Identifier
Identifier
Values
[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] => 2
[3,2,1] => 3
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 4
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 4
[3,4,1,2] => 4
[3,4,2,1] => 5
[4,1,2,3] => 3
[4,1,3,2] => 4
[4,2,1,3] => 4
[4,2,3,1] => 5
[4,3,1,2] => 5
[4,3,2,1] => 6
[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] => 2
[1,2,5,4,3] => 3
[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] => 3
[1,3,5,4,2] => 4
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 4
[1,4,5,3,2] => 5
[1,5,2,3,4] => 3
[1,5,2,4,3] => 4
[1,5,3,2,4] => 4
[1,5,3,4,2] => 5
[1,5,4,2,3] => 5
[1,5,4,3,2] => 6
[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] => 3
[2,1,5,4,3] => 4
[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] => 4
[2,3,5,4,1] => 5
[2,4,1,3,5] => 3
[2,4,1,5,3] => 4
[2,4,3,1,5] => 4
[2,4,3,5,1] => 5
[2,4,5,1,3] => 5
[2,4,5,3,1] => 6
[2,5,1,3,4] => 4
[2,5,1,4,3] => 5
[2,5,3,1,4] => 5
[2,5,3,4,1] => 6
[2,5,4,1,3] => 6
[2,5,4,3,1] => 7
[3,1,2,4,5] => 2
[3,1,2,5,4] => 3
[3,1,4,2,5] => 3
[3,1,4,5,2] => 4
[3,1,5,2,4] => 4
[3,1,5,4,2] => 5
[3,2,1,4,5] => 3
[3,2,1,5,4] => 4
[3,2,4,1,5] => 4
[3,2,4,5,1] => 5
[3,2,5,1,4] => 5
[3,2,5,4,1] => 6
[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] => 6
[3,4,5,2,1] => 7
[3,5,1,2,4] => 5
[3,5,1,4,2] => 6
[3,5,2,1,4] => 6
[3,5,2,4,1] => 7
[3,5,4,1,2] => 7
[3,5,4,2,1] => 8
[4,1,2,3,5] => 3
[4,1,2,5,3] => 4
[4,1,3,2,5] => 4
[4,1,3,5,2] => 5
[4,1,5,2,3] => 5
[4,1,5,3,2] => 6
[4,2,1,3,5] => 4
[4,2,1,5,3] => 5
[4,2,3,1,5] => 5
[4,2,3,5,1] => 6
[4,2,5,1,3] => 6
[4,2,5,3,1] => 7
[4,3,1,2,5] => 5
[4,3,1,5,2] => 6
[4,3,2,1,5] => 6
[4,3,2,5,1] => 7
[4,3,5,1,2] => 7
[4,3,5,2,1] => 8
[4,5,1,2,3] => 6
[4,5,1,3,2] => 7
[4,5,2,1,3] => 7
[4,5,2,3,1] => 8
[4,5,3,1,2] => 8
[4,5,3,2,1] => 9
[5,1,2,3,4] => 4
[5,1,2,4,3] => 5
[5,1,3,2,4] => 5
[5,1,3,4,2] => 6
[5,1,4,2,3] => 6
[5,1,4,3,2] => 7
[5,2,1,3,4] => 5
[5,2,1,4,3] => 6
[5,2,3,1,4] => 6
[5,2,3,4,1] => 7
[5,2,4,1,3] => 7
[5,2,4,3,1] => 8
[5,3,1,2,4] => 6
[5,3,1,4,2] => 7
[5,3,2,1,4] => 7
[5,3,2,4,1] => 8
[5,3,4,1,2] => 8
[5,3,4,2,1] => 9
[5,4,1,2,3] => 7
[5,4,1,3,2] => 8
[5,4,2,1,3] => 8
[5,4,2,3,1] => 9
[5,4,3,1,2] => 9
[5,4,3,2,1] => 10
[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] => 2
[1,2,3,6,5,4] => 3
[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] => 3
[1,2,4,6,5,3] => 4
[1,2,5,3,4,6] => 2
[1,2,5,3,6,4] => 3
[1,2,5,4,3,6] => 3
[1,2,5,4,6,3] => 4
[1,2,5,6,3,4] => 4
[1,2,5,6,4,3] => 5
[1,2,6,3,4,5] => 3
[1,2,6,3,5,4] => 4
[1,2,6,4,3,5] => 4
[1,2,6,4,5,3] => 5
[1,2,6,5,3,4] => 5
[1,2,6,5,4,3] => 6
[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] => 3
[1,3,2,6,5,4] => 4
[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] => 4
[1,3,4,6,5,2] => 5
[1,3,5,2,4,6] => 3
[1,3,5,2,6,4] => 4
[1,3,5,4,2,6] => 4
[1,3,5,4,6,2] => 5
[1,3,5,6,2,4] => 5
[1,3,5,6,4,2] => 6
[1,3,6,2,4,5] => 4
[1,3,6,2,5,4] => 5
[1,3,6,4,2,5] => 5
[1,3,6,4,5,2] => 6
[1,3,6,5,2,4] => 6
[1,3,6,5,4,2] => 7
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 3
[1,4,2,5,3,6] => 3
[1,4,2,5,6,3] => 4
[1,4,2,6,3,5] => 4
[1,4,2,6,5,3] => 5
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 4
[1,4,3,5,2,6] => 4
[1,4,3,5,6,2] => 5
[1,4,3,6,2,5] => 5
[1,4,3,6,5,2] => 6
[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] => 6
[1,4,5,6,3,2] => 7
[1,4,6,2,3,5] => 5
[1,4,6,2,5,3] => 6
[1,4,6,3,2,5] => 6
[1,4,6,3,5,2] => 7
[1,4,6,5,2,3] => 7
[1,4,6,5,3,2] => 8
[1,5,2,3,4,6] => 3
[1,5,2,3,6,4] => 4
[1,5,2,4,3,6] => 4
[1,5,2,4,6,3] => 5
[1,5,2,6,3,4] => 5
[1,5,2,6,4,3] => 6
[1,5,3,2,4,6] => 4
[1,5,3,2,6,4] => 5
[1,5,3,4,2,6] => 5
[1,5,3,4,6,2] => 6
[1,5,3,6,2,4] => 6
[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] => 6
[1,5,4,3,6,2] => 7
[1,5,4,6,2,3] => 7
[1,5,4,6,3,2] => 8
[1,5,6,2,3,4] => 6
[1,5,6,2,4,3] => 7
[1,5,6,3,2,4] => 7
[1,5,6,3,4,2] => 8
[1,5,6,4,2,3] => 8
[1,5,6,4,3,2] => 9
[1,6,2,3,4,5] => 4
[1,6,2,3,5,4] => 5
[1,6,2,4,3,5] => 5
[1,6,2,4,5,3] => 6
[1,6,2,5,3,4] => 6
[1,6,2,5,4,3] => 7
[1,6,3,2,4,5] => 5
[1,6,3,2,5,4] => 6
[1,6,3,4,2,5] => 6
[1,6,3,4,5,2] => 7
[1,6,3,5,2,4] => 7
[1,6,3,5,4,2] => 8
[1,6,4,2,3,5] => 6
[1,6,4,2,5,3] => 7
[1,6,4,3,2,5] => 7
[1,6,4,3,5,2] => 8
[1,6,4,5,2,3] => 8
[1,6,4,5,3,2] => 9
[1,6,5,2,3,4] => 7
[1,6,5,2,4,3] => 8
[1,6,5,3,2,4] => 8
[1,6,5,3,4,2] => 9
[1,6,5,4,2,3] => 9
[1,6,5,4,3,2] => 10
[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] => 3
[2,1,3,6,5,4] => 4
[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] => 4
[2,1,4,6,5,3] => 5
[2,1,5,3,4,6] => 3
[2,1,5,3,6,4] => 4
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 5
[2,1,5,6,3,4] => 5
[2,1,5,6,4,3] => 6
[2,1,6,3,4,5] => 4
[2,1,6,3,5,4] => 5
[2,1,6,4,3,5] => 5
[2,1,6,4,5,3] => 6
[2,1,6,5,3,4] => 6
[2,1,6,5,4,3] => 7
[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] => 4
[2,3,1,6,5,4] => 5
[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] => 5
[2,3,4,6,5,1] => 6
[2,3,5,1,4,6] => 4
[2,3,5,1,6,4] => 5
[2,3,5,4,1,6] => 5
[2,3,5,4,6,1] => 6
[2,3,5,6,1,4] => 6
[2,3,5,6,4,1] => 7
[2,3,6,1,4,5] => 5
[2,3,6,1,5,4] => 6
[2,3,6,4,1,5] => 6
[2,3,6,4,5,1] => 7
[2,3,6,5,1,4] => 7
[2,3,6,5,4,1] => 8
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 4
[2,4,1,5,3,6] => 4
[2,4,1,5,6,3] => 5
[2,4,1,6,3,5] => 5
[2,4,1,6,5,3] => 6
[2,4,3,1,5,6] => 4
[2,4,3,1,6,5] => 5
[2,4,3,5,1,6] => 5
[2,4,3,5,6,1] => 6
[2,4,3,6,1,5] => 6
[2,4,3,6,5,1] => 7
[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] => 7
[2,4,5,6,3,1] => 8
[2,4,6,1,3,5] => 6
[2,4,6,1,5,3] => 7
[2,4,6,3,1,5] => 7
[2,4,6,3,5,1] => 8
[2,4,6,5,1,3] => 8
[2,4,6,5,3,1] => 9
[2,5,1,3,4,6] => 4
[2,5,1,3,6,4] => 5
[2,5,1,4,3,6] => 5
[2,5,1,4,6,3] => 6
[2,5,1,6,3,4] => 6
[2,5,1,6,4,3] => 7
[2,5,3,1,4,6] => 5
[2,5,3,1,6,4] => 6
[2,5,3,4,1,6] => 6
[2,5,3,4,6,1] => 7
[2,5,3,6,1,4] => 7
[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] => 7
[2,5,4,3,6,1] => 8
[2,5,4,6,1,3] => 8
[2,5,4,6,3,1] => 9
[2,5,6,1,3,4] => 7
[2,5,6,1,4,3] => 8
[2,5,6,3,1,4] => 8
[2,5,6,3,4,1] => 9
[2,5,6,4,1,3] => 9
[2,5,6,4,3,1] => 10
[2,6,1,3,4,5] => 5
[2,6,1,3,5,4] => 6
[2,6,1,4,3,5] => 6
[2,6,1,4,5,3] => 7
[2,6,1,5,3,4] => 7
[2,6,1,5,4,3] => 8
[2,6,3,1,4,5] => 6
[2,6,3,1,5,4] => 7
[2,6,3,4,1,5] => 7
[2,6,3,4,5,1] => 8
[2,6,3,5,1,4] => 8
[2,6,3,5,4,1] => 9
[2,6,4,1,3,5] => 7
[2,6,4,1,5,3] => 8
[2,6,4,3,1,5] => 8
[2,6,4,3,5,1] => 9
[2,6,4,5,1,3] => 9
[2,6,4,5,3,1] => 10
[2,6,5,1,3,4] => 8
[2,6,5,1,4,3] => 9
[2,6,5,3,1,4] => 9
[2,6,5,3,4,1] => 10
[2,6,5,4,1,3] => 10
[2,6,5,4,3,1] => 11
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 3
[3,1,2,5,4,6] => 3
[3,1,2,5,6,4] => 4
[3,1,2,6,4,5] => 4
[3,1,2,6,5,4] => 5
[3,1,4,2,5,6] => 3
[3,1,4,2,6,5] => 4
[3,1,4,5,2,6] => 4
[3,1,4,5,6,2] => 5
[3,1,4,6,2,5] => 5
[3,1,4,6,5,2] => 6
[3,1,5,2,4,6] => 4
[3,1,5,2,6,4] => 5
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 6
[3,1,5,6,2,4] => 6
[3,1,5,6,4,2] => 7
[3,1,6,2,4,5] => 5
[3,1,6,2,5,4] => 6
[3,1,6,4,2,5] => 6
[3,1,6,4,5,2] => 7
[3,1,6,5,2,4] => 7
[3,1,6,5,4,2] => 8
[3,2,1,4,5,6] => 3
[3,2,1,4,6,5] => 4
[3,2,1,5,4,6] => 4
[3,2,1,5,6,4] => 5
[3,2,1,6,4,5] => 5
[3,2,1,6,5,4] => 6
[3,2,4,1,5,6] => 4
[3,2,4,1,6,5] => 5
[3,2,4,5,1,6] => 5
[3,2,4,5,6,1] => 6
[3,2,4,6,1,5] => 6
[3,2,4,6,5,1] => 7
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 6
[3,2,5,4,1,6] => 6
[3,2,5,4,6,1] => 7
[3,2,5,6,1,4] => 7
[3,2,5,6,4,1] => 8
[3,2,6,1,4,5] => 6
[3,2,6,1,5,4] => 7
[3,2,6,4,1,5] => 7
[3,2,6,4,5,1] => 8
[3,2,6,5,1,4] => 8
[3,2,6,5,4,1] => 9
[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] => 6
[3,4,1,6,5,2] => 7
[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] => 7
[3,4,2,6,5,1] => 8
[3,4,5,1,2,6] => 6
[3,4,5,1,6,2] => 7
[3,4,5,2,1,6] => 7
[3,4,5,2,6,1] => 8
[3,4,5,6,1,2] => 8
[3,4,5,6,2,1] => 9
[3,4,6,1,2,5] => 7
[3,4,6,1,5,2] => 8
[3,4,6,2,1,5] => 8
[3,4,6,2,5,1] => 9
[3,4,6,5,1,2] => 9
[3,4,6,5,2,1] => 10
[3,5,1,2,4,6] => 5
[3,5,1,2,6,4] => 6
[3,5,1,4,2,6] => 6
[3,5,1,4,6,2] => 7
[3,5,1,6,2,4] => 7
[3,5,1,6,4,2] => 8
[3,5,2,1,4,6] => 6
[3,5,2,1,6,4] => 7
[3,5,2,4,1,6] => 7
[3,5,2,4,6,1] => 8
[3,5,2,6,1,4] => 8
[3,5,2,6,4,1] => 9
[3,5,4,1,2,6] => 7
[3,5,4,1,6,2] => 8
[3,5,4,2,1,6] => 8
[3,5,4,2,6,1] => 9
[3,5,4,6,1,2] => 9
[3,5,4,6,2,1] => 10
[3,5,6,1,2,4] => 8
[3,5,6,1,4,2] => 9
[3,5,6,2,1,4] => 9
[3,5,6,2,4,1] => 10
[3,5,6,4,1,2] => 10
[3,5,6,4,2,1] => 11
[3,6,1,2,4,5] => 6
[3,6,1,2,5,4] => 7
[3,6,1,4,2,5] => 7
[3,6,1,4,5,2] => 8
[3,6,1,5,2,4] => 8
[3,6,1,5,4,2] => 9
[3,6,2,1,4,5] => 7
[3,6,2,1,5,4] => 8
[3,6,2,4,1,5] => 8
[3,6,2,4,5,1] => 9
[3,6,2,5,1,4] => 9
[3,6,2,5,4,1] => 10
[3,6,4,1,2,5] => 8
[3,6,4,1,5,2] => 9
[3,6,4,2,1,5] => 9
[3,6,4,2,5,1] => 10
[3,6,4,5,1,2] => 10
[3,6,4,5,2,1] => 11
[3,6,5,1,2,4] => 9
[3,6,5,1,4,2] => 10
[3,6,5,2,1,4] => 10
[3,6,5,2,4,1] => 11
[3,6,5,4,1,2] => 11
[3,6,5,4,2,1] => 12
[4,1,2,3,5,6] => 3
[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] => 5
[4,1,2,6,5,3] => 6
[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] => 6
[4,1,3,6,5,2] => 7
[4,1,5,2,3,6] => 5
[4,1,5,2,6,3] => 6
[4,1,5,3,2,6] => 6
[4,1,5,3,6,2] => 7
[4,1,5,6,2,3] => 7
[4,1,5,6,3,2] => 8
[4,1,6,2,3,5] => 6
[4,1,6,2,5,3] => 7
[4,1,6,3,2,5] => 7
[4,1,6,3,5,2] => 8
[4,1,6,5,2,3] => 8
[4,1,6,5,3,2] => 9
[4,2,1,3,5,6] => 4
[4,2,1,3,6,5] => 5
[4,2,1,5,3,6] => 5
[4,2,1,5,6,3] => 6
[4,2,1,6,3,5] => 6
[4,2,1,6,5,3] => 7
[4,2,3,1,5,6] => 5
[4,2,3,1,6,5] => 6
[4,2,3,5,1,6] => 6
[4,2,3,5,6,1] => 7
[4,2,3,6,1,5] => 7
[4,2,3,6,5,1] => 8
[4,2,5,1,3,6] => 6
[4,2,5,1,6,3] => 7
[4,2,5,3,1,6] => 7
[4,2,5,3,6,1] => 8
[4,2,5,6,1,3] => 8
[4,2,5,6,3,1] => 9
[4,2,6,1,3,5] => 7
[4,2,6,1,5,3] => 8
[4,2,6,3,1,5] => 8
[4,2,6,3,5,1] => 9
[4,2,6,5,1,3] => 9
[4,2,6,5,3,1] => 10
[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] => 7
[4,3,1,6,5,2] => 8
[4,3,2,1,5,6] => 6
[4,3,2,1,6,5] => 7
[4,3,2,5,1,6] => 7
[4,3,2,5,6,1] => 8
[4,3,2,6,1,5] => 8
[4,3,2,6,5,1] => 9
[4,3,5,1,2,6] => 7
[4,3,5,1,6,2] => 8
[4,3,5,2,1,6] => 8
[4,3,5,2,6,1] => 9
[4,3,5,6,1,2] => 9
[4,3,5,6,2,1] => 10
[4,3,6,1,2,5] => 8
[4,3,6,1,5,2] => 9
[4,3,6,2,1,5] => 9
[4,3,6,2,5,1] => 10
[4,3,6,5,1,2] => 10
[4,3,6,5,2,1] => 11
[4,5,1,2,3,6] => 6
[4,5,1,2,6,3] => 7
[4,5,1,3,2,6] => 7
[4,5,1,3,6,2] => 8
[4,5,1,6,2,3] => 8
[4,5,1,6,3,2] => 9
[4,5,2,1,3,6] => 7
[4,5,2,1,6,3] => 8
[4,5,2,3,1,6] => 8
[4,5,2,3,6,1] => 9
[4,5,2,6,1,3] => 9
[4,5,2,6,3,1] => 10
[4,5,3,1,2,6] => 8
[4,5,3,1,6,2] => 9
[4,5,3,2,1,6] => 9
[4,5,3,2,6,1] => 10
[4,5,3,6,1,2] => 10
[4,5,3,6,2,1] => 11
[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] => 11
[4,5,6,3,2,1] => 12
[4,6,1,2,3,5] => 7
[4,6,1,2,5,3] => 8
[4,6,1,3,2,5] => 8
[4,6,1,3,5,2] => 9
[4,6,1,5,2,3] => 9
[4,6,1,5,3,2] => 10
[4,6,2,1,3,5] => 8
[4,6,2,1,5,3] => 9
[4,6,2,3,1,5] => 9
[4,6,2,3,5,1] => 10
[4,6,2,5,1,3] => 10
[4,6,2,5,3,1] => 11
[4,6,3,1,2,5] => 9
[4,6,3,1,5,2] => 10
[4,6,3,2,1,5] => 10
[4,6,3,2,5,1] => 11
[4,6,3,5,1,2] => 11
[4,6,3,5,2,1] => 12
[4,6,5,1,2,3] => 10
[4,6,5,1,3,2] => 11
[4,6,5,2,1,3] => 11
[4,6,5,2,3,1] => 12
[4,6,5,3,1,2] => 12
[4,6,5,3,2,1] => 13
[5,1,2,3,4,6] => 4
[5,1,2,3,6,4] => 5
[5,1,2,4,3,6] => 5
[5,1,2,4,6,3] => 6
[5,1,2,6,3,4] => 6
[5,1,2,6,4,3] => 7
[5,1,3,2,4,6] => 5
[5,1,3,2,6,4] => 6
[5,1,3,4,2,6] => 6
[5,1,3,4,6,2] => 7
[5,1,3,6,2,4] => 7
[5,1,3,6,4,2] => 8
[5,1,4,2,3,6] => 6
[5,1,4,2,6,3] => 7
[5,1,4,3,2,6] => 7
[5,1,4,3,6,2] => 8
[5,1,4,6,2,3] => 8
[5,1,4,6,3,2] => 9
[5,1,6,2,3,4] => 7
[5,1,6,2,4,3] => 8
[5,1,6,3,2,4] => 8
[5,1,6,3,4,2] => 9
[5,1,6,4,2,3] => 9
[5,1,6,4,3,2] => 10
[5,2,1,3,4,6] => 5
[5,2,1,3,6,4] => 6
[5,2,1,4,3,6] => 6
[5,2,1,4,6,3] => 7
[5,2,1,6,3,4] => 7
[5,2,1,6,4,3] => 8
[5,2,3,1,4,6] => 6
[5,2,3,1,6,4] => 7
[5,2,3,4,1,6] => 7
[5,2,3,4,6,1] => 8
[5,2,3,6,1,4] => 8
[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] => 8
[5,2,4,3,6,1] => 9
[5,2,4,6,1,3] => 9
[5,2,4,6,3,1] => 10
[5,2,6,1,3,4] => 8
[5,2,6,1,4,3] => 9
[5,2,6,3,1,4] => 9
[5,2,6,3,4,1] => 10
[5,2,6,4,1,3] => 10
[5,2,6,4,3,1] => 11
[5,3,1,2,4,6] => 6
[5,3,1,2,6,4] => 7
[5,3,1,4,2,6] => 7
[5,3,1,4,6,2] => 8
[5,3,1,6,2,4] => 8
[5,3,1,6,4,2] => 9
[5,3,2,1,4,6] => 7
[5,3,2,1,6,4] => 8
[5,3,2,4,1,6] => 8
[5,3,2,4,6,1] => 9
[5,3,2,6,1,4] => 9
[5,3,2,6,4,1] => 10
[5,3,4,1,2,6] => 8
[5,3,4,1,6,2] => 9
[5,3,4,2,1,6] => 9
[5,3,4,2,6,1] => 10
[5,3,4,6,1,2] => 10
[5,3,4,6,2,1] => 11
[5,3,6,1,2,4] => 9
[5,3,6,1,4,2] => 10
[5,3,6,2,1,4] => 10
[5,3,6,2,4,1] => 11
[5,3,6,4,1,2] => 11
[5,3,6,4,2,1] => 12
[5,4,1,2,3,6] => 7
[5,4,1,2,6,3] => 8
[5,4,1,3,2,6] => 8
[5,4,1,3,6,2] => 9
[5,4,1,6,2,3] => 9
[5,4,1,6,3,2] => 10
[5,4,2,1,3,6] => 8
[5,4,2,1,6,3] => 9
[5,4,2,3,1,6] => 9
[5,4,2,3,6,1] => 10
[5,4,2,6,1,3] => 10
[5,4,2,6,3,1] => 11
[5,4,3,1,2,6] => 9
[5,4,3,1,6,2] => 10
[5,4,3,2,1,6] => 10
[5,4,3,2,6,1] => 11
[5,4,3,6,1,2] => 11
[5,4,3,6,2,1] => 12
[5,4,6,1,2,3] => 10
[5,4,6,1,3,2] => 11
[5,4,6,2,1,3] => 11
[5,4,6,2,3,1] => 12
[5,4,6,3,1,2] => 12
[5,4,6,3,2,1] => 13
[5,6,1,2,3,4] => 8
[5,6,1,2,4,3] => 9
[5,6,1,3,2,4] => 9
[5,6,1,3,4,2] => 10
[5,6,1,4,2,3] => 10
[5,6,1,4,3,2] => 11
[5,6,2,1,3,4] => 9
[5,6,2,1,4,3] => 10
[5,6,2,3,1,4] => 10
[5,6,2,3,4,1] => 11
[5,6,2,4,1,3] => 11
[5,6,2,4,3,1] => 12
[5,6,3,1,2,4] => 10
[5,6,3,1,4,2] => 11
[5,6,3,2,1,4] => 11
[5,6,3,2,4,1] => 12
[5,6,3,4,1,2] => 12
[5,6,3,4,2,1] => 13
[5,6,4,1,2,3] => 11
[5,6,4,1,3,2] => 12
[5,6,4,2,1,3] => 12
[5,6,4,2,3,1] => 13
[5,6,4,3,1,2] => 13
[5,6,4,3,2,1] => 14
[6,1,2,3,4,5] => 5
[6,1,2,3,5,4] => 6
[6,1,2,4,3,5] => 6
[6,1,2,4,5,3] => 7
[6,1,2,5,3,4] => 7
[6,1,2,5,4,3] => 8
[6,1,3,2,4,5] => 6
[6,1,3,2,5,4] => 7
[6,1,3,4,2,5] => 7
[6,1,3,4,5,2] => 8
[6,1,3,5,2,4] => 8
[6,1,3,5,4,2] => 9
[6,1,4,2,3,5] => 7
[6,1,4,2,5,3] => 8
[6,1,4,3,2,5] => 8
[6,1,4,3,5,2] => 9
[6,1,4,5,2,3] => 9
[6,1,4,5,3,2] => 10
[6,1,5,2,3,4] => 8
[6,1,5,2,4,3] => 9
[6,1,5,3,2,4] => 9
[6,1,5,3,4,2] => 10
[6,1,5,4,2,3] => 10
[6,1,5,4,3,2] => 11
[6,2,1,3,4,5] => 6
[6,2,1,3,5,4] => 7
[6,2,1,4,3,5] => 7
[6,2,1,4,5,3] => 8
[6,2,1,5,3,4] => 8
[6,2,1,5,4,3] => 9
[6,2,3,1,4,5] => 7
[6,2,3,1,5,4] => 8
[6,2,3,4,1,5] => 8
[6,2,3,4,5,1] => 9
[6,2,3,5,1,4] => 9
[6,2,3,5,4,1] => 10
[6,2,4,1,3,5] => 8
[6,2,4,1,5,3] => 9
[6,2,4,3,1,5] => 9
[6,2,4,3,5,1] => 10
[6,2,4,5,1,3] => 10
[6,2,4,5,3,1] => 11
[6,2,5,1,3,4] => 9
[6,2,5,1,4,3] => 10
[6,2,5,3,1,4] => 10
[6,2,5,3,4,1] => 11
[6,2,5,4,1,3] => 11
[6,2,5,4,3,1] => 12
[6,3,1,2,4,5] => 7
[6,3,1,2,5,4] => 8
[6,3,1,4,2,5] => 8
[6,3,1,4,5,2] => 9
[6,3,1,5,2,4] => 9
[6,3,1,5,4,2] => 10
[6,3,2,1,4,5] => 8
[6,3,2,1,5,4] => 9
[6,3,2,4,1,5] => 9
[6,3,2,4,5,1] => 10
[6,3,2,5,1,4] => 10
[6,3,2,5,4,1] => 11
[6,3,4,1,2,5] => 9
[6,3,4,1,5,2] => 10
[6,3,4,2,1,5] => 10
[6,3,4,2,5,1] => 11
[6,3,4,5,1,2] => 11
[6,3,4,5,2,1] => 12
[6,3,5,1,2,4] => 10
[6,3,5,1,4,2] => 11
[6,3,5,2,1,4] => 11
[6,3,5,2,4,1] => 12
[6,3,5,4,1,2] => 12
[6,3,5,4,2,1] => 13
[6,4,1,2,3,5] => 8
[6,4,1,2,5,3] => 9
[6,4,1,3,2,5] => 9
[6,4,1,3,5,2] => 10
[6,4,1,5,2,3] => 10
[6,4,1,5,3,2] => 11
[6,4,2,1,3,5] => 9
[6,4,2,1,5,3] => 10
[6,4,2,3,1,5] => 10
[6,4,2,3,5,1] => 11
[6,4,2,5,1,3] => 11
[6,4,2,5,3,1] => 12
[6,4,3,1,2,5] => 10
[6,4,3,1,5,2] => 11
[6,4,3,2,1,5] => 11
[6,4,3,2,5,1] => 12
[6,4,3,5,1,2] => 12
[6,4,3,5,2,1] => 13
[6,4,5,1,2,3] => 11
[6,4,5,1,3,2] => 12
[6,4,5,2,1,3] => 12
[6,4,5,2,3,1] => 13
[6,4,5,3,1,2] => 13
[6,4,5,3,2,1] => 14
[6,5,1,2,3,4] => 9
[6,5,1,2,4,3] => 10
[6,5,1,3,2,4] => 10
[6,5,1,3,4,2] => 11
[6,5,1,4,2,3] => 11
[6,5,1,4,3,2] => 12
[6,5,2,1,3,4] => 10
[6,5,2,1,4,3] => 11
[6,5,2,3,1,4] => 11
[6,5,2,3,4,1] => 12
[6,5,2,4,1,3] => 12
[6,5,2,4,3,1] => 13
[6,5,3,1,2,4] => 11
[6,5,3,1,4,2] => 12
[6,5,3,2,1,4] => 12
[6,5,3,2,4,1] => 13
[6,5,3,4,1,2] => 13
[6,5,3,4,2,1] => 14
[6,5,4,1,2,3] => 12
[6,5,4,1,3,2] => 13
[6,5,4,2,1,3] => 13
[6,5,4,2,3,1] => 14
[6,5,4,3,1,2] => 14
[6,5,4,3,2,1] => 15
click to show generating function       
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
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] List of permutations of 1,2,3,...,n for n=1,2,3,..., in lexicographic order. OEIS:A030298
[3] Array of digit sums of factorial representation of numbers 0,1,...,n!-1 for n>=1. OEIS:A139365
Code
def statistic(x):
    return x.number_of_inversions()
Created
Sep 27, 2011 at 23:29 by Christian Stump
Updated
Jan 23, 2017 at 15:55 by Martin Rubey