Identifier
Identifier
Values
[] => 0
[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] => 2
[3,2,1] => 3
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 4
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 4
[2,4,1,3] => 2
[2,4,3,1] => 4
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 3
[3,4,2,1] => 4
[4,1,2,3] => 4
[4,1,3,2] => 4
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 4
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 6
[1,2,4,3,5] => 2
[1,2,4,5,3] => 6
[1,2,5,3,4] => 6
[1,2,5,4,3] => 7
[1,3,2,4,5] => 5
[1,3,2,5,4] => 1
[1,3,4,2,5] => 5
[1,3,4,5,2] => 6
[1,3,5,2,4] => 3
[1,3,5,4,2] => 7
[1,4,2,3,5] => 5
[1,4,2,5,3] => 3
[1,4,3,2,5] => 5
[1,4,3,5,2] => 6
[1,4,5,2,3] => 5
[1,4,5,3,2] => 8
[1,5,2,3,4] => 6
[1,5,2,4,3] => 7
[1,5,3,2,4] => 6
[1,5,3,4,2] => 4
[1,5,4,2,3] => 8
[1,5,4,3,2] => 6
[2,1,3,4,5] => 1
[2,1,3,5,4] => 5
[2,1,4,3,5] => 1
[2,1,4,5,3] => 7
[2,1,5,3,4] => 7
[2,1,5,4,3] => 7
[2,3,1,4,5] => 4
[2,3,1,5,4] => 2
[2,3,4,1,5] => 4
[2,3,4,5,1] => 7
[2,3,5,1,4] => 4
[2,3,5,4,1] => 8
[2,4,1,3,5] => 6
[2,4,1,5,3] => 2
[2,4,3,1,5] => 6
[2,4,3,5,1] => 5
[2,4,5,1,3] => 4
[2,4,5,3,1] => 7
[2,5,1,3,4] => 7
[2,5,1,4,3] => 7
[2,5,3,1,4] => 7
[2,5,3,4,1] => 5
[2,5,4,1,3] => 7
[2,5,4,3,1] => 5
[3,1,2,4,5] => 4
[3,1,2,5,4] => 2
[3,1,4,2,5] => 6
[3,1,4,5,2] => 7
[3,1,5,2,4] => 2
[3,1,5,4,2] => 7
[3,2,1,4,5] => 3
[3,2,1,5,4] => 3
[3,2,4,1,5] => 5
[3,2,4,5,1] => 8
[3,2,5,1,4] => 3
[3,2,5,4,1] => 8
[3,4,1,2,5] => 8
[3,4,1,5,2] => 7
[3,4,2,1,5] => 8
[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] => 3
[3,5,2,1,4] => 7
[3,5,2,4,1] => 4
[3,5,4,1,2] => 5
[3,5,4,2,1] => 6
[4,1,2,3,5] => 4
[4,1,2,5,3] => 4
[4,1,3,2,5] => 6
[4,1,3,5,2] => 7
[4,1,5,2,3] => 4
[4,1,5,3,2] => 7
[4,2,1,3,5] => 5
[4,2,1,5,3] => 3
[4,2,3,1,5] => 7
[4,2,3,5,1] => 6
[4,2,5,1,3] => 3
[4,2,5,3,1] => 6
[4,3,1,2,5] => 8
[4,3,1,5,2] => 7
[4,3,2,1,5] => 7
[4,3,2,5,1] => 7
[4,3,5,1,2] => 7
[4,3,5,2,1] => 7
[4,5,1,2,3] => 7
[4,5,1,3,2] => 5
[4,5,2,1,3] => 7
[4,5,2,3,1] => 4
[4,5,3,1,2] => 7
[4,5,3,2,1] => 6
[5,1,2,3,4] => 7
[5,1,2,4,3] => 8
[5,1,3,2,4] => 5
[5,1,3,4,2] => 5
[5,1,4,2,3] => 7
[5,1,4,3,2] => 5
[5,2,1,3,4] => 8
[5,2,1,4,3] => 8
[5,2,3,1,4] => 6
[5,2,3,4,1] => 6
[5,2,4,1,3] => 6
[5,2,4,3,1] => 6
[5,3,1,2,4] => 6
[5,3,1,4,2] => 4
[5,3,2,1,4] => 7
[5,3,2,4,1] => 5
[5,3,4,1,2] => 4
[5,3,4,2,1] => 5
[5,4,1,2,3] => 6
[5,4,1,3,2] => 6
[5,4,2,1,3] => 7
[5,4,2,3,1] => 5
[5,4,3,1,2] => 6
[5,4,3,2,1] => 5
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 11
[1,2,3,5,4,6] => 6
[1,2,3,5,6,4] => 11
[1,2,3,6,4,5] => 11
[1,2,3,6,5,4] => 9
[1,2,4,3,5,6] => 13
[1,2,4,3,6,5] => 2
[1,2,4,5,3,6] => 11
[1,2,4,5,6,3] => 9
[1,2,4,6,3,5] => 6
[1,2,4,6,5,3] => 13
[1,2,5,3,4,6] => 11
[1,2,5,3,6,4] => 6
[1,2,5,4,3,6] => 7
[1,2,5,4,6,3] => 11
[1,2,5,6,3,4] => 12
[1,2,5,6,4,3] => 16
[1,2,6,3,4,5] => 9
[1,2,6,3,5,4] => 13
[1,2,6,4,3,5] => 11
[1,2,6,4,5,3] => 8
[1,2,6,5,3,4] => 16
[1,2,6,5,4,3] => 12
[1,3,2,4,5,6] => 5
[1,3,2,4,6,5] => 10
[1,3,2,5,4,6] => 1
[1,3,2,5,6,4] => 10
[1,3,2,6,4,5] => 10
[1,3,2,6,5,4] => 12
[1,3,4,2,5,6] => 10
[1,3,4,2,6,5] => 5
[1,3,4,5,2,6] => 6
[1,3,4,5,6,2] => 10
[1,3,4,6,2,5] => 11
[1,3,4,6,5,2] => 15
[1,3,5,2,4,6] => 12
[1,3,5,2,6,4] => 3
[1,3,5,4,2,6] => 10
[1,3,5,4,6,2] => 8
[1,3,5,6,2,4] => 7
[1,3,5,6,4,2] => 12
[1,3,6,2,4,5] => 12
[1,3,6,2,5,4] => 10
[1,3,6,4,2,5] => 15
[1,3,6,4,5,2] => 11
[1,3,6,5,2,4] => 11
[1,3,6,5,4,2] => 9
[1,4,2,3,5,6] => 10
[1,4,2,3,6,5] => 5
[1,4,2,5,3,6] => 12
[1,4,2,5,6,3] => 12
[1,4,2,6,3,5] => 3
[1,4,2,6,5,3] => 10
[1,4,3,2,5,6] => 5
[1,4,3,2,6,5] => 10
[1,4,3,5,2,6] => 11
[1,4,3,5,6,2] => 15
[1,4,3,6,2,5] => 6
[1,4,3,6,5,2] => 10
[1,4,5,2,3,6] => 12
[1,4,5,2,6,3] => 12
[1,4,5,3,2,6] => 15
[1,4,5,3,6,2] => 11
[1,4,5,6,2,3] => 11
[1,4,5,6,3,2] => 11
[1,4,6,2,3,5] => 12
[1,4,6,2,5,3] => 5
[1,4,6,3,2,5] => 10
[1,4,6,3,5,2] => 8
[1,4,6,5,2,3] => 9
[1,4,6,5,3,2] => 12
[1,5,2,3,4,6] => 6
[1,5,2,3,6,4] => 11
[1,5,2,4,3,6] => 10
[1,5,2,4,6,3] => 15
[1,5,2,6,3,4] => 7
[1,5,2,6,4,3] => 11
[1,5,3,2,4,6] => 11
[1,5,3,2,6,4] => 6
[1,5,3,4,2,6] => 11
[1,5,3,4,6,2] => 11
[1,5,3,6,2,4] => 4
[1,5,3,6,4,2] => 9
[1,5,4,2,3,6] => 15
[1,5,4,2,6,3] => 10
[1,5,4,3,2,6] => 11
[1,5,4,3,6,2] => 11
[1,5,4,6,2,3] => 11
[1,5,4,6,3,2] => 12
[1,5,6,2,3,4] => 11
[1,5,6,2,4,3] => 9
[1,5,6,3,2,4] => 11
[1,5,6,3,4,2] => 6
[1,5,6,4,2,3] => 13
[1,5,6,4,3,2] => 10
[1,6,2,3,4,5] => 10
[1,6,2,3,5,4] => 15
[1,6,2,4,3,5] => 8
[1,6,2,4,5,3] => 11
[1,6,2,5,3,4] => 12
[1,6,2,5,4,3] => 9
[1,6,3,2,4,5] => 15
[1,6,3,2,5,4] => 10
[1,6,3,4,2,5] => 11
[1,6,3,4,5,2] => 10
[1,6,3,5,2,4] => 9
[1,6,3,5,4,2] => 12
[1,6,4,2,3,5] => 11
[1,6,4,2,5,3] => 8
[1,6,4,3,2,5] => 11
[1,6,4,3,5,2] => 11
[1,6,4,5,2,3] => 6
[1,6,4,5,3,2] => 9
[1,6,5,2,3,4] => 11
[1,6,5,2,4,3] => 12
[1,6,5,3,2,4] => 12
[1,6,5,3,4,2] => 9
[1,6,5,4,2,3] => 10
[1,6,5,4,3,2] => 7
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 12
[2,1,3,5,4,6] => 5
[2,1,3,5,6,4] => 12
[2,1,3,6,4,5] => 12
[2,1,3,6,5,4] => 10
[2,1,4,3,5,6] => 12
[2,1,4,3,6,5] => 1
[2,1,4,5,3,6] => 10
[2,1,4,5,6,3] => 8
[2,1,4,6,3,5] => 7
[2,1,4,6,5,3] => 12
[2,1,5,3,4,6] => 10
[2,1,5,3,6,4] => 7
[2,1,5,4,3,6] => 8
[2,1,5,4,6,3] => 12
[2,1,5,6,3,4] => 11
[2,1,5,6,4,3] => 15
[2,1,6,3,4,5] => 8
[2,1,6,3,5,4] => 12
[2,1,6,4,3,5] => 12
[2,1,6,4,5,3] => 7
[2,1,6,5,3,4] => 15
[2,1,6,5,4,3] => 13
[2,3,1,4,5,6] => 4
[2,3,1,4,6,5] => 11
[2,3,1,5,4,6] => 2
[2,3,1,5,6,4] => 11
[2,3,1,6,4,5] => 11
[2,3,1,6,5,4] => 13
[2,3,4,1,5,6] => 11
[2,3,4,1,6,5] => 4
[2,3,4,5,1,6] => 7
[2,3,4,5,6,1] => 9
[2,3,4,6,1,5] => 10
[2,3,4,6,5,1] => 14
[2,3,5,1,4,6] => 11
[2,3,5,1,6,4] => 4
[2,3,5,4,1,6] => 9
[2,3,5,4,6,1] => 9
[2,3,5,6,1,4] => 8
[2,3,5,6,4,1] => 13
[2,3,6,1,4,5] => 11
[2,3,6,1,5,4] => 11
[2,3,6,4,1,5] => 14
[2,3,6,4,5,1] => 10
[2,3,6,5,1,4] => 12
[2,3,6,5,4,1] => 10
[2,4,1,3,5,6] => 9
[2,4,1,3,6,5] => 6
[2,4,1,5,3,6] => 11
[2,4,1,5,6,3] => 13
[2,4,1,6,3,5] => 2
[2,4,1,6,5,3] => 9
[2,4,3,1,5,6] => 6
[2,4,3,1,6,5] => 9
[2,4,3,5,1,6] => 12
[2,4,3,5,6,1] => 14
[2,4,3,6,1,5] => 5
[2,4,3,6,5,1] => 10
[2,4,5,1,3,6] => 13
[2,4,5,1,6,3] => 11
[2,4,5,3,1,6] => 14
[2,4,5,3,6,1] => 12
[2,4,5,6,1,3] => 10
[2,4,5,6,3,1] => 10
[2,4,6,1,3,5] => 11
[2,4,6,1,5,3] => 4
[2,4,6,3,1,5] => 11
[2,4,6,3,5,1] => 7
[2,4,6,5,1,3] => 8
[2,4,6,5,3,1] => 11
[2,5,1,3,4,6] => 7
[2,5,1,3,6,4] => 10
[2,5,1,4,3,6] => 9
[2,5,1,4,6,3] => 14
[2,5,1,6,3,4] => 8
[2,5,1,6,4,3] => 12
[2,5,3,1,4,6] => 10
[2,5,3,1,6,4] => 7
[2,5,3,4,1,6] => 10
[2,5,3,4,6,1] => 12
[2,5,3,6,1,4] => 5
[2,5,3,6,4,1] => 10
[2,5,4,1,3,6] => 14
[2,5,4,1,6,3] => 9
[2,5,4,3,1,6] => 12
[2,5,4,3,6,1] => 10
[2,5,4,6,1,3] => 10
[2,5,4,6,3,1] => 13
[2,5,6,1,3,4] => 12
[2,5,6,1,4,3] => 10
[2,5,6,3,1,4] => 11
[2,5,6,3,4,1] => 7
[2,5,6,4,1,3] => 13
[2,5,6,4,3,1] => 10
[2,6,1,3,4,5] => 11
[2,6,1,3,5,4] => 14
[2,6,1,4,3,5] => 7
[2,6,1,4,5,3] => 12
[2,6,1,5,3,4] => 12
[2,6,1,5,4,3] => 8
[2,6,3,1,4,5] => 14
[2,6,3,1,5,4] => 11
[2,6,3,4,1,5] => 10
[2,6,3,4,5,1] => 11
[2,6,3,5,1,4] => 10
[2,6,3,5,4,1] => 11
[2,6,4,1,3,5] => 12
[2,6,4,1,5,3] => 7
[2,6,4,3,1,5] => 12
[2,6,4,3,5,1] => 10
[2,6,4,5,1,3] => 5
[2,6,4,5,3,1] => 8
[2,6,5,1,3,4] => 10
[2,6,5,1,4,3] => 12
[2,6,5,3,1,4] => 13
[2,6,5,3,4,1] => 10
[2,6,5,4,1,3] => 10
[2,6,5,4,3,1] => 7
[3,1,2,4,5,6] => 4
[3,1,2,4,6,5] => 11
[3,1,2,5,4,6] => 2
[3,1,2,5,6,4] => 11
[3,1,2,6,4,5] => 11
[3,1,2,6,5,4] => 13
[3,1,4,2,5,6] => 9
[3,1,4,2,6,5] => 6
[3,1,4,5,2,6] => 7
[3,1,4,5,6,2] => 11
[3,1,4,6,2,5] => 10
[3,1,4,6,5,2] => 14
[3,1,5,2,4,6] => 11
[3,1,5,2,6,4] => 2
[3,1,5,4,2,6] => 9
[3,1,5,4,6,2] => 7
[3,1,5,6,2,4] => 8
[3,1,5,6,4,2] => 12
[3,1,6,2,4,5] => 13
[3,1,6,2,5,4] => 9
[3,1,6,4,2,5] => 14
[3,1,6,4,5,2] => 12
[3,1,6,5,2,4] => 12
[3,1,6,5,4,2] => 8
[3,2,1,4,5,6] => 3
[3,2,1,4,6,5] => 12
[3,2,1,5,4,6] => 3
[3,2,1,5,6,4] => 12
[3,2,1,6,4,5] => 12
[3,2,1,6,5,4] => 12
[3,2,4,1,5,6] => 10
[3,2,4,1,6,5] => 5
[3,2,4,5,1,6] => 8
[3,2,4,5,6,1] => 10
[3,2,4,6,1,5] => 9
[3,2,4,6,5,1] => 14
[3,2,5,1,4,6] => 12
[3,2,5,1,6,4] => 3
[3,2,5,4,1,6] => 8
[3,2,5,4,6,1] => 8
[3,2,5,6,1,4] => 9
[3,2,5,6,4,1] => 13
[3,2,6,1,4,5] => 12
[3,2,6,1,5,4] => 10
[3,2,6,4,1,5] => 13
[3,2,6,4,5,1] => 11
[3,2,6,5,1,4] => 13
[3,2,6,5,4,1] => 9
[3,4,1,2,5,6] => 8
[3,4,1,2,6,5] => 9
[3,4,1,5,2,6] => 8
[3,4,1,5,6,2] => 13
[3,4,1,6,2,5] => 7
[3,4,1,6,5,2] => 12
[3,4,2,1,5,6] => 9
[3,4,2,1,6,5] => 8
[3,4,2,5,1,6] => 9
[3,4,2,5,6,1] => 13
[3,4,2,6,1,5] => 6
[3,4,2,6,5,1] => 11
[3,4,5,1,2,6] => 13
[3,4,5,1,6,2] => 8
[3,4,5,2,1,6] => 13
[3,4,5,2,6,1] => 9
[3,4,5,6,1,2] => 11
[3,4,5,6,2,1] => 12
[3,4,6,1,2,5] => 13
[3,4,6,1,5,2] => 9
[3,4,6,2,1,5] => 12
[3,4,6,2,5,1] => 8
[3,4,6,5,1,2] => 12
[3,4,6,5,2,1] => 11
[3,5,1,2,4,6] => 8
[3,5,1,2,6,4] => 7
[3,5,1,4,2,6] => 10
[3,5,1,4,6,2] => 12
[3,5,1,6,2,4] => 3
[3,5,1,6,4,2] => 8
[3,5,2,1,4,6] => 7
[3,5,2,1,6,4] => 8
[3,5,2,4,1,6] => 11
[3,5,2,4,6,1] => 13
[3,5,2,6,1,4] => 4
[3,5,2,6,4,1] => 9
[3,5,4,1,2,6] => 12
[3,5,4,1,6,2] => 10
[3,5,4,2,1,6] => 13
[3,5,4,2,6,1] => 11
[3,5,4,6,1,2] => 9
[3,5,4,6,2,1] => 10
[3,5,6,1,2,4] => 10
[3,5,6,1,4,2] => 5
[3,5,6,2,1,4] => 11
[3,5,6,2,4,1] => 6
[3,5,6,4,1,2] => 9
[3,5,6,4,2,1] => 10
[3,6,1,2,4,5] => 13
[3,6,1,2,5,4] => 12
[3,6,1,4,2,5] => 12
[3,6,1,4,5,2] => 9
[3,6,1,5,2,4] => 8
[3,6,1,5,4,2] => 11
[3,6,2,1,4,5] => 12
[3,6,2,1,5,4] => 13
[3,6,2,4,1,5] => 11
[3,6,2,4,5,1] => 10
[3,6,2,5,1,4] => 9
[3,6,2,5,4,1] => 12
[3,6,4,1,2,5] => 11
[3,6,4,1,5,2] => 12
[3,6,4,2,1,5] => 12
[3,6,4,2,5,1] => 11
[3,6,4,5,1,2] => 9
[3,6,4,5,2,1] => 8
[3,6,5,1,2,4] => 11
[3,6,5,1,4,2] => 8
[3,6,5,2,1,4] => 11
[3,6,5,2,4,1] => 9
[3,6,5,4,1,2] => 6
[3,6,5,4,2,1] => 7
[4,1,2,3,5,6] => 11
[4,1,2,3,6,5] => 4
[4,1,2,5,3,6] => 11
[4,1,2,5,6,3] => 11
[4,1,2,6,3,5] => 4
[4,1,2,6,5,3] => 11
[4,1,3,2,5,6] => 6
[4,1,3,2,6,5] => 9
[4,1,3,5,2,6] => 10
[4,1,3,5,6,2] => 14
[4,1,3,6,2,5] => 7
[4,1,3,6,5,2] => 11
[4,1,5,2,3,6] => 13
[4,1,5,2,6,3] => 11
[4,1,5,3,2,6] => 14
[4,1,5,3,6,2] => 12
[4,1,5,6,2,3] => 12
[4,1,5,6,3,2] => 10
[4,1,6,2,3,5] => 11
[4,1,6,2,5,3] => 4
[4,1,6,3,2,5] => 9
[4,1,6,3,5,2] => 7
[4,1,6,5,2,3] => 10
[4,1,6,5,3,2] => 12
[4,2,1,3,5,6] => 10
[4,2,1,3,6,5] => 5
[4,2,1,5,3,6] => 12
[4,2,1,5,6,3] => 12
[4,2,1,6,3,5] => 3
[4,2,1,6,5,3] => 10
[4,2,3,1,5,6] => 7
[4,2,3,1,6,5] => 8
[4,2,3,5,1,6] => 11
[4,2,3,5,6,1] => 13
[4,2,3,6,1,5] => 6
[4,2,3,6,5,1] => 11
[4,2,5,1,3,6] => 13
[4,2,5,1,6,3] => 10
[4,2,5,3,1,6] => 14
[4,2,5,3,6,1] => 11
[4,2,5,6,1,3] => 11
[4,2,5,6,3,1] => 9
[4,2,6,1,3,5] => 10
[4,2,6,1,5,3] => 3
[4,2,6,3,1,5] => 10
[4,2,6,3,5,1] => 6
[4,2,6,5,1,3] => 9
[4,2,6,5,3,1] => 11
[4,3,1,2,5,6] => 9
[4,3,1,2,6,5] => 8
[4,3,1,5,2,6] => 7
[4,3,1,5,6,2] => 12
[4,3,1,6,2,5] => 8
[4,3,1,6,5,2] => 13
[4,3,2,1,5,6] => 10
[4,3,2,1,6,5] => 7
[4,3,2,5,1,6] => 8
[4,3,2,5,6,1] => 12
[4,3,2,6,1,5] => 7
[4,3,2,6,5,1] => 12
[4,3,5,1,2,6] => 13
[4,3,5,1,6,2] => 9
[4,3,5,2,1,6] => 14
[4,3,5,2,6,1] => 10
[4,3,5,6,1,2] => 12
[4,3,5,6,2,1] => 13
[4,3,6,1,2,5] => 12
[4,3,6,1,5,2] => 8
[4,3,6,2,1,5] => 12
[4,3,6,2,5,1] => 7
[4,3,6,5,1,2] => 11
[4,3,6,5,2,1] => 10
[4,5,1,2,3,6] => 13
[4,5,1,2,6,3] => 13
[4,5,1,3,2,6] => 12
[4,5,1,3,6,2] => 11
[4,5,1,6,2,3] => 10
[4,5,1,6,3,2] => 11
[4,5,2,1,3,6] => 13
[4,5,2,1,6,3] => 12
[4,5,2,3,1,6] => 12
[4,5,2,3,6,1] => 10
[4,5,2,6,1,3] => 9
[4,5,2,6,3,1] => 10
[4,5,3,1,2,6] => 13
[4,5,3,1,6,2] => 12
[4,5,3,2,1,6] => 12
[4,5,3,2,6,1] => 12
[4,5,3,6,1,2] => 9
[4,5,3,6,2,1] => 10
[4,5,6,1,2,3] => 10
[4,5,6,1,3,2] => 10
[4,5,6,2,1,3] => 10
[4,5,6,2,3,1] => 9
[4,5,6,3,1,2] => 8
[4,5,6,3,2,1] => 7
[4,6,1,2,3,5] => 8
[4,6,1,2,5,3] => 9
[4,6,1,3,2,5] => 10
[4,6,1,3,5,2] => 12
[4,6,1,5,2,3] => 5
[4,6,1,5,3,2] => 8
[4,6,2,1,3,5] => 9
[4,6,2,1,5,3] => 8
[4,6,2,3,1,5] => 9
[4,6,2,3,5,1] => 11
[4,6,2,5,1,3] => 4
[4,6,2,5,3,1] => 7
[4,6,3,1,2,5] => 12
[4,6,3,1,5,2] => 10
[4,6,3,2,1,5] => 11
[4,6,3,2,5,1] => 9
[4,6,3,5,1,2] => 9
[4,6,3,5,2,1] => 8
[4,6,5,1,2,3] => 10
[4,6,5,1,3,2] => 7
[4,6,5,2,1,3] => 9
[4,6,5,2,3,1] => 6
[4,6,5,3,1,2] => 10
[4,6,5,3,2,1] => 10
[5,1,2,3,4,6] => 7
[5,1,2,3,6,4] => 10
[5,1,2,4,3,6] => 9
[5,1,2,4,6,3] => 14
[5,1,2,6,3,4] => 8
[5,1,2,6,4,3] => 12
[5,1,3,2,4,6] => 12
[5,1,3,2,6,4] => 5
[5,1,3,4,2,6] => 10
[5,1,3,4,6,2] => 10
[5,1,3,6,2,4] => 5
[5,1,3,6,4,2] => 10
[5,1,4,2,3,6] => 14
[5,1,4,2,6,3] => 11
[5,1,4,3,2,6] => 12
[5,1,4,3,6,2] => 12
[5,1,4,6,2,3] => 11
[5,1,4,6,3,2] => 13
[5,1,6,2,3,4] => 10
[5,1,6,2,4,3] => 8
[5,1,6,3,2,4] => 10
[5,1,6,3,4,2] => 5
[5,1,6,4,2,3] => 13
[5,1,6,4,3,2] => 10
[5,2,1,3,4,6] => 8
[5,2,1,3,6,4] => 9
[5,2,1,4,3,6] => 8
[5,2,1,4,6,3] => 13
[5,2,1,6,3,4] => 9
[5,2,1,6,4,3] => 13
[5,2,3,1,4,6] => 11
[5,2,3,1,6,4] => 6
[5,2,3,4,1,6] => 9
[5,2,3,4,6,1] => 11
[5,2,3,6,1,4] => 6
[5,2,3,6,4,1] => 11
[5,2,4,1,3,6] => 14
[5,2,4,1,6,3] => 10
[5,2,4,3,1,6] => 13
[5,2,4,3,6,1] => 11
[5,2,4,6,1,3] => 11
[5,2,4,6,3,1] => 12
[5,2,6,1,3,4] => 11
[5,2,6,1,4,3] => 9
[5,2,6,3,1,4] => 11
[5,2,6,3,4,1] => 6
[5,2,6,4,1,3] => 12
[5,2,6,4,3,1] => 9
[5,3,1,2,4,6] => 9
[5,3,1,2,6,4] => 6
[5,3,1,4,2,6] => 11
[5,3,1,4,6,2] => 11
[5,3,1,6,2,4] => 4
[5,3,1,6,4,2] => 9
[5,3,2,1,4,6] => 8
[5,3,2,1,6,4] => 7
[5,3,2,4,1,6] => 12
[5,3,2,4,6,1] => 12
[5,3,2,6,1,4] => 5
[5,3,2,6,4,1] => 10
[5,3,4,1,2,6] => 12
[5,3,4,1,6,2] => 9
[5,3,4,2,1,6] => 13
[5,3,4,2,6,1] => 10
[5,3,4,6,1,2] => 10
[5,3,4,6,2,1] => 10
[5,3,6,1,2,4] => 9
[5,3,6,1,4,2] => 4
[5,3,6,2,1,4] => 10
[5,3,6,2,4,1] => 5
[5,3,6,4,1,2] => 10
[5,3,6,4,2,1] => 10
[5,4,1,2,3,6] => 13
[5,4,1,2,6,3] => 12
[5,4,1,3,2,6] => 13
[5,4,1,3,6,2] => 12
[5,4,1,6,2,3] => 11
[5,4,1,6,3,2] => 11
[5,4,2,1,3,6] => 14
[5,4,2,1,6,3] => 12
[5,4,2,3,1,6] => 13
[5,4,2,3,6,1] => 11
[5,4,2,6,1,3] => 10
[5,4,2,6,3,1] => 11
[5,4,3,1,2,6] => 12
[5,4,3,1,6,2] => 11
[5,4,3,2,1,6] => 11
[5,4,3,2,6,1] => 11
[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] => 9
[5,4,6,2,1,3] => 9
[5,4,6,2,3,1] => 8
[5,4,6,3,1,2] => 9
[5,4,6,3,2,1] => 8
[5,6,1,2,3,4] => 11
[5,6,1,2,4,3] => 12
[5,6,1,3,2,4] => 9
[5,6,1,3,4,2] => 9
[5,6,1,4,2,3] => 9
[5,6,1,4,3,2] => 6
[5,6,2,1,3,4] => 12
[5,6,2,1,4,3] => 11
[5,6,2,3,1,4] => 10
[5,6,2,3,4,1] => 8
[5,6,2,4,1,3] => 10
[5,6,2,4,3,1] => 7
[5,6,3,1,2,4] => 9
[5,6,3,1,4,2] => 9
[5,6,3,2,1,4] => 8
[5,6,3,2,4,1] => 10
[5,6,3,4,1,2] => 5
[5,6,3,4,2,1] => 6
[5,6,4,1,2,3] => 8
[5,6,4,1,3,2] => 10
[5,6,4,2,1,3] => 9
[5,6,4,2,3,1] => 9
[5,6,4,3,1,2] => 8
[5,6,4,3,2,1] => 7
[6,1,2,3,4,5] => 9
[6,1,2,3,5,4] => 14
[6,1,2,4,3,5] => 9
[6,1,2,4,5,3] => 10
[6,1,2,5,3,4] => 13
[6,1,2,5,4,3] => 10
[6,1,3,2,4,5] => 14
[6,1,3,2,5,4] => 10
[6,1,3,4,2,5] => 12
[6,1,3,4,5,2] => 11
[6,1,3,5,2,4] => 10
[6,1,3,5,4,2] => 11
[6,1,4,2,3,5] => 12
[6,1,4,2,5,3] => 7
[6,1,4,3,2,5] => 10
[6,1,4,3,5,2] => 10
[6,1,4,5,2,3] => 7
[6,1,4,5,3,2] => 10
[6,1,5,2,3,4] => 10
[6,1,5,2,4,3] => 11
[6,1,5,3,2,4] => 13
[6,1,5,3,4,2] => 8
[6,1,5,4,2,3] => 10
[6,1,5,4,3,2] => 7
[6,2,1,3,4,5] => 10
[6,2,1,3,5,4] => 14
[6,2,1,4,3,5] => 8
[6,2,1,4,5,3] => 11
[6,2,1,5,3,4] => 13
[6,2,1,5,4,3] => 9
[6,2,3,1,4,5] => 13
[6,2,3,1,5,4] => 11
[6,2,3,4,1,5] => 11
[6,2,3,4,5,1] => 12
[6,2,3,5,1,4] => 11
[6,2,3,5,4,1] => 12
[6,2,4,1,3,5] => 11
[6,2,4,1,5,3] => 6
[6,2,4,3,1,5] => 11
[6,2,4,3,5,1] => 9
[6,2,4,5,1,3] => 6
[6,2,4,5,3,1] => 9
[6,2,5,1,3,4] => 9
[6,2,5,1,4,3] => 11
[6,2,5,3,1,4] => 12
[6,2,5,3,4,1] => 9
[6,2,5,4,1,3] => 9
[6,2,5,4,3,1] => 6
[6,3,1,2,4,5] => 13
[6,3,1,2,5,4] => 11
[6,3,1,4,2,5] => 13
[6,3,1,4,5,2] => 10
[6,3,1,5,2,4] => 9
[6,3,1,5,4,2] => 12
[6,3,2,1,4,5] => 12
[6,3,2,1,5,4] => 12
[6,3,2,4,1,5] => 12
[6,3,2,4,5,1] => 11
[6,3,2,5,1,4] => 10
[6,3,2,5,4,1] => 12
[6,3,4,1,2,5] => 10
[6,3,4,1,5,2] => 11
[6,3,4,2,1,5] => 11
[6,3,4,2,5,1] => 10
[6,3,4,5,1,2] => 8
[6,3,4,5,2,1] => 7
[6,3,5,1,2,4] => 10
[6,3,5,1,4,2] => 7
[6,3,5,2,1,4] => 11
[6,3,5,2,4,1] => 8
[6,3,5,4,1,2] => 7
[6,3,5,4,2,1] => 8
[6,4,1,2,3,5] => 9
[6,4,1,2,5,3] => 8
[6,4,1,3,2,5] => 11
[6,4,1,3,5,2] => 11
[6,4,1,5,2,3] => 6
[6,4,1,5,3,2] => 9
[6,4,2,1,3,5] => 10
[6,4,2,1,5,3] => 7
[6,4,2,3,1,5] => 10
[6,4,2,3,5,1] => 10
[6,4,2,5,1,3] => 5
[6,4,2,5,3,1] => 8
[6,4,3,1,2,5] => 12
[6,4,3,1,5,2] => 9
[6,4,3,2,1,5] => 11
[6,4,3,2,5,1] => 8
[6,4,3,5,1,2] => 10
[6,4,3,5,2,1] => 9
[6,4,5,1,2,3] => 9
[6,4,5,1,3,2] => 6
[6,4,5,2,1,3] => 8
[6,4,5,2,3,1] => 5
[6,4,5,3,1,2] => 9
[6,4,5,3,2,1] => 10
[6,5,1,2,3,4] => 12
[6,5,1,2,4,3] => 11
[6,5,1,3,2,4] => 10
[6,5,1,3,4,2] => 8
[6,5,1,4,2,3] => 10
[6,5,1,4,3,2] => 7
[6,5,2,1,3,4] => 13
[6,5,2,1,4,3] => 10
[6,5,2,3,1,4] => 10
[6,5,2,3,4,1] => 7
[6,5,2,4,1,3] => 10
[6,5,2,4,3,1] => 8
[6,5,3,1,2,4] => 10
[6,5,3,1,4,2] => 8
[6,5,3,2,1,4] => 9
[6,5,3,2,4,1] => 9
[6,5,3,4,1,2] => 6
[6,5,3,4,2,1] => 7
[6,5,4,1,2,3] => 7
[6,5,4,1,3,2] => 10
[6,5,4,2,1,3] => 8
[6,5,4,2,3,1] => 10
[6,5,4,3,1,2] => 7
[6,5,4,3,2,1] => 6
click to show generating function       
Description
The minimal length of a factorization of a permutation using the permutations (12)(34)..., (23)(45)..., and (12).
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k} \},$$
where, with $m_1$ the largest even number at most $n$ and $m_2$ the largest odd number at most $n$, each factor $\tau_i$ is one of the three permutations $(1,2)(3,4)\cdots(m_1-1,m_1)$ or $(2,3)(4,5)\cdots(m_2-1,m_2)$ or $(1,2)$.
References
[1] Pak, I. "Natural" generating sets for symmetric groups. MathOverflow:24128
Code
def statistic(pi):
    def gens(n):
        n1 = n if is_even(n) else n-1
        n2 = n-2 if is_even(n) else n-1
        return [Permutation([i+2 if is_even(i) else i for i in range(n1)]),
                Permutation([1] + [i+3 if is_even(i) else i+1 for i in range(n2)]), Permutation([2,1])]

    G = PermutationGroup(gens(len(pi)))
    pi = G(pi)
    w = gap.Factorization(G._gap_(), pi._gap_())
    return w.Length()

Created
Jan 08, 2018 at 23:38 by Martin Rubey
Updated
Jan 09, 2018 at 19:40 by Martin Rubey