Identifier
Identifier
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 5
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 3
[2,3,4,1] => 6
[2,4,1,3] => 3
[2,4,3,1] => 4
[3,1,2,4] => 1
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 5
[3,4,1,2] => 3
[3,4,2,1] => 4
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 4
[4,3,2,1] => 5
[1,2,3,4,5] => 0
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 7
[1,2,5,3,4] => 3
[1,2,5,4,3] => 4
[1,3,2,4,5] => 2
[1,3,2,5,4] => 6
[1,3,4,2,5] => 5
[1,3,4,5,2] => 9
[1,3,5,2,4] => 5
[1,3,5,4,2] => 6
[1,4,2,3,5] => 2
[1,4,2,5,3] => 6
[1,4,3,2,5] => 3
[1,4,3,5,2] => 7
[1,4,5,2,3] => 5
[1,4,5,3,2] => 6
[1,5,2,3,4] => 2
[1,5,2,4,3] => 3
[1,5,3,2,4] => 3
[1,5,3,4,2] => 4
[1,5,4,2,3] => 6
[1,5,4,3,2] => 7
[2,1,3,4,5] => 1
[2,1,3,5,4] => 5
[2,1,4,3,5] => 4
[2,1,4,5,3] => 8
[2,1,5,3,4] => 4
[2,1,5,4,3] => 5
[2,3,1,4,5] => 3
[2,3,1,5,4] => 7
[2,3,4,1,5] => 6
[2,3,4,5,1] => 10
[2,3,5,1,4] => 6
[2,3,5,4,1] => 7
[2,4,1,3,5] => 3
[2,4,1,5,3] => 7
[2,4,3,1,5] => 4
[2,4,3,5,1] => 8
[2,4,5,1,3] => 6
[2,4,5,3,1] => 7
[2,5,1,3,4] => 3
[2,5,1,4,3] => 4
[2,5,3,1,4] => 4
[2,5,3,4,1] => 5
[2,5,4,1,3] => 7
[2,5,4,3,1] => 8
[3,1,2,4,5] => 1
[3,1,2,5,4] => 5
[3,1,4,2,5] => 4
[3,1,4,5,2] => 8
[3,1,5,2,4] => 4
[3,1,5,4,2] => 5
[3,2,1,4,5] => 2
[3,2,1,5,4] => 6
[3,2,4,1,5] => 5
[3,2,4,5,1] => 9
[3,2,5,1,4] => 5
[3,2,5,4,1] => 6
[3,4,1,2,5] => 3
[3,4,1,5,2] => 7
[3,4,2,1,5] => 4
[3,4,2,5,1] => 8
[3,4,5,1,2] => 6
[3,4,5,2,1] => 7
[3,5,1,2,4] => 3
[3,5,1,4,2] => 4
[3,5,2,1,4] => 4
[3,5,2,4,1] => 5
[3,5,4,1,2] => 7
[3,5,4,2,1] => 8
[4,1,2,3,5] => 1
[4,1,2,5,3] => 5
[4,1,3,2,5] => 2
[4,1,3,5,2] => 6
[4,1,5,2,3] => 4
[4,1,5,3,2] => 5
[4,2,1,3,5] => 2
[4,2,1,5,3] => 6
[4,2,3,1,5] => 3
[4,2,3,5,1] => 7
[4,2,5,1,3] => 5
[4,2,5,3,1] => 6
[4,3,1,2,5] => 4
[4,3,1,5,2] => 8
[4,3,2,1,5] => 5
[4,3,2,5,1] => 9
[4,3,5,1,2] => 7
[4,3,5,2,1] => 8
[4,5,1,2,3] => 3
[4,5,1,3,2] => 4
[4,5,2,1,3] => 4
[4,5,2,3,1] => 5
[4,5,3,1,2] => 5
[4,5,3,2,1] => 6
[5,1,2,3,4] => 1
[5,1,2,4,3] => 2
[5,1,3,2,4] => 2
[5,1,3,4,2] => 3
[5,1,4,2,3] => 5
[5,1,4,3,2] => 6
[5,2,1,3,4] => 2
[5,2,1,4,3] => 3
[5,2,3,1,4] => 3
[5,2,3,4,1] => 4
[5,2,4,1,3] => 6
[5,2,4,3,1] => 7
[5,3,1,2,4] => 4
[5,3,1,4,2] => 5
[5,3,2,1,4] => 5
[5,3,2,4,1] => 6
[5,3,4,1,2] => 8
[5,3,4,2,1] => 9
[5,4,1,2,3] => 4
[5,4,1,3,2] => 5
[5,4,2,1,3] => 5
[5,4,2,3,1] => 6
[5,4,3,1,2] => 6
[5,4,3,2,1] => 7
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 5
[1,2,3,5,4,6] => 4
[1,2,3,5,6,4] => 9
[1,2,3,6,4,5] => 4
[1,2,3,6,5,4] => 5
[1,2,4,3,5,6] => 3
[1,2,4,3,6,5] => 8
[1,2,4,5,3,6] => 7
[1,2,4,5,6,3] => 12
[1,2,4,6,3,5] => 7
[1,2,4,6,5,3] => 8
[1,2,5,3,4,6] => 3
[1,2,5,3,6,4] => 8
[1,2,5,4,3,6] => 4
[1,2,5,4,6,3] => 9
[1,2,5,6,3,4] => 7
[1,2,5,6,4,3] => 8
[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] => 8
[1,2,6,5,4,3] => 9
[1,3,2,4,5,6] => 2
[1,3,2,4,6,5] => 7
[1,3,2,5,4,6] => 6
[1,3,2,5,6,4] => 11
[1,3,2,6,4,5] => 6
[1,3,2,6,5,4] => 7
[1,3,4,2,5,6] => 5
[1,3,4,2,6,5] => 10
[1,3,4,5,2,6] => 9
[1,3,4,5,6,2] => 14
[1,3,4,6,2,5] => 9
[1,3,4,6,5,2] => 10
[1,3,5,2,4,6] => 5
[1,3,5,2,6,4] => 10
[1,3,5,4,2,6] => 6
[1,3,5,4,6,2] => 11
[1,3,5,6,2,4] => 9
[1,3,5,6,4,2] => 10
[1,3,6,2,4,5] => 5
[1,3,6,2,5,4] => 6
[1,3,6,4,2,5] => 6
[1,3,6,4,5,2] => 7
[1,3,6,5,2,4] => 10
[1,3,6,5,4,2] => 11
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 7
[1,4,2,5,3,6] => 6
[1,4,2,5,6,3] => 11
[1,4,2,6,3,5] => 6
[1,4,2,6,5,3] => 7
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 8
[1,4,3,5,2,6] => 7
[1,4,3,5,6,2] => 12
[1,4,3,6,2,5] => 7
[1,4,3,6,5,2] => 8
[1,4,5,2,3,6] => 5
[1,4,5,2,6,3] => 10
[1,4,5,3,2,6] => 6
[1,4,5,3,6,2] => 11
[1,4,5,6,2,3] => 9
[1,4,5,6,3,2] => 10
[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] => 10
[1,4,6,5,3,2] => 11
[1,5,2,3,4,6] => 2
[1,5,2,3,6,4] => 7
[1,5,2,4,3,6] => 3
[1,5,2,4,6,3] => 8
[1,5,2,6,3,4] => 6
[1,5,2,6,4,3] => 7
[1,5,3,2,4,6] => 3
[1,5,3,2,6,4] => 8
[1,5,3,4,2,6] => 4
[1,5,3,4,6,2] => 9
[1,5,3,6,2,4] => 7
[1,5,3,6,4,2] => 8
[1,5,4,2,3,6] => 6
[1,5,4,2,6,3] => 11
[1,5,4,3,2,6] => 7
[1,5,4,3,6,2] => 12
[1,5,4,6,2,3] => 10
[1,5,4,6,3,2] => 11
[1,5,6,2,3,4] => 5
[1,5,6,2,4,3] => 6
[1,5,6,3,2,4] => 6
[1,5,6,3,4,2] => 7
[1,5,6,4,2,3] => 7
[1,5,6,4,3,2] => 8
[1,6,2,3,4,5] => 2
[1,6,2,3,5,4] => 3
[1,6,2,4,3,5] => 3
[1,6,2,4,5,3] => 4
[1,6,2,5,3,4] => 7
[1,6,2,5,4,3] => 8
[1,6,3,2,4,5] => 3
[1,6,3,2,5,4] => 4
[1,6,3,4,2,5] => 4
[1,6,3,4,5,2] => 5
[1,6,3,5,2,4] => 8
[1,6,3,5,4,2] => 9
[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] => 11
[1,6,4,5,3,2] => 12
[1,6,5,2,3,4] => 6
[1,6,5,2,4,3] => 7
[1,6,5,3,2,4] => 7
[1,6,5,3,4,2] => 8
[1,6,5,4,2,3] => 8
[1,6,5,4,3,2] => 9
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 6
[2,1,3,5,4,6] => 5
[2,1,3,5,6,4] => 10
[2,1,3,6,4,5] => 5
[2,1,3,6,5,4] => 6
[2,1,4,3,5,6] => 4
[2,1,4,3,6,5] => 9
[2,1,4,5,3,6] => 8
[2,1,4,5,6,3] => 13
[2,1,4,6,3,5] => 8
[2,1,4,6,5,3] => 9
[2,1,5,3,4,6] => 4
[2,1,5,3,6,4] => 9
[2,1,5,4,3,6] => 5
[2,1,5,4,6,3] => 10
[2,1,5,6,3,4] => 8
[2,1,5,6,4,3] => 9
[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] => 9
[2,1,6,5,4,3] => 10
[2,3,1,4,5,6] => 3
[2,3,1,4,6,5] => 8
[2,3,1,5,4,6] => 7
[2,3,1,5,6,4] => 12
[2,3,1,6,4,5] => 7
[2,3,1,6,5,4] => 8
[2,3,4,1,5,6] => 6
[2,3,4,1,6,5] => 11
[2,3,4,5,1,6] => 10
[2,3,4,5,6,1] => 15
[2,3,4,6,1,5] => 10
[2,3,4,6,5,1] => 11
[2,3,5,1,4,6] => 6
[2,3,5,1,6,4] => 11
[2,3,5,4,1,6] => 7
[2,3,5,4,6,1] => 12
[2,3,5,6,1,4] => 10
[2,3,5,6,4,1] => 11
[2,3,6,1,4,5] => 6
[2,3,6,1,5,4] => 7
[2,3,6,4,1,5] => 7
[2,3,6,4,5,1] => 8
[2,3,6,5,1,4] => 11
[2,3,6,5,4,1] => 12
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 8
[2,4,1,5,3,6] => 7
[2,4,1,5,6,3] => 12
[2,4,1,6,3,5] => 7
[2,4,1,6,5,3] => 8
[2,4,3,1,5,6] => 4
[2,4,3,1,6,5] => 9
[2,4,3,5,1,6] => 8
[2,4,3,5,6,1] => 13
[2,4,3,6,1,5] => 8
[2,4,3,6,5,1] => 9
[2,4,5,1,3,6] => 6
[2,4,5,1,6,3] => 11
[2,4,5,3,1,6] => 7
[2,4,5,3,6,1] => 12
[2,4,5,6,1,3] => 10
[2,4,5,6,3,1] => 11
[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] => 11
[2,4,6,5,3,1] => 12
[2,5,1,3,4,6] => 3
[2,5,1,3,6,4] => 8
[2,5,1,4,3,6] => 4
[2,5,1,4,6,3] => 9
[2,5,1,6,3,4] => 7
[2,5,1,6,4,3] => 8
[2,5,3,1,4,6] => 4
[2,5,3,1,6,4] => 9
[2,5,3,4,1,6] => 5
[2,5,3,4,6,1] => 10
[2,5,3,6,1,4] => 8
[2,5,3,6,4,1] => 9
[2,5,4,1,3,6] => 7
[2,5,4,1,6,3] => 12
[2,5,4,3,1,6] => 8
[2,5,4,3,6,1] => 13
[2,5,4,6,1,3] => 11
[2,5,4,6,3,1] => 12
[2,5,6,1,3,4] => 6
[2,5,6,1,4,3] => 7
[2,5,6,3,1,4] => 7
[2,5,6,3,4,1] => 8
[2,5,6,4,1,3] => 8
[2,5,6,4,3,1] => 9
[2,6,1,3,4,5] => 3
[2,6,1,3,5,4] => 4
[2,6,1,4,3,5] => 4
[2,6,1,4,5,3] => 5
[2,6,1,5,3,4] => 8
[2,6,1,5,4,3] => 9
[2,6,3,1,4,5] => 4
[2,6,3,1,5,4] => 5
[2,6,3,4,1,5] => 5
[2,6,3,4,5,1] => 6
[2,6,3,5,1,4] => 9
[2,6,3,5,4,1] => 10
[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] => 12
[2,6,4,5,3,1] => 13
[2,6,5,1,3,4] => 7
[2,6,5,1,4,3] => 8
[2,6,5,3,1,4] => 8
[2,6,5,3,4,1] => 9
[2,6,5,4,1,3] => 9
[2,6,5,4,3,1] => 10
[3,1,2,4,5,6] => 1
[3,1,2,4,6,5] => 6
[3,1,2,5,4,6] => 5
[3,1,2,5,6,4] => 10
[3,1,2,6,4,5] => 5
[3,1,2,6,5,4] => 6
[3,1,4,2,5,6] => 4
[3,1,4,2,6,5] => 9
[3,1,4,5,2,6] => 8
[3,1,4,5,6,2] => 13
[3,1,4,6,2,5] => 8
[3,1,4,6,5,2] => 9
[3,1,5,2,4,6] => 4
[3,1,5,2,6,4] => 9
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 10
[3,1,5,6,2,4] => 8
[3,1,5,6,4,2] => 9
[3,1,6,2,4,5] => 4
[3,1,6,2,5,4] => 5
[3,1,6,4,2,5] => 5
[3,1,6,4,5,2] => 6
[3,1,6,5,2,4] => 9
[3,1,6,5,4,2] => 10
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 7
[3,2,1,5,4,6] => 6
[3,2,1,5,6,4] => 11
[3,2,1,6,4,5] => 6
[3,2,1,6,5,4] => 7
[3,2,4,1,5,6] => 5
[3,2,4,1,6,5] => 10
[3,2,4,5,1,6] => 9
[3,2,4,5,6,1] => 14
[3,2,4,6,1,5] => 9
[3,2,4,6,5,1] => 10
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 10
[3,2,5,4,1,6] => 6
[3,2,5,4,6,1] => 11
[3,2,5,6,1,4] => 9
[3,2,5,6,4,1] => 10
[3,2,6,1,4,5] => 5
[3,2,6,1,5,4] => 6
[3,2,6,4,1,5] => 6
[3,2,6,4,5,1] => 7
[3,2,6,5,1,4] => 10
[3,2,6,5,4,1] => 11
[3,4,1,2,5,6] => 3
[3,4,1,2,6,5] => 8
[3,4,1,5,2,6] => 7
[3,4,1,5,6,2] => 12
[3,4,1,6,2,5] => 7
[3,4,1,6,5,2] => 8
[3,4,2,1,5,6] => 4
[3,4,2,1,6,5] => 9
[3,4,2,5,1,6] => 8
[3,4,2,5,6,1] => 13
[3,4,2,6,1,5] => 8
[3,4,2,6,5,1] => 9
[3,4,5,1,2,6] => 6
[3,4,5,1,6,2] => 11
[3,4,5,2,1,6] => 7
[3,4,5,2,6,1] => 12
[3,4,5,6,1,2] => 10
[3,4,5,6,2,1] => 11
[3,4,6,1,2,5] => 6
[3,4,6,1,5,2] => 7
[3,4,6,2,1,5] => 7
[3,4,6,2,5,1] => 8
[3,4,6,5,1,2] => 11
[3,4,6,5,2,1] => 12
[3,5,1,2,4,6] => 3
[3,5,1,2,6,4] => 8
[3,5,1,4,2,6] => 4
[3,5,1,4,6,2] => 9
[3,5,1,6,2,4] => 7
[3,5,1,6,4,2] => 8
[3,5,2,1,4,6] => 4
[3,5,2,1,6,4] => 9
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 10
[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] => 12
[3,5,4,2,1,6] => 8
[3,5,4,2,6,1] => 13
[3,5,4,6,1,2] => 11
[3,5,4,6,2,1] => 12
[3,5,6,1,2,4] => 6
[3,5,6,1,4,2] => 7
[3,5,6,2,1,4] => 7
[3,5,6,2,4,1] => 8
[3,5,6,4,1,2] => 8
[3,5,6,4,2,1] => 9
[3,6,1,2,4,5] => 3
[3,6,1,2,5,4] => 4
[3,6,1,4,2,5] => 4
[3,6,1,4,5,2] => 5
[3,6,1,5,2,4] => 8
[3,6,1,5,4,2] => 9
[3,6,2,1,4,5] => 4
[3,6,2,1,5,4] => 5
[3,6,2,4,1,5] => 5
[3,6,2,4,5,1] => 6
[3,6,2,5,1,4] => 9
[3,6,2,5,4,1] => 10
[3,6,4,1,2,5] => 7
[3,6,4,1,5,2] => 8
[3,6,4,2,1,5] => 8
[3,6,4,2,5,1] => 9
[3,6,4,5,1,2] => 12
[3,6,4,5,2,1] => 13
[3,6,5,1,2,4] => 7
[3,6,5,1,4,2] => 8
[3,6,5,2,1,4] => 8
[3,6,5,2,4,1] => 9
[3,6,5,4,1,2] => 9
[3,6,5,4,2,1] => 10
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 6
[4,1,2,5,3,6] => 5
[4,1,2,5,6,3] => 10
[4,1,2,6,3,5] => 5
[4,1,2,6,5,3] => 6
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 7
[4,1,3,5,2,6] => 6
[4,1,3,5,6,2] => 11
[4,1,3,6,2,5] => 6
[4,1,3,6,5,2] => 7
[4,1,5,2,3,6] => 4
[4,1,5,2,6,3] => 9
[4,1,5,3,2,6] => 5
[4,1,5,3,6,2] => 10
[4,1,5,6,2,3] => 8
[4,1,5,6,3,2] => 9
[4,1,6,2,3,5] => 4
[4,1,6,2,5,3] => 5
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 6
[4,1,6,5,2,3] => 9
[4,1,6,5,3,2] => 10
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 7
[4,2,1,5,3,6] => 6
[4,2,1,5,6,3] => 11
[4,2,1,6,3,5] => 6
[4,2,1,6,5,3] => 7
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 8
[4,2,3,5,1,6] => 7
[4,2,3,5,6,1] => 12
[4,2,3,6,1,5] => 7
[4,2,3,6,5,1] => 8
[4,2,5,1,3,6] => 5
[4,2,5,1,6,3] => 10
[4,2,5,3,1,6] => 6
[4,2,5,3,6,1] => 11
[4,2,5,6,1,3] => 9
[4,2,5,6,3,1] => 10
[4,2,6,1,3,5] => 5
[4,2,6,1,5,3] => 6
[4,2,6,3,1,5] => 6
[4,2,6,3,5,1] => 7
[4,2,6,5,1,3] => 10
[4,2,6,5,3,1] => 11
[4,3,1,2,5,6] => 4
[4,3,1,2,6,5] => 9
[4,3,1,5,2,6] => 8
[4,3,1,5,6,2] => 13
[4,3,1,6,2,5] => 8
[4,3,1,6,5,2] => 9
[4,3,2,1,5,6] => 5
[4,3,2,1,6,5] => 10
[4,3,2,5,1,6] => 9
[4,3,2,5,6,1] => 14
[4,3,2,6,1,5] => 9
[4,3,2,6,5,1] => 10
[4,3,5,1,2,6] => 7
[4,3,5,1,6,2] => 12
[4,3,5,2,1,6] => 8
[4,3,5,2,6,1] => 13
[4,3,5,6,1,2] => 11
[4,3,5,6,2,1] => 12
[4,3,6,1,2,5] => 7
[4,3,6,1,5,2] => 8
[4,3,6,2,1,5] => 8
[4,3,6,2,5,1] => 9
[4,3,6,5,1,2] => 12
[4,3,6,5,2,1] => 13
[4,5,1,2,3,6] => 3
[4,5,1,2,6,3] => 8
[4,5,1,3,2,6] => 4
[4,5,1,3,6,2] => 9
[4,5,1,6,2,3] => 7
[4,5,1,6,3,2] => 8
[4,5,2,1,3,6] => 4
[4,5,2,1,6,3] => 9
[4,5,2,3,1,6] => 5
[4,5,2,3,6,1] => 10
[4,5,2,6,1,3] => 8
[4,5,2,6,3,1] => 9
[4,5,3,1,2,6] => 5
[4,5,3,1,6,2] => 10
[4,5,3,2,1,6] => 6
[4,5,3,2,6,1] => 11
[4,5,3,6,1,2] => 9
[4,5,3,6,2,1] => 10
[4,5,6,1,2,3] => 6
[4,5,6,1,3,2] => 7
[4,5,6,2,1,3] => 7
[4,5,6,2,3,1] => 8
[4,5,6,3,1,2] => 8
[4,5,6,3,2,1] => 9
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 4
[4,6,1,3,2,5] => 4
[4,6,1,3,5,2] => 5
[4,6,1,5,2,3] => 8
[4,6,1,5,3,2] => 9
[4,6,2,1,3,5] => 4
[4,6,2,1,5,3] => 5
[4,6,2,3,1,5] => 5
[4,6,2,3,5,1] => 6
[4,6,2,5,1,3] => 9
[4,6,2,5,3,1] => 10
[4,6,3,1,2,5] => 5
[4,6,3,1,5,2] => 6
[4,6,3,2,1,5] => 6
[4,6,3,2,5,1] => 7
[4,6,3,5,1,2] => 10
[4,6,3,5,2,1] => 11
[4,6,5,1,2,3] => 7
[4,6,5,1,3,2] => 8
[4,6,5,2,1,3] => 8
[4,6,5,2,3,1] => 9
[4,6,5,3,1,2] => 9
[4,6,5,3,2,1] => 10
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 6
[5,1,2,4,3,6] => 2
[5,1,2,4,6,3] => 7
[5,1,2,6,3,4] => 5
[5,1,2,6,4,3] => 6
[5,1,3,2,4,6] => 2
[5,1,3,2,6,4] => 7
[5,1,3,4,2,6] => 3
[5,1,3,4,6,2] => 8
[5,1,3,6,2,4] => 6
[5,1,3,6,4,2] => 7
[5,1,4,2,3,6] => 5
[5,1,4,2,6,3] => 10
[5,1,4,3,2,6] => 6
[5,1,4,3,6,2] => 11
[5,1,4,6,2,3] => 9
[5,1,4,6,3,2] => 10
[5,1,6,2,3,4] => 4
[5,1,6,2,4,3] => 5
[5,1,6,3,2,4] => 5
[5,1,6,3,4,2] => 6
[5,1,6,4,2,3] => 6
[5,1,6,4,3,2] => 7
[5,2,1,3,4,6] => 2
[5,2,1,3,6,4] => 7
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 8
[5,2,1,6,3,4] => 6
[5,2,1,6,4,3] => 7
[5,2,3,1,4,6] => 3
[5,2,3,1,6,4] => 8
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 9
[5,2,3,6,1,4] => 7
[5,2,3,6,4,1] => 8
[5,2,4,1,3,6] => 6
[5,2,4,1,6,3] => 11
[5,2,4,3,1,6] => 7
[5,2,4,3,6,1] => 12
[5,2,4,6,1,3] => 10
[5,2,4,6,3,1] => 11
[5,2,6,1,3,4] => 5
[5,2,6,1,4,3] => 6
[5,2,6,3,1,4] => 6
[5,2,6,3,4,1] => 7
[5,2,6,4,1,3] => 7
[5,2,6,4,3,1] => 8
[5,3,1,2,4,6] => 4
[5,3,1,2,6,4] => 9
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 10
[5,3,1,6,2,4] => 8
[5,3,1,6,4,2] => 9
[5,3,2,1,4,6] => 5
[5,3,2,1,6,4] => 10
[5,3,2,4,1,6] => 6
[5,3,2,4,6,1] => 11
[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] => 13
[5,3,4,2,1,6] => 9
[5,3,4,2,6,1] => 14
[5,3,4,6,1,2] => 12
[5,3,4,6,2,1] => 13
[5,3,6,1,2,4] => 7
[5,3,6,1,4,2] => 8
[5,3,6,2,1,4] => 8
[5,3,6,2,4,1] => 9
[5,3,6,4,1,2] => 9
[5,3,6,4,2,1] => 10
[5,4,1,2,3,6] => 4
[5,4,1,2,6,3] => 9
[5,4,1,3,2,6] => 5
[5,4,1,3,6,2] => 10
[5,4,1,6,2,3] => 8
[5,4,1,6,3,2] => 9
[5,4,2,1,3,6] => 5
[5,4,2,1,6,3] => 10
[5,4,2,3,1,6] => 6
[5,4,2,3,6,1] => 11
[5,4,2,6,1,3] => 9
[5,4,2,6,3,1] => 10
[5,4,3,1,2,6] => 6
[5,4,3,1,6,2] => 11
[5,4,3,2,1,6] => 7
[5,4,3,2,6,1] => 12
[5,4,3,6,1,2] => 10
[5,4,3,6,2,1] => 11
[5,4,6,1,2,3] => 7
[5,4,6,1,3,2] => 8
[5,4,6,2,1,3] => 8
[5,4,6,2,3,1] => 9
[5,4,6,3,1,2] => 9
[5,4,6,3,2,1] => 10
[5,6,1,2,3,4] => 3
[5,6,1,2,4,3] => 4
[5,6,1,3,2,4] => 4
[5,6,1,3,4,2] => 5
[5,6,1,4,2,3] => 5
[5,6,1,4,3,2] => 6
[5,6,2,1,3,4] => 4
[5,6,2,1,4,3] => 5
[5,6,2,3,1,4] => 5
[5,6,2,3,4,1] => 6
[5,6,2,4,1,3] => 6
[5,6,2,4,3,1] => 7
[5,6,3,1,2,4] => 5
[5,6,3,1,4,2] => 6
[5,6,3,2,1,4] => 6
[5,6,3,2,4,1] => 7
[5,6,3,4,1,2] => 7
[5,6,3,4,2,1] => 8
[5,6,4,1,2,3] => 8
[5,6,4,1,3,2] => 9
[5,6,4,2,1,3] => 9
[5,6,4,2,3,1] => 10
[5,6,4,3,1,2] => 10
[5,6,4,3,2,1] => 11
[6,1,2,3,4,5] => 1
[6,1,2,3,5,4] => 2
[6,1,2,4,3,5] => 2
[6,1,2,4,5,3] => 3
[6,1,2,5,3,4] => 6
[6,1,2,5,4,3] => 7
[6,1,3,2,4,5] => 2
[6,1,3,2,5,4] => 3
[6,1,3,4,2,5] => 3
[6,1,3,4,5,2] => 4
[6,1,3,5,2,4] => 7
[6,1,3,5,4,2] => 8
[6,1,4,2,3,5] => 5
[6,1,4,2,5,3] => 6
[6,1,4,3,2,5] => 6
[6,1,4,3,5,2] => 7
[6,1,4,5,2,3] => 10
[6,1,4,5,3,2] => 11
[6,1,5,2,3,4] => 5
[6,1,5,2,4,3] => 6
[6,1,5,3,2,4] => 6
[6,1,5,3,4,2] => 7
[6,1,5,4,2,3] => 7
[6,1,5,4,3,2] => 8
[6,2,1,3,4,5] => 2
[6,2,1,3,5,4] => 3
[6,2,1,4,3,5] => 3
[6,2,1,4,5,3] => 4
[6,2,1,5,3,4] => 7
[6,2,1,5,4,3] => 8
[6,2,3,1,4,5] => 3
[6,2,3,1,5,4] => 4
[6,2,3,4,1,5] => 4
[6,2,3,4,5,1] => 5
[6,2,3,5,1,4] => 8
[6,2,3,5,4,1] => 9
[6,2,4,1,3,5] => 6
[6,2,4,1,5,3] => 7
[6,2,4,3,1,5] => 7
[6,2,4,3,5,1] => 8
[6,2,4,5,1,3] => 11
[6,2,4,5,3,1] => 12
[6,2,5,1,3,4] => 6
[6,2,5,1,4,3] => 7
[6,2,5,3,1,4] => 7
[6,2,5,3,4,1] => 8
[6,2,5,4,1,3] => 8
[6,2,5,4,3,1] => 9
[6,3,1,2,4,5] => 4
[6,3,1,2,5,4] => 5
[6,3,1,4,2,5] => 5
[6,3,1,4,5,2] => 6
[6,3,1,5,2,4] => 9
[6,3,1,5,4,2] => 10
[6,3,2,1,4,5] => 5
[6,3,2,1,5,4] => 6
[6,3,2,4,1,5] => 6
[6,3,2,4,5,1] => 7
[6,3,2,5,1,4] => 10
[6,3,2,5,4,1] => 11
[6,3,4,1,2,5] => 8
[6,3,4,1,5,2] => 9
[6,3,4,2,1,5] => 9
[6,3,4,2,5,1] => 10
[6,3,4,5,1,2] => 13
[6,3,4,5,2,1] => 14
[6,3,5,1,2,4] => 8
[6,3,5,1,4,2] => 9
[6,3,5,2,1,4] => 9
[6,3,5,2,4,1] => 10
[6,3,5,4,1,2] => 10
[6,3,5,4,2,1] => 11
[6,4,1,2,3,5] => 4
[6,4,1,2,5,3] => 5
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 6
[6,4,1,5,2,3] => 9
[6,4,1,5,3,2] => 10
[6,4,2,1,3,5] => 5
[6,4,2,1,5,3] => 6
[6,4,2,3,1,5] => 6
[6,4,2,3,5,1] => 7
[6,4,2,5,1,3] => 10
[6,4,2,5,3,1] => 11
[6,4,3,1,2,5] => 6
[6,4,3,1,5,2] => 7
[6,4,3,2,1,5] => 7
[6,4,3,2,5,1] => 8
[6,4,3,5,1,2] => 11
[6,4,3,5,2,1] => 12
[6,4,5,1,2,3] => 8
[6,4,5,1,3,2] => 9
[6,4,5,2,1,3] => 9
[6,4,5,2,3,1] => 10
[6,4,5,3,1,2] => 10
[6,4,5,3,2,1] => 11
[6,5,1,2,3,4] => 4
[6,5,1,2,4,3] => 5
[6,5,1,3,2,4] => 5
[6,5,1,3,4,2] => 6
[6,5,1,4,2,3] => 6
[6,5,1,4,3,2] => 7
[6,5,2,1,3,4] => 5
[6,5,2,1,4,3] => 6
[6,5,2,3,1,4] => 6
[6,5,2,3,4,1] => 7
[6,5,2,4,1,3] => 7
[6,5,2,4,3,1] => 8
[6,5,3,1,2,4] => 6
[6,5,3,1,4,2] => 7
[6,5,3,2,1,4] => 7
[6,5,3,2,4,1] => 8
[6,5,3,4,1,2] => 8
[6,5,3,4,2,1] => 9
[6,5,4,1,2,3] => 9
[6,5,4,1,3,2] => 10
[6,5,4,2,1,3] => 10
[6,5,4,2,3,1] => 11
[6,5,4,3,1,2] => 11
[6,5,4,3,2,1] => 12
click to show generating function       
Description
The Denert index of a permutation.
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
References
[1] Denert, M. The genus zeta function of hereditary orders in central simple algebras over global fields MathSciNet:0993928
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian MathSciNet:1061147
[3] 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
Code
def statistic(pi):
    n = pi.size()
    A = sum( 1 for l,k in Subsets(range(n),2) if pi[k] <  pi[l] <= k+1   )
    B = sum( 1 for l,k in Subsets(range(n),2) if pi[l] <= k+1   <  pi[k] )
    C = sum( 1 for l,k in Subsets(range(n),2) if k+1   <  pi[k] <  pi[l] )
    return A+B+C
Created
Jul 24, 2013 at 12:25 by Christian Stump
Updated
May 31, 2015 at 21:39 by Christian Stump