,4] => 8
[1,6,5,2,4,3] => 8
[1,6,5,3,2,4] => 9
[1,6,5,3,4,2] => 7
[1,6,5,4,2,3] => 7
[1,6,5,4,3,2] => 6
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 2
[2,1,3,5,4,6] => 2
[2,1,3,5,6,4] => 3
[2,1,3,6,4,5] => 4
[2,1,3,6,5,4] => 3
[2,1,4,3,5,6] => 2
[2,1,4,3,6,5] => 3
[2,1,4,5,3,6] => 3
[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] => 4
[2,1,5,3,6,4] => 5
[2,1,5,4,3,6] => 3
[2,1,5,4,6,3] => 4
[2,1,5,6,3,4] => 5
[2,1,5,6,4,3] => 6
[2,1,6,3,4,5] => 7
[2,1,6,3,5,4] => 5
[2,1,6,4,3,5] => 6
[2,1,6,4,5,3] => 4
[2,1,6,5,3,4] => 6
[2,1,6,5,4,3] => 5
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 3
[2,3,1,5,4,6] => 3
[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] => 4
[2,3,4,5,6,1] => 5
[2,3,4,6,1,5] => 6
[2,3,4,6,5,1] => 5
[2,3,5,1,4,6] => 5
[2,3,5,1,6,4] => 6
[2,3,5,4,1,6] => 4
[2,3,5,4,6,1] => 5
[2,3,5,6,1,4] => 6
[2,3,5,6,4,1] => 7
[2,3,6,1,4,5] => 8
[2,3,6,1,5,4] => 6
[2,3,6,4,1,5] => 7
[2,3,6,4,5,1] => 5
[2,3,6,5,1,4] => 7
[2,3,6,5,4,1] => 6
[2,4,1,3,5,6] => 4
[2,4,1,3,6,5] => 5
[2,4,1,5,3,6] => 5
[2,4,1,5,6,3] => 6
[2,4,1,6,3,5] => 7
[2,4,1,6,5,3] => 6
[2,4,3,1,5,6] => 3
[2,4,3,1,6,5] => 4
[2,4,3,5,1,6] => 4
[2,4,3,5,6,1] => 5
[2,4,3,6,1,5] => 6
[2,4,3,6,5,1] => 5
[2,4,5,1,3,6] => 5
[2,4,5,1,6,3] => 6
[2,4,5,3,1,6] => 6
[2,4,5,3,6,1] => 7
[2,4,5,6,1,3] => 8
[2,4,5,6,3,1] => 7
[2,4,6,1,3,5] => 8
[2,4,6,1,5,3] => 6
[2,4,6,3,1,5] => 9
[2,4,6,3,5,1] => 7
[2,4,6,5,1,3] => 7
[2,4,6,5,3,1] => 8
[2,5,1,3,4,6] => 7
[2,5,1,3,6,4] => 8
[2,5,1,4,3,6] => 5
[2,5,1,4,6,3] => 6
[2,5,1,6,3,4] => 7
[2,5,1,6,4,3] => 9
[2,5,3,1,4,6] => 6
[2,5,3,1,6,4] => 7
[2,5,3,4,1,6] => 4
[2,5,3,4,6,1] => 5
[2,5,3,6,1,4] => 6
[2,5,3,6,4,1] => 8
[2,5,4,1,3,6] => 6
[2,5,4,1,6,3] => 7
[2,5,4,3,1,6] => 5
[2,5,4,3,6,1] => 6
[2,5,4,6,1,3] => 7
[2,5,4,6,3,1] => 8
[2,5,6,1,3,4] => 9
[2,5,6,1,4,3] => 9
[2,5,6,3,1,4] => 8
[2,5,6,3,4,1] => 10
[2,5,6,4,1,3] => 7
[2,5,6,4,3,1] => 8
[2,6,1,3,4,5] => 11
[2,6,1,3,5,4] => 8
[2,6,1,4,3,5] => 9
[2,6,1,4,5,3] => 6
[2,6,1,5,3,4] => 9
[2,6,1,5,4,3] => 7
[2,6,3,1,4,5] => 10
[2,6,3,1,5,4] => 7
[2,6,3,4,1,5] => 8
[2,6,3,4,5,1] => 5
[2,6,3,5,1,4] => 8
[2,6,3,5,4,1] => 6
[2,6,4,1,3,5] => 10
[2,6,4,1,5,3] => 7
[2,6,4,3,1,5] => 9
[2,6,4,3,5,1] => 6
[2,6,4,5,1,3] => 8
[2,6,4,5,3,1] => 7
[2,6,5,1,3,4] => 9
[2,6,5,1,4,3] => 9
[2,6,5,3,1,4] => 10
[2,6,5,3,4,1] => 8
[2,6,5,4,1,3] => 8
[2,6,5,4,3,1] => 7
[3,1,2,4,5,6] => 3
[3,1,2,4,6,5] => 4
[3,1,2,5,4,6] => 4
[3,1,2,5,6,4] => 5
[3,1,2,6,4,5] => 6
[3,1,2,6,5,4] => 5
[3,1,4,2,5,6] => 4
[3,1,4,2,6,5] => 5
[3,1,4,5,2,6] => 5
[3,1,4,5,6,2] => 6
[3,1,4,6,2,5] => 7
[3,1,4,6,5,2] => 6
[3,1,5,2,4,6] => 6
[3,1,5,2,6,4] => 7
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 6
[3,1,5,6,2,4] => 7
[3,1,5,6,4,2] => 8
[3,1,6,2,4,5] => 9
[3,1,6,2,5,4] => 7
[3,1,6,4,2,5] => 8
[3,1,6,4,5,2] => 6
[3,1,6,5,2,4] => 8
[3,1,6,5,4,2] => 7
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 3
[3,2,1,5,4,6] => 3
[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] => 3
[3,2,4,1,6,5] => 4
[3,2,4,5,1,6] => 4
[3,2,4,5,6,1] => 5
[3,2,4,6,1,5] => 6
[3,2,4,6,5,1] => 5
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 6
[3,2,5,4,1,6] => 4
[3,2,5,4,6,1] => 5
[3,2,5,6,1,4] => 6
[3,2,5,6,4,1] => 7
[3,2,6,1,4,5] => 8
[3,2,6,1,5,4] => 6
[3,2,6,4,1,5] => 7
[3,2,6,4,5,1] => 5
[3,2,6,5,1,4] => 7
[3,2,6,5,4,1] => 6
[3,4,1,2,5,6] => 4
[3,4,1,2,6,5] => 5
[3,4,1,5,2,6] => 5
[3,4,1,5,6,2] => 6
[3,4,1,6,2,5] => 7
[3,4,1,6,5,2] => 6
[3,4,2,1,5,6] => 5
[3,4,2,1,6,5] => 6
[3,4,2,5,1,6] => 6
[3,4,2,5,6,1] => 7
[3,4,2,6,1,5] => 8
[3,4,2,6,5,1] => 7
[3,4,5,1,2,6] => 7
[3,4,5,1,6,2] => 8
[3,4,5,2,1,6] => 6
[3,4,5,2,6,1] => 7
[3,4,5,6,1,2] => 8
[3,4,5,6,2,1] => 9
[3,4,6,1,2,5] => 10
[3,4,6,1,5,2] => 8
[3,4,6,2,1,5] => 9
[3,4,6,2,5,1] => 7
[3,4,6,5,1,2] => 9
[3,4,6,5,2,1] => 8
[3,5,1,2,4,6] => 7
[3,5,1,2,6,4] => 8
[3,5,1,4,2,6] => 5
[3,5,1,4,6,2] => 6
[3,5,1,6,2,4] => 7
[3,5,1,6,4,2] => 9
[3,5,2,1,4,6] => 8
[3,5,2,1,6,4] => 9
[3,5,2,4,1,6] => 6
[3,5,2,4,6,1] => 7
[3,5,2,6,1,4] => 8
[3,5,2,6,4,1] => 10
[3,5,4,1,2,6] => 6
[3,5,4,1,6,2] => 7
[3,5,4,2,1,6] => 7
[3,5,4,2,6,1] => 8
[3,5,4,6,1,2] => 9
[3,5,4,6,2,1] => 8
[3,5,6,1,2,4] => 9
[3,5,6,1,4,2] => 11
[3,5,6,2,1,4] => 10
[3,5,6,2,4,1] => 10
[3,5,6,4,1,2] => 9
[3,5,6,4,2,1] => 8
[3,6,1,2,4,5] => 11
[3,6,1,2,5,4] => 8
[3,6,1,4,2,5] => 9
[3,6,1,4,5,2] => 6
[3,6,1,5,2,4] => 9
[3,6,1,5,4,2] => 7
[3,6,2,1,4,5] => 12
[3,6,2,1,5,4] => 9
[3,6,2,4,1,5] => 10
[3,6,2,4,5,1] => 7
[3,6,2,5,1,4] => 10
[3,6,2,5,4,1] => 8
[3,6,4,1,2,5] => 10
[3,6,4,1,5,2] => 7
[3,6,4,2,1,5] => 11
[3,6,4,2,5,1] => 8
[3,6,4,5,1,2] => 8
[3,6,4,5,2,1] => 9
[3,6,5,1,2,4] => 11
[3,6,5,1,4,2] => 9
[3,6,5,2,1,4] => 10
[3,6,5,2,4,1] => 10
[3,6,5,4,1,2] => 8
[3,6,5,4,2,1] => 9
[4,1,2,3,5,6] => 6
[4,1,2,3,6,5] => 7
[4,1,2,5,3,6] => 7
[4,1,2,5,6,3] => 8
[4,1,2,6,3,5] => 9
[4,1,2,6,5,3] => 8
[4,1,3,2,5,6] => 4
[4,1,3,2,6,5] => 5
[4,1,3,5,2,6] => 5
[4,1,3,5,6,2] => 6
[4,1,3,6,2,5] => 7
[4,1,3,6,5,2] => 6
[4,1,5,2,3,6] => 6
[4,1,5,2,6,3] => 7
[4,1,5,3,2,6] => 8
[4,1,5,3,6,2] => 9
[4,1,5,6,2,3] => 10
[4,1,5,6,3,2] => 8
[4,1,6,2,3,5] => 9
[4,1,6,2,5,3] => 7
[4,1,6,3,2,5] => 11
[4,1,6,3,5,2] => 9
[4,1,6,5,2,3] => 8
[4,1,6,5,3,2] => 10
[4,2,1,3,5,6] => 5
[4,2,1,3,6,5] => 6
[4,2,1,5,3,6] => 6
[4,2,1,5,6,3] => 7
[4,2,1,6,3,5] => 8
[4,2,1,6,5,3] => 7
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 4
[4,2,3,5,1,6] => 4
[4,2,3,5,6,1] => 5
[4,2,3,6,1,5] => 6
[4,2,3,6,5,1] => 5
[4,2,5,1,3,6] => 5
[4,2,5,1,6,3] => 6
[4,2,5,3,1,6] => 7
[4,2,5,3,6,1] => 8
[4,2,5,6,1,3] => 9
[4,2,5,6,3,1] => 7
[4,2,6,1,3,5] => 8
[4,2,6,1,5,3] => 6
[4,2,6,3,1,5] => 10
[4,2,6,3,5,1] => 8
[4,2,6,5,1,3] => 7
[4,2,6,5,3,1] => 9
[4,3,1,2,5,6] => 5
[4,3,1,2,6,5] => 6
[4,3,1,5,2,6] => 6
[4,3,1,5,6,2] => 7
[4,3,1,6,2,5] => 8
[4,3,1,6,5,2] => 7
[4,3,2,1,5,6] => 4
[4,3,2,1,6,5] => 5
[4,3,2,5,1,6] => 5
[4,3,2,5,6,1] => 6
[4,3,2,6,1,5] => 7
[4,3,2,6,5,1] => 6
[4,3,5,1,2,6] => 6
[4,3,5,1,6,2] => 7
[4,3,5,2,1,6] => 7
[4,3,5,2,6,1] => 8
[4,3,5,6,1,2] => 9
[4,3,5,6,2,1] => 8
[4,3,6,1,2,5] => 9
[4,3,6,1,5,2] => 7
[4,3,6,2,1,5] => 10
[4,3,6,2,5,1] => 8
[4,3,6,5,1,2] => 8
[4,3,6,5,2,1] => 9
[4,5,1,2,3,6] => 8
[4,5,1,2,6,3] => 9
[4,5,1,3,2,6] => 8
[4,5,1,3,6,2] => 9
[4,5,1,6,2,3] => 10
[4,5,1,6,3,2] => 10
[4,5,2,1,3,6] => 7
[4,5,2,1,6,3] => 8
[4,5,2,3,1,6] => 9
[4,5,2,3,6,1] => 10
[4,5,2,6,1,3] => 11
[4,5,2,6,3,1] => 9
[4,5,3,1,2,6] => 6
[4,5,3,1,6,2] => 7
[4,5,3,2,1,6] => 7
[4,5,3,2,6,1] => 8
[4,5,3,6,1,2] => 9
[4,5,3,6,2,1] => 8
[4,5,6,1,2,3] => 9
[4,5,6,1,3,2] => 10
[4,5,6,2,1,3] => 10
[4,5,6,2,3,1] => 11
[4,5,6,3,1,2] => 12
[4,5,6,3,2,1] => 11
[4,6,1,2,3,5] => 12
[4,6,1,2,5,3] => 9
[4,6,1,3,2,5] => 12
[4,6,1,3,5,2] => 9
[4,6,1,5,2,3] => 10
[4,6,1,5,3,2] => 10
[4,6,2,1,3,5] => 11
[4,6,2,1,5,3] => 8
[4,6,2,3,1,5] => 13
[4,6,2,3,5,1] => 10
[4,6,2,5,1,3] => 9
[4,6,2,5,3,1] => 11
[4,6,3,1,2,5] => 10
[4,6,3,1,5,2] => 7
[4,6,3,2,1,5] => 11
[4,6,3,2,5,1] => 8
[4,6,3,5,1,2] => 8
[4,6,3,5,2,1] => 9
[4,6,5,1,2,3] => 10
[4,6,5,1,3,2] => 9
[4,6,5,2,1,3] => 11
[4,6,5,2,3,1] => 10
[4,6,5,3,1,2] => 11
[4,6,5,3,2,1] => 12
[5,1,2,3,4,6] => 10
[5,1,2,3,6,4] => 11
[5,1,2,4,3,6] => 7
[5,1,2,4,6,3] => 8
[5,1,2,6,3,4] => 9
[5,1,2,6,4,3] => 12
[5,1,3,2,4,6] => 8
[5,1,3,2,6,4] => 9
[5,1,3,4,2,6] => 5
[5,1,3,4,6,2] => 6
[5,1,3,6,2,4] => 7
[5,1,3,6,4,2] => 10
[5,1,4,2,3,6] => 8
[5,1,4,2,6,3] => 9
[5,1,4,3,2,6] => 6
[5,1,4,3,6,2] => 7
[5,1,4,6,2,3] => 8
[5,1,4,6,3,2] => 10
[5,1,6,2,3,4] => 11
[5,1,6,2,4,3] => 11
[5,1,6,3,2,4] => 9
[5,1,6,3,4,2] => 13
[5,1,6,4,2,3] => 8
[5,1,6,4,3,2] => 10
[5,2,1,3,4,6] => 9
[5,2,1,3,6,4] => 10
[5,2,1,4,3,6] => 6
[5,2,1,4,6,3] => 7
[5,2,1,6,3,4] => 8
[5,2,1,6,4,3] => 11
[5,2,3,1,4,6] => 7
[5,2,3,1,6,4] => 8
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 5
[5,2,3,6,1,4] => 6
[5,2,3,6,4,1] => 9
[5,2,4,1,3,6] => 7
[5,2,4,1,6,3] => 8
[5,2,4,3,1,6] => 5
[5,2,4,3,6,1] => 6
[5,2,4,6,1,3] => 7
[5,2,4,6,3,1] => 9
[5,2,6,1,3,4] => 10
[5,2,6,1,4,3] => 10
[5,2,6,3,1,4] => 8
[5,2,6,3,4,1] => 12
[5,2,6,4,1,3] => 7
[5,2,6,4,3,1] => 9
[5,3,1,2,4,6] => 9
[5,3,1,2,6,4] => 10
[5,3,1,4,2,6] => 6
[5,3,1,4,6,2] => 7
[5,3,1,6,2,4] => 8
[5,3,1,6,4,2] => 11
[5,3,2,1,4,6] => 8
[5,3,2,1,6,4] => 9
[5,3,2,4,1,6] => 5
[5,3,2,4,6,1] => 6
[5,3,2,6,1,4] => 7
[5,3,2,6,4,1] => 10
[5,3,4,1,2,6] => 7
[5,3,4,1,6,2] => 8
[5,3,4,2,1,6] => 6
[5,3,4,2,6,1] => 7
[5,3,4,6,1,2] => 8
[5,3,4,6,2,1] => 9
[5,3,6,1,2,4] => 10
[5,3,6,1,4,2] => 11
[5,3,6,2,1,4] => 9
[5,3,6,2,4,1] => 12
[5,3,6,4,1,2] => 8
[5,3,6,4,2,1] => 9
[5,4,1,2,3,6] => 8
[5,4,1,2,6,3] => 9
[5,4,1,3,2,6] => 8
[5,4,1,3,6,2] => 9
[5,4,1,6,2,3] => 10
[5,4,1,6,3,2] => 10
[5,4,2,1,3,6] => 9
[5,4,2,1,6,3] => 10
[5,4,2,3,1,6] => 7
[5,4,2,3,6,1] => 8
[5,4,2,6,1,3] => 9
[5,4,2,6,3,1] => 11
[5,4,3,1,2,6] => 7
[5,4,3,1,6,2] => 8
[5,4,3,2,1,6] => 6
[5,4,3,2,6,1] => 7
[5,4,3,6,1,2] => 8
[5,4,3,6,2,1] => 9
[5,4,6,1,2,3] => 10
[5,4,6,1,3,2] => 12
[5,4,6,2,1,3] => 9
[5,4,6,2,3,1] => 11
[5,4,6,3,1,2] => 10
[5,4,6,3,2,1] => 11
[5,6,1,2,3,4] => 12
[5,6,1,2,4,3] => 13
[5,6,1,3,2,4] => 12
[5,6,1,3,4,2] => 13
[5,6,1,4,2,3] => 10
[5,6,1,4,3,2] => 10
[5,6,2,1,3,4] => 13
[5,6,2,1,4,3] => 12
[5,6,2,3,1,4] => 11
[5,6,2,3,4,1] => 14
[5,6,2,4,1,3] => 9
[5,6,2,4,3,1] => 11
[5,6,3,1,2,4] => 11
[5,6,3,1,4,2] => 11
[5,6,3,2,1,4] => 10
[5,6,3,2,4,1] => 12
[5,6,3,4,1,2] => 8
[5,6,3,4,2,1] => 9
[5,6,4,1,2,3] => 11
[5,6,4,1,3,2] => 11
[5,6,4,2,1,3] => 10
[5,6,4,2,3,1] => 12
[5,6,4,3,1,2] => 9
[5,6,4,3,2,1] => 10
[6,1,2,3,4,5] => 15
[6,1,2,3,5,4] => 11
[6,1,2,4,3,5] => 12
[6,1,2,4,5,3] => 8
[6,1,2,5,3,4] => 12
[6,1,2,5,4,3] => 9
[6,1,3,2,4,5] => 13
[6,1,3,2,5,4] => 9
[6,1,3,4,2,5] => 10
[6,1,3,4,5,2] => 6
[6,1,3,5,2,4] => 10
[6,1,3,5,4,2] => 7
[6,1,4,2,3,5] => 13
[6,1,4,2,5,3] => 9
[6,1,4,3,2,5] => 11
[6,1,4,3,5,2] => 7
[6,1,4,5,2,3] => 10
[6,1,4,5,3,2] => 8
[6,1,5,2,3,4] => 11
[6,1,5,2,4,3] => 11
[6,1,5,3,2,4] => 13
[6,1,5,3,4,2] => 9
[6,1,5,4,2,3] => 10
[6,1,5,4,3,2] => 8
[6,2,1,3,4,5] => 14
[6,2,1,3,5,4] => 10
[6,2,1,4,3,5] => 11
[6,2,1,4,5,3] => 7
[6,2,1,5,3,4] => 11
[6,2,1,5,4,3] => 8
[6,2,3,1,4,5] => 12
[6,2,3,1,5,4] => 8
[6,2,3,4,1,5] => 9
[6,2,3,4,5,1] => 5
[6,2,3,5,1,4] => 9
[6,2,3,5,4,1] => 6
[6,2,4,1,3,5] => 12
[6,2,4,1,5,3] => 8
[6,2,4,3,1,5] => 10
[6,2,4,3,5,1] => 6
[6,2,4,5,1,3] => 9
[6,2,4,5,3,1] => 7
[6,2,5,1,3,4] => 10
[6,2,5,1,4,3] => 10
[6,2,5,3,1,4] => 12
[6,2,5,3,4,1] => 8
[6,2,5,4,1,3] => 9
[6,2,5,4,3,1] => 7
[6,3,1,2,4,5] => 14
[6,3,1,2,5,4] => 10
[6,3,1,4,2,5] => 11
[6,3,1,4,5,2] => 7
[6,3,1,5,2,4] => 11
[6,3,1,5,4,2] => 8
[6,3,2,1,4,5] => 13
[6,3,2,1,5,4] => 9
[6,3,2,4,1,5] => 10
[6,3,2,4,5,1] => 6
[6,3,2,5,1,4] => 10
[6,3,2,5,4,1] => 7
[6,3,4,1,2,5] => 12
[6,3,4,1,5,2] => 8
[6,3,4,2,1,5] => 11
[6,3,4,2,5,1] => 7
[6,3,4,5,1,2] => 9
[6,3,4,5,2,1] => 8
[6,3,5,1,2,4] => 11
[6,3,5,1,4,2] => 10
[6,3,5,2,1,4] => 12
[6,3,5,2,4,1] => 9
[6,3,5,4,1,2] => 9
[6,3,5,4,2,1] => 8
[6,4,1,2,3,5] => 13
[6,4,1,2,5,3] => 9
[6,4,1,3,2,5] => 13
[6,4,1,3,5,2] => 9
[6,4,1,5,2,3] => 10
[6,4,1,5,3,2] => 10
[6,4,2,1,3,5] => 14
[6,4,2,1,5,3] => 10
[6,4,2,3,1,5] => 12
[6,4,2,3,5,1] => 8
[6,4,2,5,1,3] => 11
[6,4,2,5,3,1] => 9
[6,4,3,1,2,5] => 12
[6,4,3,1,5,2] => 8
[6,4,3,2,1,5] => 11
[6,4,3,2,5,1] => 7
[6,4,3,5,1,2] => 9
[6,4,3,5,2,1] => 8
[6,4,5,1,2,3] => 12
[6,4,5,1,3,2] => 10
[6,4,5,2,1,3] => 11
[6,4,5,2,3,1] => 9
[6,4,5,3,1,2] => 11
[6,4,5,3,2,1] => 10
[6,5,1,2,3,4] => 13
[6,5,1,2,4,3] => 12
[6,5,1,3,2,4] => 13
[6,5,1,3,4,2] => 12
[6,5,1,4,2,3] => 10
[6,5,1,4,3,2] => 10
[6,5,2,1,3,4] => 12
[6,5,2,1,4,3] => 13
[6,5,2,3,1,4] => 14
[6,5,2,3,4,1] => 11
[6,5,2,4,1,3] => 11
[6,5,2,4,3,1] => 9
[6,5,3,1,2,4] => 11
[6,5,3,1,4,2] => 11
[6,5,3,2,1,4] => 12
[6,5,3,2,4,1] => 10
[6,5,3,4,1,2] => 9
[6,5,3,4,2,1] => 8
[6,5,4,1,2,3] => 11
[6,5,4,1,3,2] => 11
[6,5,4,2,1,3] => 12
[6,5,4,2,3,1] => 10
[6,5,4,3,1,2] => 10
[6,5,4,3,2,1] => 9
click to show generating function       
Description
The sorting index of a permutation.
The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols.
Compare this to St000018The number of inversions of a permutation., the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.
References
[1] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). OEIS:A008302
[2] Petersen, T. K. The sorting index MathSciNet:2822205
Code
def statistic(pi):
    n = len(pi)
    if n == 0:
        return 0
    else:
        x = pi.index(n) + 1
        if x == n:
            return statistic(pi[:n-1])
        else:
            return (n-x) + statistic(Permutation([(n, x)])*pi)
Created
Sep 30, 2014 at 09:17 by Martin Rubey
Updated
Aug 30, 2015 at 04:04 by Daniel Cullina