Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 3
[2,3,4,1] => 4
[2,4,1,3] => 5
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 5
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 6
[3,4,2,1] => 5
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 5
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 4
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[1,3,5,2,4] => 5
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 5
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 6
[1,4,5,3,2] => 5
[1,5,2,3,4] => 4
[1,5,2,4,3] => 3
[1,5,3,2,4] => 3
[1,5,3,4,2] => 2
[1,5,4,2,3] => 5
[1,5,4,3,2] => 4
[2,1,3,4,5] => 2
[2,1,3,5,4] => 4
[2,1,4,3,5] => 4
[2,1,4,5,3] => 6
[2,1,5,3,4] => 6
[2,1,5,4,3] => 4
[2,3,1,4,5] => 3
[2,3,1,5,4] => 6
[2,3,4,1,5] => 4
[2,3,4,5,1] => 5
[2,3,5,1,4] => 7
[2,3,5,4,1] => 4
[2,4,1,3,5] => 5
[2,4,1,5,3] => 8
[2,4,3,1,5] => 3
[2,4,3,5,1] => 4
[2,4,5,1,3] => 9
[2,4,5,3,1] => 7
[2,5,1,3,4] => 7
[2,5,1,4,3] => 5
[2,5,3,1,4] => 5
[2,5,3,4,1] => 3
[2,5,4,1,3] => 8
[2,5,4,3,1] => 6
[3,1,2,4,5] => 3
[3,1,2,5,4] => 6
[3,1,4,2,5] => 5
[3,1,4,5,2] => 7
[3,1,5,2,4] => 8
[3,1,5,4,2] => 5
[3,2,1,4,5] => 2
[3,2,1,5,4] => 4
[3,2,4,1,5] => 3
[3,2,4,5,1] => 4
[3,2,5,1,4] => 5
[3,2,5,4,1] => 3
[3,4,1,2,5] => 6
[3,4,1,5,2] => 9
[3,4,2,1,5] => 5
[3,4,2,5,1] => 7
[3,4,5,1,2] => 10
[3,4,5,2,1] => 9
[3,5,1,2,4] => 9
[3,5,1,4,2] => 6
[3,5,2,1,4] => 8
[3,5,2,4,1] => 5
[3,5,4,1,2] => 9
[3,5,4,2,1] => 8
[4,1,2,3,5] => 4
[4,1,2,5,3] => 7
[4,1,3,2,5] => 3
[4,1,3,5,2] => 5
[4,1,5,2,3] => 9
[4,1,5,3,2] => 8
[4,2,1,3,5] => 3
[4,2,1,5,3] => 5
[4,2,3,1,5] => 2
[4,2,3,5,1] => 3
[4,2,5,1,3] => 6
[4,2,5,3,1] => 5
[4,3,1,2,5] => 5
[4,3,1,5,2] => 8
[4,3,2,1,5] => 4
[4,3,2,5,1] => 6
[4,3,5,1,2] => 9
[4,3,5,2,1] => 8
[4,5,1,2,3] => 10
[4,5,1,3,2] => 9
[4,5,2,1,3] => 9
[4,5,2,3,1] => 7
[4,5,3,1,2] => 6
[4,5,3,2,1] => 5
[5,1,2,3,4] => 5
[5,1,2,4,3] => 4
[5,1,3,2,4] => 4
[5,1,3,4,2] => 3
[5,1,4,2,3] => 7
[5,1,4,3,2] => 6
[5,2,1,3,4] => 4
[5,2,1,4,3] => 3
[5,2,3,1,4] => 3
[5,2,3,4,1] => 2
[5,2,4,1,3] => 5
[5,2,4,3,1] => 4
[5,3,1,2,4] => 7
[5,3,1,4,2] => 5
[5,3,2,1,4] => 6
[5,3,2,4,1] => 4
[5,3,4,1,2] => 7
[5,3,4,2,1] => 6
[5,4,1,2,3] => 9
[5,4,1,3,2] => 8
[5,4,2,1,3] => 8
[5,4,2,3,1] => 6
[5,4,3,1,2] => 5
[5,4,3,2,1] => 4
[1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => 2
[1,2,3,5,4,6] => 2
[1,2,3,5,6,4] => 3
[1,2,3,6,4,5] => 3
[1,2,3,6,5,4] => 2
[1,2,4,3,5,6] => 2
[1,2,4,3,6,5] => 4
[1,2,4,5,3,6] => 3
[1,2,4,5,6,3] => 4
[1,2,4,6,3,5] => 5
[1,2,4,6,5,3] => 3
[1,2,5,3,4,6] => 3
[1,2,5,3,6,4] => 5
[1,2,5,4,3,6] => 2
[1,2,5,4,6,3] => 3
[1,2,5,6,3,4] => 6
[1,2,5,6,4,3] => 5
[1,2,6,3,4,5] => 4
[1,2,6,3,5,4] => 3
[1,2,6,4,3,5] => 3
[1,2,6,4,5,3] => 2
[1,2,6,5,3,4] => 5
[1,2,6,5,4,3] => 4
[1,3,2,4,5,6] => 2
[1,3,2,4,6,5] => 4
[1,3,2,5,4,6] => 4
[1,3,2,5,6,4] => 6
[1,3,2,6,4,5] => 6
[1,3,2,6,5,4] => 4
[1,3,4,2,5,6] => 3
[1,3,4,2,6,5] => 6
[1,3,4,5,2,6] => 4
[1,3,4,5,6,2] => 5
[1,3,4,6,2,5] => 7
[1,3,4,6,5,2] => 4
[1,3,5,2,4,6] => 5
[1,3,5,2,6,4] => 8
[1,3,5,4,2,6] => 3
[1,3,5,4,6,2] => 4
[1,3,5,6,2,4] => 9
[1,3,5,6,4,2] => 7
[1,3,6,2,4,5] => 7
[1,3,6,2,5,4] => 5
[1,3,6,4,2,5] => 5
[1,3,6,4,5,2] => 3
[1,3,6,5,2,4] => 8
[1,3,6,5,4,2] => 6
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 6
[1,4,2,5,3,6] => 5
[1,4,2,5,6,3] => 7
[1,4,2,6,3,5] => 8
[1,4,2,6,5,3] => 5
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 4
[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] => 3
[1,4,5,2,3,6] => 6
[1,4,5,2,6,3] => 9
[1,4,5,3,2,6] => 5
[1,4,5,3,6,2] => 7
[1,4,5,6,2,3] => 10
[1,4,5,6,3,2] => 9
[1,4,6,2,3,5] => 9
[1,4,6,2,5,3] => 6
[1,4,6,3,2,5] => 8
[1,4,6,3,5,2] => 5
[1,4,6,5,2,3] => 9
[1,4,6,5,3,2] => 8
[1,5,2,3,4,6] => 4
[1,5,2,3,6,4] => 7
[1,5,2,4,3,6] => 3
[1,5,2,4,6,3] => 5
[1,5,2,6,3,4] => 9
[1,5,2,6,4,3] => 8
[1,5,3,2,4,6] => 3
[1,5,3,2,6,4] => 5
[1,5,3,4,2,6] => 2
[1,5,3,4,6,2] => 3
[1,5,3,6,2,4] => 6
[1,5,3,6,4,2] => 5
[1,5,4,2,3,6] => 5
[1,5,4,2,6,3] => 8
[1,5,4,3,2,6] => 4
[1,5,4,3,6,2] => 6
[1,5,4,6,2,3] => 9
[1,5,4,6,3,2] => 8
[1,5,6,2,3,4] => 10
[1,5,6,2,4,3] => 9
[1,5,6,3,2,4] => 9
[1,5,6,3,4,2] => 7
[1,5,6,4,2,3] => 6
[1,5,6,4,3,2] => 5
[1,6,2,3,4,5] => 5
[1,6,2,3,5,4] => 4
[1,6,2,4,3,5] => 4
[1,6,2,4,5,3] => 3
[1,6,2,5,3,4] => 7
[1,6,2,5,4,3] => 6
[1,6,3,2,4,5] => 4
[1,6,3,2,5,4] => 3
[1,6,3,4,2,5] => 3
[1,6,3,4,5,2] => 2
[1,6,3,5,2,4] => 5
[1,6,3,5,4,2] => 4
[1,6,4,2,3,5] => 7
[1,6,4,2,5,3] => 5
[1,6,4,3,2,5] => 6
[1,6,4,3,5,2] => 4
[1,6,4,5,2,3] => 7
[1,6,4,5,3,2] => 6
[1,6,5,2,3,4] => 9
[1,6,5,2,4,3] => 8
[1,6,5,3,2,4] => 8
[1,6,5,3,4,2] => 6
[1,6,5,4,2,3] => 5
[1,6,5,4,3,2] => 4
[2,1,3,4,5,6] => 2
[2,1,3,4,6,5] => 4
[2,1,3,5,4,6] => 4
[2,1,3,5,6,4] => 6
[2,1,3,6,4,5] => 6
[2,1,3,6,5,4] => 4
[2,1,4,3,5,6] => 4
[2,1,4,3,6,5] => 8
[2,1,4,5,3,6] => 6
[2,1,4,5,6,3] => 8
[2,1,4,6,3,5] => 10
[2,1,4,6,5,3] => 6
[2,1,5,3,4,6] => 6
[2,1,5,3,6,4] => 10
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 6
[2,1,5,6,3,4] => 12
[2,1,5,6,4,3] => 10
[2,1,6,3,4,5] => 8
[2,1,6,3,5,4] => 6
[2,1,6,4,3,5] => 6
[2,1,6,4,5,3] => 4
[2,1,6,5,3,4] => 10
[2,1,6,5,4,3] => 8
[2,3,1,4,5,6] => 3
[2,3,1,4,6,5] => 6
[2,3,1,5,4,6] => 6
[2,3,1,5,6,4] => 9
[2,3,1,6,4,5] => 9
[2,3,1,6,5,4] => 6
[2,3,4,1,5,6] => 4
[2,3,4,1,6,5] => 8
[2,3,4,5,1,6] => 5
[2,3,4,5,6,1] => 6
[2,3,4,6,1,5] => 9
[2,3,4,6,5,1] => 5
[2,3,5,1,4,6] => 7
[2,3,5,1,6,4] => 11
[2,3,5,4,1,6] => 4
[2,3,5,4,6,1] => 5
[2,3,5,6,1,4] => 12
[2,3,5,6,4,1] => 9
[2,3,6,1,4,5] => 10
[2,3,6,1,5,4] => 7
[2,3,6,4,1,5] => 7
[2,3,6,4,5,1] => 4
[2,3,6,5,1,4] => 11
[2,3,6,5,4,1] => 8
[2,4,1,3,5,6] => 5
[2,4,1,3,6,5] => 10
[2,4,1,5,3,6] => 8
[2,4,1,5,6,3] => 11
[2,4,1,6,3,5] => 13
[2,4,1,6,5,3] => 8
[2,4,3,1,5,6] => 3
[2,4,3,1,6,5] => 6
[2,4,3,5,1,6] => 4
[2,4,3,5,6,1] => 5
[2,4,3,6,1,5] => 7
[2,4,3,6,5,1] => 4
[2,4,5,1,3,6] => 9
[2,4,5,1,6,3] => 13
[2,4,5,3,1,6] => 7
[2,4,5,3,6,1] => 9
[2,4,5,6,1,3] => 14
[2,4,5,6,3,1] => 12
[2,4,6,1,3,5] => 14
[2,4,6,1,5,3] => 9
[2,4,6,3,1,5] => 12
[2,4,6,3,5,1] => 7
[2,4,6,5,1,3] => 13
[2,4,6,5,3,1] => 11
[2,5,1,3,4,6] => 7
[2,5,1,3,6,4] => 12
[2,5,1,4,3,6] => 5
[2,5,1,4,6,3] => 8
[2,5,1,6,3,4] => 15
[2,5,1,6,4,3] => 13
[2,5,3,1,4,6] => 5
[2,5,3,1,6,4] => 8
[2,5,3,4,1,6] => 3
[2,5,3,4,6,1] => 4
[2,5,3,6,1,4] => 9
[2,5,3,6,4,1] => 7
[2,5,4,1,3,6] => 8
[2,5,4,1,6,3] => 12
[2,5,4,3,1,6] => 6
[2,5,4,3,6,1] => 8
[2,5,4,6,1,3] => 13
[2,5,4,6,3,1] => 11
[2,5,6,1,3,4] => 16
[2,5,6,1,4,3] => 14
[2,5,6,3,1,4] => 14
[2,5,6,3,4,1] => 10
[2,5,6,4,1,3] => 9
[2,5,6,4,3,1] => 7
[2,6,1,3,4,5] => 9
[2,6,1,3,5,4] => 7
[2,6,1,4,3,5] => 7
[2,6,1,4,5,3] => 5
[2,6,1,5,3,4] => 12
[2,6,1,5,4,3] => 10
[2,6,3,1,4,5] => 7
[2,6,3,1,5,4] => 5
[2,6,3,4,1,5] => 5
[2,6,3,4,5,1] => 3
[2,6,3,5,1,4] => 8
[2,6,3,5,4,1] => 6
[2,6,4,1,3,5] => 12
[2,6,4,1,5,3] => 8
[2,6,4,3,1,5] => 10
[2,6,4,3,5,1] => 6
[2,6,4,5,1,3] => 11
[2,6,4,5,3,1] => 9
[2,6,5,1,3,4] => 15
[2,6,5,1,4,3] => 13
[2,6,5,3,1,4] => 13
[2,6,5,3,4,1] => 9
[2,6,5,4,1,3] => 8
[2,6,5,4,3,1] => 6
[3,1,2,4,5,6] => 3
[3,1,2,4,6,5] => 6
[3,1,2,5,4,6] => 6
[3,1,2,5,6,4] => 9
[3,1,2,6,4,5] => 9
[3,1,2,6,5,4] => 6
[3,1,4,2,5,6] => 5
[3,1,4,2,6,5] => 10
[3,1,4,5,2,6] => 7
[3,1,4,5,6,2] => 9
[3,1,4,6,2,5] => 12
[3,1,4,6,5,2] => 7
[3,1,5,2,4,6] => 8
[3,1,5,2,6,4] => 13
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 7
[3,1,5,6,2,4] => 15
[3,1,5,6,4,2] => 12
[3,1,6,2,4,5] => 11
[3,1,6,2,5,4] => 8
[3,1,6,4,2,5] => 8
[3,1,6,4,5,2] => 5
[3,1,6,5,2,4] => 13
[3,1,6,5,4,2] => 10
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 4
[3,2,1,5,4,6] => 4
[3,2,1,5,6,4] => 6
[3,2,1,6,4,5] => 6
[3,2,1,6,5,4] => 4
[3,2,4,1,5,6] => 3
[3,2,4,1,6,5] => 6
[3,2,4,5,1,6] => 4
[3,2,4,5,6,1] => 5
[3,2,4,6,1,5] => 7
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 8
[3,2,5,4,1,6] => 3
[3,2,5,4,6,1] => 4
[3,2,5,6,1,4] => 9
[3,2,5,6,4,1] => 7
[3,2,6,1,4,5] => 7
[3,2,6,1,5,4] => 5
[3,2,6,4,1,5] => 5
[3,2,6,4,5,1] => 3
[3,2,6,5,1,4] => 8
[3,2,6,5,4,1] => 6
[3,4,1,2,5,6] => 6
[3,4,1,2,6,5] => 12
[3,4,1,5,2,6] => 9
[3,4,1,5,6,2] => 12
[3,4,1,6,2,5] => 15
[3,4,1,6,5,2] => 9
[3,4,2,1,5,6] => 5
[3,4,2,1,6,5] => 10
[3,4,2,5,1,6] => 7
[3,4,2,5,6,1] => 9
[3,4,2,6,1,5] => 12
[3,4,2,6,5,1] => 7
[3,4,5,1,2,6] => 10
[3,4,5,1,6,2] => 14
[3,4,5,2,1,6] => 9
[3,4,5,2,6,1] => 12
[3,4,5,6,1,2] => 15
[3,4,5,6,2,1] => 14
[3,4,6,1,2,5] => 16
[3,4,6,1,5,2] => 10
[3,4,6,2,1,5] => 15
[3,4,6,2,5,1] => 9
[3,4,6,5,1,2] => 14
[3,4,6,5,2,1] => 13
[3,5,1,2,4,6] => 9
[3,5,1,2,6,4] => 15
[3,5,1,4,2,6] => 6
[3,5,1,4,6,2] => 9
[3,5,1,6,2,4] => 18
[3,5,1,6,4,2] => 15
[3,5,2,1,4,6] => 8
[3,5,2,1,6,4] => 13
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 7
[3,5,2,6,1,4] => 15
[3,5,2,6,4,1] => 12
[3,5,4,1,2,6] => 9
[3,5,4,1,6,2] => 13
[3,5,4,2,1,6] => 8
[3,5,4,2,6,1] => 11
[3,5,4,6,1,2] => 14
[3,5,4,6,2,1] => 13
[3,5,6,1,2,4] => 19
[3,5,6,1,4,2] => 16
[3,5,6,2,1,4] => 18
[3,5,6,2,4,1] => 14
[3,5,6,4,1,2] => 10
[3,5,6,4,2,1] => 9
[3,6,1,2,4,5] => 12
[3,6,1,2,5,4] => 9
[3,6,1,4,2,5] => 9
[3,6,1,4,5,2] => 6
[3,6,1,5,2,4] => 15
[3,6,1,5,4,2] => 12
[3,6,2,1,4,5] => 11
[3,6,2,1,5,4] => 8
[3,6,2,4,1,5] => 8
[3,6,2,4,5,1] => 5
[3,6,2,5,1,4] => 13
[3,6,2,5,4,1] => 10
[3,6,4,1,2,5] => 14
[3,6,4,1,5,2] => 9
[3,6,4,2,1,5] => 13
[3,6,4,2,5,1] => 8
[3,6,4,5,1,2] => 12
[3,6,4,5,2,1] => 11
[3,6,5,1,2,4] => 18
[3,6,5,1,4,2] => 15
[3,6,5,2,1,4] => 17
[3,6,5,2,4,1] => 13
[3,6,5,4,1,2] => 9
[3,6,5,4,2,1] => 8
[4,1,2,3,5,6] => 4
[4,1,2,3,6,5] => 8
[4,1,2,5,3,6] => 7
[4,1,2,5,6,3] => 10
[4,1,2,6,3,5] => 11
[4,1,2,6,5,3] => 7
[4,1,3,2,5,6] => 3
[4,1,3,2,6,5] => 6
[4,1,3,5,2,6] => 5
[4,1,3,5,6,2] => 7
[4,1,3,6,2,5] => 8
[4,1,3,6,5,2] => 5
[4,1,5,2,3,6] => 9
[4,1,5,2,6,3] => 14
[4,1,5,3,2,6] => 8
[4,1,5,3,6,2] => 12
[4,1,5,6,2,3] => 16
[4,1,5,6,3,2] => 15
[4,1,6,2,3,5] => 13
[4,1,6,2,5,3] => 9
[4,1,6,3,2,5] => 12
[4,1,6,3,5,2] => 8
[4,1,6,5,2,3] => 14
[4,1,6,5,3,2] => 13
[4,2,1,3,5,6] => 3
[4,2,1,3,6,5] => 6
[4,2,1,5,3,6] => 5
[4,2,1,5,6,3] => 7
[4,2,1,6,3,5] => 8
[4,2,1,6,5,3] => 5
[4,2,3,1,5,6] => 2
[4,2,3,1,6,5] => 4
[4,2,3,5,1,6] => 3
[4,2,3,5,6,1] => 4
[4,2,3,6,1,5] => 5
[4,2,3,6,5,1] => 3
[4,2,5,1,3,6] => 6
[4,2,5,1,6,3] => 9
[4,2,5,3,1,6] => 5
[4,2,5,3,6,1] => 7
[4,2,5,6,1,3] => 10
[4,2,5,6,3,1] => 9
[4,2,6,1,3,5] => 9
[4,2,6,1,5,3] => 6
[4,2,6,3,1,5] => 8
[4,2,6,3,5,1] => 5
[4,2,6,5,1,3] => 9
[4,2,6,5,3,1] => 8
[4,3,1,2,5,6] => 5
[4,3,1,2,6,5] => 10
[4,3,1,5,2,6] => 8
[4,3,1,5,6,2] => 11
[4,3,1,6,2,5] => 13
[4,3,1,6,5,2] => 8
[4,3,2,1,5,6] => 4
[4,3,2,1,6,5] => 8
[4,3,2,5,1,6] => 6
[4,3,2,5,6,1] => 8
[4,3,2,6,1,5] => 10
[4,3,2,6,5,1] => 6
[4,3,5,1,2,6] => 9
[4,3,5,1,6,2] => 13
[4,3,5,2,1,6] => 8
[4,3,5,2,6,1] => 11
[4,3,5,6,1,2] => 14
[4,3,5,6,2,1] => 13
[4,3,6,1,2,5] => 14
[4,3,6,1,5,2] => 9
[4,3,6,2,1,5] => 13
[4,3,6,2,5,1] => 8
[4,3,6,5,1,2] => 13
[4,3,6,5,2,1] => 12
[4,5,1,2,3,6] => 10
[4,5,1,2,6,3] => 16
[4,5,1,3,2,6] => 9
[4,5,1,3,6,2] => 14
[4,5,1,6,2,3] => 19
[4,5,1,6,3,2] => 18
[4,5,2,1,3,6] => 9
[4,5,2,1,6,3] => 14
[4,5,2,3,1,6] => 7
[4,5,2,3,6,1] => 10
[4,5,2,6,1,3] => 16
[4,5,2,6,3,1] => 14
[4,5,3,1,2,6] => 6
[4,5,3,1,6,2] => 9
[4,5,3,2,1,6] => 5
[4,5,3,2,6,1] => 7
[4,5,3,6,1,2] => 10
[4,5,3,6,2,1] => 9
[4,5,6,1,2,3] => 20
[4,5,6,1,3,2] => 19
[4,5,6,2,1,3] => 19
[4,5,6,2,3,1] => 16
[4,5,6,3,1,2] => 16
[4,5,6,3,2,1] => 14
[4,6,1,2,3,5] => 14
[4,6,1,2,5,3] => 10
[4,6,1,3,2,5] => 13
[4,6,1,3,5,2] => 9
[4,6,1,5,2,3] => 16
[4,6,1,5,3,2] => 15
[4,6,2,1,3,5] => 13
[4,6,2,1,5,3] => 9
[4,6,2,3,1,5] => 11
[4,6,2,3,5,1] => 7
[4,6,2,5,1,3] => 14
[4,6,2,5,3,1] => 12
[4,6,3,1,2,5] => 9
[4,6,3,1,5,2] => 6
[4,6,3,2,1,5] => 8
[4,6,3,2,5,1] => 5
[4,6,3,5,1,2] => 9
[4,6,3,5,2,1] => 8
[4,6,5,1,2,3] => 19
[4,6,5,1,3,2] => 18
[4,6,5,2,1,3] => 18
[4,6,5,2,3,1] => 15
[4,6,5,3,1,2] => 15
[4,6,5,3,2,1] => 13
[5,1,2,3,4,6] => 5
[5,1,2,3,6,4] => 9
[5,1,2,4,3,6] => 4
[5,1,2,4,6,3] => 7
[5,1,2,6,3,4] => 12
[5,1,2,6,4,3] => 11
[5,1,3,2,4,6] => 4
[5,1,3,2,6,4] => 7
[5,1,3,4,2,6] => 3
[5,1,3,4,6,2] => 5
[5,1,3,6,2,4] => 9
[5,1,3,6,4,2] => 8
[5,1,4,2,3,6] => 7
[5,1,4,2,6,3] => 12
[5,1,4,3,2,6] => 6
[5,1,4,3,6,2] => 10
[5,1,4,6,2,3] => 14
[5,1,4,6,3,2] => 13
[5,1,6,2,3,4] => 14
[5,1,6,2,4,3] => 13
[5,1,6,3,2,4] => 13
[5,1,6,3,4,2] => 11
[5,1,6,4,2,3] => 9
[5,1,6,4,3,2] => 8
[5,2,1,3,4,6] => 4
[5,2,1,3,6,4] => 7
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 5
[5,2,1,6,3,4] => 9
[5,2,1,6,4,3] => 8
[5,2,3,1,4,6] => 3
[5,2,3,1,6,4] => 5
[5,2,3,4,1,6] => 2
[5,2,3,4,6,1] => 3
[5,2,3,6,1,4] => 6
[5,2,3,6,4,1] => 5
[5,2,4,1,3,6] => 5
[5,2,4,1,6,3] => 8
[5,2,4,3,1,6] => 4
[5,2,4,3,6,1] => 6
[5,2,4,6,1,3] => 9
[5,2,4,6,3,1] => 8
[5,2,6,1,3,4] => 10
[5,2,6,1,4,3] => 9
[5,2,6,3,1,4] => 9
[5,2,6,3,4,1] => 7
[5,2,6,4,1,3] => 6
[5,2,6,4,3,1] => 5
[5,3,1,2,4,6] => 7
[5,3,1,2,6,4] => 12
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 8
[5,3,1,6,2,4] => 15
[5,3,1,6,4,2] => 13
[5,3,2,1,4,6] => 6
[5,3,2,1,6,4] => 10
[5,3,2,4,1,6] => 4
[5,3,2,4,6,1] => 6
[5,3,2,6,1,4] => 12
[5,3,2,6,4,1] => 10
[5,3,4,1,2,6] => 7
[5,3,4,1,6,2] => 11
[5,3,4,2,1,6] => 6
[5,3,4,2,6,1] => 9
[5,3,4,6,1,2] => 12
[5,3,4,6,2,1] => 11
[5,3,6,1,2,4] => 16
[5,3,6,1,4,2] => 14
[5,3,6,2,1,4] => 15
[5,3,6,2,4,1] => 12
[5,3,6,4,1,2] => 9
[5,3,6,4,2,1] => 8
[5,4,1,2,3,6] => 9
[5,4,1,2,6,3] => 15
[5,4,1,3,2,6] => 8
[5,4,1,3,6,2] => 13
[5,4,1,6,2,3] => 18
[5,4,1,6,3,2] => 17
[5,4,2,1,3,6] => 8
[5,4,2,1,6,3] => 13
[5,4,2,3,1,6] => 6
[5,4,2,3,6,1] => 9
[5,4,2,6,1,3] => 15
[5,4,2,6,3,1] => 13
[5,4,3,1,2,6] => 5
[5,4,3,1,6,2] => 8
[5,4,3,2,1,6] => 4
[5,4,3,2,6,1] => 6
[5,4,3,6,1,2] => 9
[5,4,3,6,2,1] => 8
[5,4,6,1,2,3] => 19
[5,4,6,1,3,2] => 18
[5,4,6,2,1,3] => 18
[5,4,6,2,3,1] => 15
[5,4,6,3,1,2] => 15
[5,4,6,3,2,1] => 13
[5,6,1,2,3,4] => 15
[5,6,1,2,4,3] => 14
[5,6,1,3,2,4] => 14
[5,6,1,3,4,2] => 12
[5,6,1,4,2,3] => 10
[5,6,1,4,3,2] => 9
[5,6,2,1,3,4] => 14
[5,6,2,1,4,3] => 13
[5,6,2,3,1,4] => 12
[5,6,2,3,4,1] => 9
[5,6,2,4,1,3] => 9
[5,6,2,4,3,1] => 7
[5,6,3,1,2,4] => 10
[5,6,3,1,4,2] => 9
[5,6,3,2,1,4] => 9
[5,6,3,2,4,1] => 7
[5,6,3,4,1,2] => 6
[5,6,3,4,2,1] => 5
[5,6,4,1,2,3] => 16
[5,6,4,1,3,2] => 15
[5,6,4,2,1,3] => 15
[5,6,4,2,3,1] => 12
[5,6,4,3,1,2] => 12
[5,6,4,3,2,1] => 10
[6,1,2,3,4,5] => 6
[6,1,2,3,5,4] => 5
[6,1,2,4,3,5] => 5
[6,1,2,4,5,3] => 4
[6,1,2,5,3,4] => 9
[6,1,2,5,4,3] => 8
[6,1,3,2,4,5] => 5
[6,1,3,2,5,4] => 4
[6,1,3,4,2,5] => 4
[6,1,3,4,5,2] => 3
[6,1,3,5,2,4] => 7
[6,1,3,5,4,2] => 6
[6,1,4,2,3,5] => 9
[6,1,4,2,5,3] => 7
[6,1,4,3,2,5] => 8
[6,1,4,3,5,2] => 6
[6,1,4,5,2,3] => 10
[6,1,4,5,3,2] => 9
[6,1,5,2,3,4] => 12
[6,1,5,2,4,3] => 11
[6,1,5,3,2,4] => 11
[6,1,5,3,4,2] => 9
[6,1,5,4,2,3] => 7
[6,1,5,4,3,2] => 6
[6,2,1,3,4,5] => 5
[6,2,1,3,5,4] => 4
[6,2,1,4,3,5] => 4
[6,2,1,4,5,3] => 3
[6,2,1,5,3,4] => 7
[6,2,1,5,4,3] => 6
[6,2,3,1,4,5] => 4
[6,2,3,1,5,4] => 3
[6,2,3,4,1,5] => 3
[6,2,3,4,5,1] => 2
[6,2,3,5,1,4] => 5
[6,2,3,5,4,1] => 4
[6,2,4,1,3,5] => 7
[6,2,4,1,5,3] => 5
[6,2,4,3,1,5] => 6
[6,2,4,3,5,1] => 4
[6,2,4,5,1,3] => 7
[6,2,4,5,3,1] => 6
[6,2,5,1,3,4] => 9
[6,2,5,1,4,3] => 8
[6,2,5,3,1,4] => 8
[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] => 9
[6,3,1,2,5,4] => 7
[6,3,1,4,2,5] => 7
[6,3,1,4,5,2] => 5
[6,3,1,5,2,4] => 12
[6,3,1,5,4,2] => 10
[6,3,2,1,4,5] => 8
[6,3,2,1,5,4] => 6
[6,3,2,4,1,5] => 6
[6,3,2,4,5,1] => 4
[6,3,2,5,1,4] => 10
[6,3,2,5,4,1] => 8
[6,3,4,1,2,5] => 10
[6,3,4,1,5,2] => 7
[6,3,4,2,1,5] => 9
[6,3,4,2,5,1] => 6
[6,3,4,5,1,2] => 9
[6,3,4,5,2,1] => 8
[6,3,5,1,2,4] => 14
[6,3,5,1,4,2] => 12
[6,3,5,2,1,4] => 13
[6,3,5,2,4,1] => 10
[6,3,5,4,1,2] => 7
[6,3,5,4,2,1] => 6
[6,4,1,2,3,5] => 12
[6,4,1,2,5,3] => 9
[6,4,1,3,2,5] => 11
[6,4,1,3,5,2] => 8
[6,4,1,5,2,3] => 14
[6,4,1,5,3,2] => 13
[6,4,2,1,3,5] => 11
[6,4,2,1,5,3] => 8
[6,4,2,3,1,5] => 9
[6,4,2,3,5,1] => 6
[6,4,2,5,1,3] => 12
[6,4,2,5,3,1] => 10
[6,4,3,1,2,5] => 7
[6,4,3,1,5,2] => 5
[6,4,3,2,1,5] => 6
[6,4,3,2,5,1] => 4
[6,4,3,5,1,2] => 7
[6,4,3,5,2,1] => 6
[6,4,5,1,2,3] => 16
[6,4,5,1,3,2] => 15
[6,4,5,2,1,3] => 15
[6,4,5,2,3,1] => 12
[6,4,5,3,1,2] => 12
[6,4,5,3,2,1] => 10
[6,5,1,2,3,4] => 14
[6,5,1,2,4,3] => 13
[6,5,1,3,2,4] => 13
[6,5,1,3,4,2] => 11
[6,5,1,4,2,3] => 9
[6,5,1,4,3,2] => 8
[6,5,2,1,3,4] => 13
[6,5,2,1,4,3] => 12
[6,5,2,3,1,4] => 11
[6,5,2,3,4,1] => 8
[6,5,2,4,1,3] => 8
[6,5,2,4,3,1] => 6
[6,5,3,1,2,4] => 9
[6,5,3,1,4,2] => 8
[6,5,3,2,1,4] => 8
[6,5,3,2,4,1] => 6
[6,5,3,4,1,2] => 5
[6,5,3,4,2,1] => 4
[6,5,4,1,2,3] => 14
[6,5,4,1,3,2] => 13
[6,5,4,2,1,3] => 13
[6,5,4,2,3,1] => 10
[6,5,4,3,1,2] => 10
[6,5,4,3,2,1] => 8
[1,2,3,4,5,6,7] => 1
[1,2,3,4,5,7,6] => 2
[1,2,3,4,6,5,7] => 2
[1,2,3,4,6,7,5] => 3
[1,2,3,4,7,5,6] => 3
[1,2,3,4,7,6,5] => 2
[1,2,3,5,4,6,7] => 2
[1,2,3,5,4,7,6] => 4
[1,2,3,5,6,4,7] => 3
[1,2,3,5,6,7,4] => 4
[1,2,3,5,7,4,6] => 5
[1,2,3,5,7,6,4] => 3
[1,2,3,6,4,5,7] => 3
[1,2,3,6,4,7,5] => 5
[1,2,3,6,5,4,7] => 2
[1,2,3,6,5,7,4] => 3
[1,2,3,6,7,4,5] => 6
[1,2,3,6,7,5,4] => 5
[1,2,3,7,4,5,6] => 4
[1,2,3,7,4,6,5] => 3
[1,2,3,7,5,4,6] => 3
[1,2,3,7,5,6,4] => 2
[1,2,3,7,6,4,5] => 5
[1,2,3,7,6,5,4] => 4
[1,2,4,3,5,6,7] => 2
[1,2,4,3,5,7,6] => 4
[1,2,4,3,6,5,7] => 4
[1,2,4,3,6,7,5] => 6
[1,2,4,3,7,5,6] => 6
[1,2,4,3,7,6,5] => 4
[1,2,4,5,3,6,7] => 3
[1,2,4,5,3,7,6] => 6
[1,2,4,5,6,3,7] => 4
[1,2,4,5,6,7,3] => 5
[1,2,4,5,7,3,6] => 7
[1,2,4,5,7,6,3] => 4
[1,2,4,6,3,5,7] => 5
[1,2,4,6,3,7,5] => 8
[1,2,4,6,5,3,7] => 3
[1,2,4,6,5,7,3] => 4
[1,2,4,6,7,3,5] => 9
[1,2,4,6,7,5,3] => 7
[1,2,4,7,3,5,6] => 7
[1,2,4,7,3,6,5] => 5
[1,2,4,7,5,3,6] => 5
[1,2,4,7,5,6,3] => 3
[1,2,4,7,6,3,5] => 8
[1,2,4,7,6,5,3] => 6
[1,2,5,3,4,6,7] => 3
[1,2,5,3,4,7,6] => 6
[1,2,5,3,6,4,7] => 5
[1,2,5,3,6,7,4] => 7
[1,2,5,3,7,4,6] => 8
[1,2,5,3,7,6,4] => 5
[1,2,5,4,3,6,7] => 2
[1,2,5,4,3,7,6] => 4
[1,2,5,4,6,3,7] => 3
[1,2,5,4,6,7,3] => 4
[1,2,5,4,7,3,6] => 5
[1,2,5,4,7,6,3] => 3
[1,2,5,6,3,4,7] => 6
[1,2,5,6,3,7,4] => 9
[1,2,5,6,4,3,7] => 5
[1,2,5,6,4,7,3] => 7
[1,2,5,6,7,3,4] => 10
[1,2,5,6,7,4,3] => 9
[1,2,5,7,3,4,6] => 9
[1,2,5,7,3,6,4] => 6
[1,2,5,7,4,3,6] => 8
[1,2,5,7,4,6,3] => 5
[1,2,5,7,6,3,4] => 9
[1,2,5,7,6,4,3] => 8
[1,2,6,3,4,5,7] => 4
[1,2,6,3,4,7,5] => 7
[1,2,6,3,5,4,7] => 3
[1,2,6,3,5,7,4] => 5
[1,2,6,3,7,4,5] => 9
[1,2,6,3,7,5,4] => 8
[1,2,6,4,3,5,7] => 3
[1,2,6,4,3,7,5] => 5
[1,2,6,4,5,3,7] => 2
[1,2,6,4,5,7,3] => 3
[1,2,6,4,7,3,5] => 6
[1,2,6,4,7,5,3] => 5
[1,2,6,5,3,4,7] => 5
[1,2,6,5,3,7,4] => 8
[1,2,6,5,4,3,7] => 4
[1,2,6,5,4,7,3] => 6
[1,2,6,5,7,3,4] => 9
[1,2,6,5,7,4,3] => 8
[1,2,6,7,3,4,5] => 10
[1,2,6,7,3,5,4] => 9
[1,2,6,7,4,3,5] => 9
[1,2,6,7,4,5,3] => 7
[1,2,6,7,5,3,4] => 6
[1,2,6,7,5,4,3] => 5
[1,2,7,3,4,5,6] => 5
[1,2,7,3,4,6,5] => 4
[1,2,7,3,5,4,6] => 4
[1,2,7,3,5,6,4] => 3
[1,2,7,3,6,4,5] => 7
[1,2,7,3,6,5,4] => 6
[1,2,7,4,3,5,6] => 4
[1,2,7,4,3,6,5] => 3
[1,2,7,4,5,3,6] => 3
[1,2,7,4,5,6,3] => 2
[1,2,7,4,6,3,5] => 5
[1,2,7,4,6,5,3] => 4
[1,2,7,5,3,4,6] => 7
[1,2,7,5,3,6,4] => 5
[1,2,7,5,4,3,6] => 6
[1,2,7,5,4,6,3] => 4
[1,2,7,5,6,3,4] => 7
[1,2,7,5,6,4,3] => 6
[1,2,7,6,3,4,5] => 9
[1,2,7,6,3,5,4] => 8
[1,2,7,6,4,3,5] => 8
[1,2,7,6,4,5,3] => 6
[1,2,7,6,5,3,4] => 5
[1,2,7,6,5,4,3] => 4
[1,3,2,4,5,6,7] => 2
[1,3,2,4,5,7,6] => 4
[1,3,2,4,6,5,7] => 4
[1,3,2,4,6,7,5] => 6
[1,3,2,4,7,5,6] => 6
[1,3,2,4,7,6,5] => 4
[1,3,2,5,4,6,7] => 4
[1,3,2,5,4,7,6] => 8
[1,3,2,5,6,4,7] => 6
[1,3,2,5,6,7,4] => 8
[1,3,2,5,7,4,6] => 10
[1,3,2,5,7,6,4] => 6
[1,3,2,6,4,5,7] => 6
[1,3,2,6,4,7,5] => 10
[1,3,2,6,5,4,7] => 4
[1,3,2,6,5,7,4] => 6
[1,3,2,6,7,4,5] => 12
[1,3,2,6,7,5,4] => 10
[1,3,2,7,4,5,6] => 8
[1,3,2,7,4,6,5] => 6
[1,3,2,7,5,4,6] => 6
[1,3,2,7,5,6,4] => 4
[1,3,2,7,6,4,5] => 10
[1,3,2,7,6,5,4] => 8
[1,3,4,2,5,6,7] => 3
[1,3,4,2,5,7,6] => 6
[1,3,4,2,6,5,7] => 6
[1,3,4,2,6,7,5] => 9
[1,3,4,2,7,5,6] => 9
[1,3,4,2,7,6,5] => 6
[1,3,4,5,2,6,7] => 4
[1,3,4,5,2,7,6] => 8
[1,3,4,5,6,2,7] => 5
[1,3,4,5,6,7,2] => 6
[1,3,4,5,7,2,6] => 9
[1,3,4,5,7,6,2] => 5
[1,3,4,6,2,5,7] => 7
[1,3,4,6,2,7,5] => 11
[1,3,4,6,5,2,7] => 4
[1,3,4,6,5,7,2] => 5
[1,3,4,6,7,2,5] => 12
[1,3,4,6,7,5,2] => 9
[1,3,4,7,2,5,6] => 10
[1,3,4,7,2,6,5] => 7
[1,3,4,7,5,2,6] => 7
[1,3,4,7,5,6,2] => 4
[1,3,4,7,6,2,5] => 11
[1,3,4,7,6,5,2] => 8
[1,3,5,2,4,6,7] => 5
[1,3,5,2,4,7,6] => 10
[1,3,5,2,6,4,7] => 8
[1,3,5,2,6,7,4] => 11
[1,3,5,2,7,4,6] => 13
[1,3,5,2,7,6,4] => 8
[1,3,5,4,2,6,7] => 3
[1,3,5,4,2,7,6] => 6
[1,3,5,4,6,2,7] => 4
[1,3,5,4,6,7,2] => 5
[1,3,5,4,7,2,6] => 7
[1,3,5,4,7,6,2] => 4
[1,3,5,6,2,4,7] => 9
[1,3,5,6,2,7,4] => 13
[1,3,5,6,4,2,7] => 7
[1,3,5,6,4,7,2] => 9
[1,3,5,6,7,2,4] => 14
[1,3,5,6,7,4,2] => 12
[1,3,5,7,2,4,6] => 14
[1,3,5,7,2,6,4] => 9
[1,3,5,7,4,2,6] => 12
[1,3,5,7,4,6,2] => 7
[1,3,5,7,6,2,4] => 13
[1,3,5,7,6,4,2] => 11
[1,3,6,2,4,5,7] => 7
[1,3,6,2,4,7,5] => 12
[1,3,6,2,5,4,7] => 5
[1,3,6,2,5,7,4] => 8
[1,3,6,2,7,4,5] => 15
[1,3,6,2,7,5,4] => 13
[1,3,6,4,2,5,7] => 5
[1,3,6,4,2,7,5] => 8
[1,3,6,4,5,2,7] => 3
[1,3,6,4,5,7,2] => 4
[1,3,6,4,7,2,5] => 9
[1,3,6,4,7,5,2] => 7
[1,3,6,5,2,4,7] => 8
[1,3,6,5,2,7,4] => 12
[1,3,6,5,4,2,7] => 6
[1,3,6,5,4,7,2] => 8
[1,3,6,5,7,2,4] => 13
[1,3,6,5,7,4,2] => 11
[1,3,6,7,2,4,5] => 16
[1,3,6,7,2,5,4] => 14
[1,3,6,7,4,2,5] => 14
[1,3,6,7,4,5,2] => 10
[1,3,6,7,5,2,4] => 9
[1,3,6,7,5,4,2] => 7
[1,3,7,2,4,5,6] => 9
[1,3,7,2,4,6,5] => 7
[1,3,7,2,5,4,6] => 7
[1,3,7,2,5,6,4] => 5
[1,3,7,2,6,4,5] => 12
[1,3,7,2,6,5,4] => 10
[1,3,7,4,2,5,6] => 7
[1,3,7,4,2,6,5] => 5
[1,3,7,4,5,2,6] => 5
[1,3,7,4,5,6,2] => 3
[1,3,7,4,6,2,5] => 8
[1,3,7,4,6,5,2] => 6
[1,3,7,5,2,4,6] => 12
[1,3,7,5,2,6,4] => 8
[1,3,7,5,4,2,6] => 10
[1,3,7,5,4,6,2] => 6
[1,3,7,5,6,2,4] => 11
[1,3,7,5,6,4,2] => 9
[1,3,7,6,2,4,5] => 15
[1,3,7,6,2,5,4] => 13
[1,3,7,6,4,2,5] => 13
[1,3,7,6,4,5,2] => 9
[1,3,7,6,5,2,4] => 8
[1,3,7,6,5,4,2] => 6
[1,4,2,3,5,6,7] => 3
[1,4,2,3,5,7,6] => 6
[1,4,2,3,6,5,7] => 6
[1,4,2,3,6,7,5] => 9
[1,4,2,3,7,5,6] => 9
[1,4,2,3,7,6,5] => 6
[1,4,2,5,3,6,7] => 5
[1,4,2,5,3,7,6] => 10
[1,4,2,5,6,3,7] => 7
[1,4,2,5,6,7,3] => 9
[1,4,2,5,7,3,6] => 12
[1,4,2,5,7,6,3] => 7
[1,4,2,6,3,5,7] => 8
[1,4,2,6,3,7,5] => 13
[1,4,2,6,5,3,7] => 5
[1,4,2,6,5,7,3] => 7
[1,4,2,6,7,3,5] => 15
[1,4,2,6,7,5,3] => 12
[1,4,2,7,3,5,6] => 11
[1,4,2,7,3,6,5] => 8
[1,4,2,7,5,3,6] => 8
[1,4,2,7,5,6,3] => 5
[1,4,2,7,6,3,5] => 13
[1,4,2,7,6,5,3] => 10
[1,4,3,2,5,6,7] => 2
[1,4,3,2,5,7,6] => 4
[1,4,3,2,6,5,7] => 4
[1,4,3,2,6,7,5] => 6
[1,4,3,2,7,5,6] => 6
[1,4,3,2,7,6,5] => 4
[1,4,3,5,2,6,7] => 3
[1,4,3,5,2,7,6] => 6
[1,4,3,5,6,2,7] => 4
[1,4,3,5,6,7,2] => 5
[1,4,3,5,7,2,6] => 7
[1,4,3,5,7,6,2] => 4
[1,4,3,6,2,5,7] => 5
[1,4,3,6,2,7,5] => 8
[1,4,3,6,5,2,7] => 3
[1,4,3,6,5,7,2] => 4
[1,4,3,6,7,2,5] => 9
[1,4,3,6,7,5,2] => 7
[1,4,3,7,2,5,6] => 7
[1,4,3,7,2,6,5] => 5
[1,4,3,7,5,2,6] => 5
[1,4,3,7,5,6,2] => 3
[1,4,3,7,6,2,5] => 8
[1,4,3,7,6,5,2] => 6
[1,4,5,2,3,6,7] => 6
[1,4,5,2,3,7,6] => 12
[1,4,5,2,6,3,7] => 9
[1,4,5,2,6,7,3] => 12
[1,4,5,2,7,3,6] => 15
[1,4,5,2,7,6,3] => 9
[1,4,5,3,2,6,7] => 5
[1,4,5,3,2,7,6] => 10
[1,4,5,3,6,2,7] => 7
[1,4,5,3,6,7,2] => 9
[1,4,5,3,7,2,6] => 12
[1,4,5,3,7,6,2] => 7
[1,4,5,6,2,3,7] => 10
[1,4,5,6,2,7,3] => 14
[1,4,5,6,3,2,7] => 9
[1,4,5,6,3,7,2] => 12
[1,4,5,6,7,2,3] => 15
[1,4,5,6,7,3,2] => 14
[1,4,5,7,2,3,6] => 16
[1,4,5,7,2,6,3] => 10
[1,4,5,7,3,2,6] => 15
[1,4,5,7,3,6,2] => 9
[1,4,5,7,6,2,3] => 14
[1,4,5,7,6,3,2] => 13
[1,4,6,2,3,5,7] => 9
[1,4,6,2,3,7,5] => 15
[1,4,6,2,5,3,7] => 6
[1,4,6,2,5,7,3] => 9
[1,4,6,2,7,3,5] => 18
[1,4,6,2,7,5,3] => 15
[1,4,6,3,2,5,7] => 8
[1,4,6,3,2,7,5] => 13
[1,4,6,3,5,2,7] => 5
[1,4,6,3,5,7,2] => 7
[1,4,6,3,7,2,5] => 15
[1,4,6,3,7,5,2] => 12
[1,4,6,5,2,3,7] => 9
[1,4,6,5,2,7,3] => 13
[1,4,6,5,3,2,7] => 8
click to show generating function       
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
References
[1] Ardila, F., Rincón, F., Williams, L. Positroids and non-crossing partitions arXiv:1308.2698
Code
def i_less(a, b, i):
    """
    Return whether a < b for i < i+1 < ... < n < 1 < ... < i-1.
    """
    if min(a, b) >= i or max(a, b) < i:
        return a < b
    return a >= i > b

def Grassmann_necklace(pi):
    """
    Return the Grassmann necklace corresponding to pi.

    sage: pi = Permutation([3,4,2,1])
    sage: Grassmann_necklace(pi)
    [{1, 2}, {2, 4}, {3, 4}, {1, 4}]
    """
    pi = Permutation(pi)
    N = []
    n = len(pi)
    for i in range(1,n+1):
        iWEX = [j for j in range(1,n+1) if j == pi(j) or i_less(j, pi(j), i)]
        N.append(set(iWEX))
    return N

def Gale_less(S, T, i):
    cmp = lambda a,b: 0 if a == b else int(-1) if i_less(a,b,i) else int(1)
    S_sorted = sorted(S, cmp=cmp)
    T_sorted = sorted(T, cmp=cmp)
    return all(s == t or i_less(s, t, i) for s,t in zip(S_sorted, T_sorted))

def positroid(N):
    """
    Return the positroid corresponding to the Grassmann necklace.

    sage: pi = Permutation([3,4,2,1])
    sage: N = Grassmann_necklace(pi)
    sage: positroid(N)
    [{1, 2}, {1, 3}, {1, 4}, {2, 4}, {3, 4}]

    sage: [set([len(positroid(Grassmann_necklace(p))) for p in orbit(pi)]) for pi in Permutations(4)]


    """
    n = len(N)
    d = len(N[0])
    result = []
    for B in Subsets(range(1,n+1), d):
        if all(B == N[j] or Gale_less(N[j], B, j+1) for j in range(n)):
            result.append(B)

    return result

def statistic(pi):
    return len(positroid(Grassmann_necklace(pi)))

Created
Aug 12, 2019 at 16:34 by Martin Rubey
Updated
Aug 12, 2019 at 16:34 by Martin Rubey