Identifier
Identifier
Values
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[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] => 1
[2,1,4,3] => 4
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 4
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[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] => 24
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[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] => 24
[1,4,5,3,2] => 4
[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] => 4
[1,5,4,3,2] => 24
[2,1,3,4,5] => 1
[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] => 1
[2,3,1,5,4] => 6
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 1
[2,3,5,4,1] => 1
[2,4,1,3,5] => 1
[2,4,1,5,3] => 1
[2,4,3,1,5] => 1
[2,4,3,5,1] => 1
[2,4,5,1,3] => 6
[2,4,5,3,1] => 1
[2,5,1,3,4] => 1
[2,5,1,4,3] => 1
[2,5,3,1,4] => 1
[2,5,3,4,1] => 1
[2,5,4,1,3] => 1
[2,5,4,3,1] => 6
[3,1,2,4,5] => 1
[3,1,2,5,4] => 6
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 1
[3,1,5,4,2] => 1
[3,2,1,4,5] => 1
[3,2,1,5,4] => 4
[3,2,4,1,5] => 1
[3,2,4,5,1] => 1
[3,2,5,1,4] => 1
[3,2,5,4,1] => 1
[3,4,1,2,5] => 4
[3,4,1,5,2] => 6
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 1
[3,4,5,2,1] => 6
[3,5,1,2,4] => 6
[3,5,1,4,2] => 4
[3,5,2,1,4] => 1
[3,5,2,4,1] => 1
[3,5,4,1,2] => 6
[3,5,4,2,1] => 1
[4,1,2,3,5] => 1
[4,1,2,5,3] => 1
[4,1,3,2,5] => 1
[4,1,3,5,2] => 1
[4,1,5,2,3] => 6
[4,1,5,3,2] => 1
[4,2,1,3,5] => 1
[4,2,1,5,3] => 1
[4,2,3,1,5] => 1
[4,2,3,5,1] => 1
[4,2,5,1,3] => 4
[4,2,5,3,1] => 1
[4,3,1,2,5] => 1
[4,3,1,5,2] => 1
[4,3,2,1,5] => 4
[4,3,2,5,1] => 6
[4,3,5,1,2] => 6
[4,3,5,2,1] => 1
[4,5,1,2,3] => 1
[4,5,1,3,2] => 6
[4,5,2,1,3] => 6
[4,5,2,3,1] => 1
[4,5,3,1,2] => 4
[4,5,3,2,1] => 1
[5,1,2,3,4] => 1
[5,1,2,4,3] => 1
[5,1,3,2,4] => 1
[5,1,3,4,2] => 1
[5,1,4,2,3] => 1
[5,1,4,3,2] => 6
[5,2,1,3,4] => 1
[5,2,1,4,3] => 1
[5,2,3,1,4] => 1
[5,2,3,4,1] => 1
[5,2,4,1,3] => 1
[5,2,4,3,1] => 4
[5,3,1,2,4] => 1
[5,3,1,4,2] => 1
[5,3,2,1,4] => 6
[5,3,2,4,1] => 4
[5,3,4,1,2] => 1
[5,3,4,2,1] => 6
[5,4,1,2,3] => 6
[5,4,1,3,2] => 1
[5,4,2,1,3] => 1
[5,4,2,3,1] => 6
[5,4,3,1,2] => 1
[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] => 24
[1,2,4,5,3,6] => 3
[1,2,4,5,6,3] => 4
[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] => 24
[1,2,5,6,4,3] => 4
[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] => 4
[1,2,6,5,4,3] => 24
[1,3,2,4,5,6] => 2
[1,3,2,4,6,5] => 24
[1,3,2,5,4,6] => 24
[1,3,2,5,6,4] => 42
[1,3,2,6,4,5] => 42
[1,3,2,6,5,4] => 24
[1,3,4,2,5,6] => 3
[1,3,4,2,6,5] => 42
[1,3,4,5,2,6] => 4
[1,3,4,5,6,2] => 5
[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] => 42
[1,3,5,6,4,2] => 5
[1,3,6,2,4,5] => 5
[1,3,6,2,5,4] => 4
[1,3,6,4,2,5] => 4
[1,3,6,4,5,2] => 3
[1,3,6,5,2,4] => 5
[1,3,6,5,4,2] => 42
[1,4,2,3,5,6] => 3
[1,4,2,3,6,5] => 42
[1,4,2,5,3,6] => 4
[1,4,2,5,6,3] => 5
[1,4,2,6,3,5] => 5
[1,4,2,6,5,3] => 4
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 24
[1,4,3,5,2,6] => 3
[1,4,3,5,6,2] => 4
[1,4,3,6,2,5] => 4
[1,4,3,6,5,2] => 3
[1,4,5,2,3,6] => 24
[1,4,5,2,6,3] => 42
[1,4,5,3,2,6] => 4
[1,4,5,3,6,2] => 5
[1,4,5,6,2,3] => 5
[1,4,5,6,3,2] => 42
[1,4,6,2,3,5] => 42
[1,4,6,2,5,3] => 24
[1,4,6,3,2,5] => 5
[1,4,6,3,5,2] => 4
[1,4,6,5,2,3] => 42
[1,4,6,5,3,2] => 5
[1,5,2,3,4,6] => 4
[1,5,2,3,6,4] => 5
[1,5,2,4,3,6] => 3
[1,5,2,4,6,3] => 4
[1,5,2,6,3,4] => 42
[1,5,2,6,4,3] => 5
[1,5,3,2,4,6] => 3
[1,5,3,2,6,4] => 4
[1,5,3,4,2,6] => 2
[1,5,3,4,6,2] => 3
[1,5,3,6,2,4] => 24
[1,5,3,6,4,2] => 4
[1,5,4,2,3,6] => 4
[1,5,4,2,6,3] => 5
[1,5,4,3,2,6] => 24
[1,5,4,3,6,2] => 42
[1,5,4,6,2,3] => 42
[1,5,4,6,3,2] => 5
[1,5,6,2,3,4] => 5
[1,5,6,2,4,3] => 42
[1,5,6,3,2,4] => 42
[1,5,6,3,4,2] => 5
[1,5,6,4,2,3] => 24
[1,5,6,4,3,2] => 4
[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] => 5
[1,6,2,5,4,3] => 42
[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] => 4
[1,6,3,5,4,2] => 24
[1,6,4,2,3,5] => 5
[1,6,4,2,5,3] => 4
[1,6,4,3,2,5] => 42
[1,6,4,3,5,2] => 24
[1,6,4,5,2,3] => 5
[1,6,4,5,3,2] => 42
[1,6,5,2,3,4] => 42
[1,6,5,2,4,3] => 5
[1,6,5,3,2,4] => 5
[1,6,5,3,4,2] => 42
[1,6,5,4,2,3] => 4
[1,6,5,4,3,2] => 24
[2,1,3,4,5,6] => 1
[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] => 56
[2,1,4,5,3,6] => 6
[2,1,4,5,6,3] => 8
[2,1,4,6,3,5] => 8
[2,1,4,6,5,3] => 6
[2,1,5,3,4,6] => 6
[2,1,5,3,6,4] => 8
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 6
[2,1,5,6,3,4] => 56
[2,1,5,6,4,3] => 8
[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] => 8
[2,1,6,5,4,3] => 56
[2,3,1,4,5,6] => 1
[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] => 1
[2,3,4,1,6,5] => 8
[2,3,4,5,1,6] => 1
[2,3,4,5,6,1] => 1
[2,3,4,6,1,5] => 1
[2,3,4,6,5,1] => 1
[2,3,5,1,4,6] => 1
[2,3,5,1,6,4] => 1
[2,3,5,4,1,6] => 1
[2,3,5,4,6,1] => 1
[2,3,5,6,1,4] => 8
[2,3,5,6,4,1] => 1
[2,3,6,1,4,5] => 1
[2,3,6,1,5,4] => 1
[2,3,6,4,1,5] => 1
[2,3,6,4,5,1] => 1
[2,3,6,5,1,4] => 1
[2,3,6,5,4,1] => 8
[2,4,1,3,5,6] => 1
[2,4,1,3,6,5] => 8
[2,4,1,5,3,6] => 1
[2,4,1,5,6,3] => 1
[2,4,1,6,3,5] => 1
[2,4,1,6,5,3] => 1
[2,4,3,1,5,6] => 1
[2,4,3,1,6,5] => 6
[2,4,3,5,1,6] => 1
[2,4,3,5,6,1] => 1
[2,4,3,6,1,5] => 1
[2,4,3,6,5,1] => 1
[2,4,5,1,3,6] => 6
[2,4,5,1,6,3] => 9
[2,4,5,3,1,6] => 1
[2,4,5,3,6,1] => 1
[2,4,5,6,1,3] => 1
[2,4,5,6,3,1] => 8
[2,4,6,1,3,5] => 9
[2,4,6,1,5,3] => 6
[2,4,6,3,1,5] => 1
[2,4,6,3,5,1] => 1
[2,4,6,5,1,3] => 8
[2,4,6,5,3,1] => 1
[2,5,1,3,4,6] => 1
[2,5,1,3,6,4] => 1
[2,5,1,4,3,6] => 1
[2,5,1,4,6,3] => 1
[2,5,1,6,3,4] => 8
[2,5,1,6,4,3] => 1
[2,5,3,1,4,6] => 1
[2,5,3,1,6,4] => 1
[2,5,3,4,1,6] => 1
[2,5,3,4,6,1] => 1
[2,5,3,6,1,4] => 6
[2,5,3,6,4,1] => 1
[2,5,4,1,3,6] => 1
[2,5,4,1,6,3] => 1
[2,5,4,3,1,6] => 6
[2,5,4,3,6,1] => 8
[2,5,4,6,1,3] => 9
[2,5,4,6,3,1] => 1
[2,5,6,1,3,4] => 1
[2,5,6,1,4,3] => 8
[2,5,6,3,1,4] => 9
[2,5,6,3,4,1] => 1
[2,5,6,4,1,3] => 6
[2,5,6,4,3,1] => 1
[2,6,1,3,4,5] => 1
[2,6,1,3,5,4] => 1
[2,6,1,4,3,5] => 1
[2,6,1,4,5,3] => 1
[2,6,1,5,3,4] => 1
[2,6,1,5,4,3] => 8
[2,6,3,1,4,5] => 1
[2,6,3,1,5,4] => 1
[2,6,3,4,1,5] => 1
[2,6,3,4,5,1] => 1
[2,6,3,5,1,4] => 1
[2,6,3,5,4,1] => 6
[2,6,4,1,3,5] => 1
[2,6,4,1,5,3] => 1
[2,6,4,3,1,5] => 8
[2,6,4,3,5,1] => 6
[2,6,4,5,1,3] => 1
[2,6,4,5,3,1] => 9
[2,6,5,1,3,4] => 8
[2,6,5,1,4,3] => 1
[2,6,5,3,1,4] => 1
[2,6,5,3,4,1] => 9
[2,6,5,4,1,3] => 1
[2,6,5,4,3,1] => 6
[3,1,2,4,5,6] => 1
[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] => 1
[3,1,4,2,6,5] => 8
[3,1,4,5,2,6] => 1
[3,1,4,5,6,2] => 1
[3,1,4,6,2,5] => 1
[3,1,4,6,5,2] => 1
[3,1,5,2,4,6] => 1
[3,1,5,2,6,4] => 1
[3,1,5,4,2,6] => 1
[3,1,5,4,6,2] => 1
[3,1,5,6,2,4] => 8
[3,1,5,6,4,2] => 1
[3,1,6,2,4,5] => 1
[3,1,6,2,5,4] => 1
[3,1,6,4,2,5] => 1
[3,1,6,4,5,2] => 1
[3,1,6,5,2,4] => 1
[3,1,6,5,4,2] => 8
[3,2,1,4,5,6] => 1
[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] => 1
[3,2,4,1,6,5] => 6
[3,2,4,5,1,6] => 1
[3,2,4,5,6,1] => 1
[3,2,4,6,1,5] => 1
[3,2,4,6,5,1] => 1
[3,2,5,1,4,6] => 1
[3,2,5,1,6,4] => 1
[3,2,5,4,1,6] => 1
[3,2,5,4,6,1] => 1
[3,2,5,6,1,4] => 6
[3,2,5,6,4,1] => 1
[3,2,6,1,4,5] => 1
[3,2,6,1,5,4] => 1
[3,2,6,4,1,5] => 1
[3,2,6,4,5,1] => 1
[3,2,6,5,1,4] => 1
[3,2,6,5,4,1] => 6
[3,4,1,2,5,6] => 4
[3,4,1,2,6,5] => 56
[3,4,1,5,2,6] => 6
[3,4,1,5,6,2] => 8
[3,4,1,6,2,5] => 8
[3,4,1,6,5,2] => 6
[3,4,2,1,5,6] => 1
[3,4,2,1,6,5] => 8
[3,4,2,5,1,6] => 1
[3,4,2,5,6,1] => 1
[3,4,2,6,1,5] => 1
[3,4,2,6,5,1] => 1
[3,4,5,1,2,6] => 1
[3,4,5,1,6,2] => 1
[3,4,5,2,1,6] => 6
[3,4,5,2,6,1] => 8
[3,4,5,6,1,2] => 9
[3,4,5,6,2,1] => 1
[3,4,6,1,2,5] => 1
[3,4,6,1,5,2] => 1
[3,4,6,2,1,5] => 8
[3,4,6,2,5,1] => 6
[3,4,6,5,1,2] => 1
[3,4,6,5,2,1] => 9
[3,5,1,2,4,6] => 6
[3,5,1,2,6,4] => 8
[3,5,1,4,2,6] => 4
[3,5,1,4,6,2] => 6
[3,5,1,6,2,4] => 56
[3,5,1,6,4,2] => 8
[3,5,2,1,4,6] => 1
[3,5,2,1,6,4] => 1
[3,5,2,4,1,6] => 1
[3,5,2,4,6,1] => 1
[3,5,2,6,1,4] => 8
[3,5,2,6,4,1] => 1
[3,5,4,1,2,6] => 6
[3,5,4,1,6,2] => 9
[3,5,4,2,1,6] => 1
[3,5,4,2,6,1] => 1
[3,5,4,6,1,2] => 1
[3,5,4,6,2,1] => 8
[3,5,6,1,2,4] => 8
[3,5,6,1,4,2] => 1
[3,5,6,2,1,4] => 1
[3,5,6,2,4,1] => 9
[3,5,6,4,1,2] => 1
[3,5,6,4,2,1] => 6
[3,6,1,2,4,5] => 8
[3,6,1,2,5,4] => 6
[3,6,1,4,2,5] => 6
[3,6,1,4,5,2] => 4
[3,6,1,5,2,4] => 8
[3,6,1,5,4,2] => 56
[3,6,2,1,4,5] => 1
[3,6,2,1,5,4] => 1
[3,6,2,4,1,5] => 1
[3,6,2,4,5,1] => 1
[3,6,2,5,1,4] => 1
[3,6,2,5,4,1] => 8
[3,6,4,1,2,5] => 9
[3,6,4,1,5,2] => 6
[3,6,4,2,1,5] => 1
[3,6,4,2,5,1] => 1
[3,6,4,5,1,2] => 8
[3,6,4,5,2,1] => 1
[3,6,5,1,2,4] => 1
[3,6,5,1,4,2] => 8
[3,6,5,2,1,4] => 9
[3,6,5,2,4,1] => 1
[3,6,5,4,1,2] => 6
[3,6,5,4,2,1] => 1
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 8
[4,1,2,5,3,6] => 1
[4,1,2,5,6,3] => 1
[4,1,2,6,3,5] => 1
[4,1,2,6,5,3] => 1
[4,1,3,2,5,6] => 1
[4,1,3,2,6,5] => 6
[4,1,3,5,2,6] => 1
[4,1,3,5,6,2] => 1
[4,1,3,6,2,5] => 1
[4,1,3,6,5,2] => 1
[4,1,5,2,3,6] => 6
[4,1,5,2,6,3] => 9
[4,1,5,3,2,6] => 1
[4,1,5,3,6,2] => 1
[4,1,5,6,2,3] => 1
[4,1,5,6,3,2] => 8
[4,1,6,2,3,5] => 9
[4,1,6,2,5,3] => 6
[4,1,6,3,2,5] => 1
[4,1,6,3,5,2] => 1
[4,1,6,5,2,3] => 8
[4,1,6,5,3,2] => 1
[4,2,1,3,5,6] => 1
[4,2,1,3,6,5] => 6
[4,2,1,5,3,6] => 1
[4,2,1,5,6,3] => 1
[4,2,1,6,3,5] => 1
[4,2,1,6,5,3] => 1
[4,2,3,1,5,6] => 1
[4,2,3,1,6,5] => 4
[4,2,3,5,1,6] => 1
[4,2,3,5,6,1] => 1
[4,2,3,6,1,5] => 1
[4,2,3,6,5,1] => 1
[4,2,5,1,3,6] => 4
[4,2,5,1,6,3] => 6
[4,2,5,3,1,6] => 1
[4,2,5,3,6,1] => 1
[4,2,5,6,1,3] => 1
[4,2,5,6,3,1] => 6
[4,2,6,1,3,5] => 6
[4,2,6,1,5,3] => 4
[4,2,6,3,1,5] => 1
[4,2,6,3,5,1] => 1
[4,2,6,5,1,3] => 6
[4,2,6,5,3,1] => 1
[4,3,1,2,5,6] => 1
[4,3,1,2,6,5] => 8
[4,3,1,5,2,6] => 1
[4,3,1,5,6,2] => 1
[4,3,1,6,2,5] => 1
[4,3,1,6,5,2] => 1
[4,3,2,1,5,6] => 4
[4,3,2,1,6,5] => 56
[4,3,2,5,1,6] => 6
[4,3,2,5,6,1] => 8
[4,3,2,6,1,5] => 8
[4,3,2,6,5,1] => 6
[4,3,5,1,2,6] => 6
[4,3,5,1,6,2] => 8
[4,3,5,2,1,6] => 1
[4,3,5,2,6,1] => 1
[4,3,5,6,1,2] => 1
[4,3,5,6,2,1] => 9
[4,3,6,1,2,5] => 8
[4,3,6,1,5,2] => 6
[4,3,6,2,1,5] => 1
[4,3,6,2,5,1] => 1
[4,3,6,5,1,2] => 9
[4,3,6,5,2,1] => 1
[4,5,1,2,3,6] => 1
[4,5,1,2,6,3] => 1
[4,5,1,3,2,6] => 6
[4,5,1,3,6,2] => 9
[4,5,1,6,2,3] => 8
[4,5,1,6,3,2] => 1
[4,5,2,1,3,6] => 6
[4,5,2,1,6,3] => 8
[4,5,2,3,1,6] => 1
[4,5,2,3,6,1] => 1
[4,5,2,6,1,3] => 1
[4,5,2,6,3,1] => 9
[4,5,3,1,2,6] => 4
[4,5,3,1,6,2] => 6
[4,5,3,2,1,6] => 1
[4,5,3,2,6,1] => 1
[4,5,3,6,1,2] => 1
[4,5,3,6,2,1] => 6
[4,5,6,1,2,3] => 56
[4,5,6,1,3,2] => 8
[4,5,6,2,1,3] => 8
[4,5,6,2,3,1] => 1
[4,5,6,3,1,2] => 1
[4,5,6,3,2,1] => 8
[4,6,1,2,3,5] => 1
[4,6,1,2,5,3] => 1
[4,6,1,3,2,5] => 9
[4,6,1,3,5,2] => 6
[4,6,1,5,2,3] => 1
[4,6,1,5,3,2] => 8
[4,6,2,1,3,5] => 8
[4,6,2,1,5,3] => 6
[4,6,2,3,1,5] => 1
[4,6,2,3,5,1] => 1
[4,6,2,5,1,3] => 9
[4,6,2,5,3,1] => 1
[4,6,3,1,2,5] => 6
[4,6,3,1,5,2] => 4
[4,6,3,2,1,5] => 1
[4,6,3,2,5,1] => 1
[4,6,3,5,1,2] => 6
[4,6,3,5,2,1] => 1
[4,6,5,1,2,3] => 8
[4,6,5,1,3,2] => 56
[4,6,5,2,1,3] => 1
[4,6,5,2,3,1] => 8
[4,6,5,3,1,2] => 8
[4,6,5,3,2,1] => 1
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 1
[5,1,2,4,3,6] => 1
[5,1,2,4,6,3] => 1
[5,1,2,6,3,4] => 8
[5,1,2,6,4,3] => 1
[5,1,3,2,4,6] => 1
[5,1,3,2,6,4] => 1
[5,1,3,4,2,6] => 1
[5,1,3,4,6,2] => 1
[5,1,3,6,2,4] => 6
[5,1,3,6,4,2] => 1
[5,1,4,2,3,6] => 1
[5,1,4,2,6,3] => 1
[5,1,4,3,2,6] => 6
[5,1,4,3,6,2] => 8
[5,1,4,6,2,3] => 9
[5,1,4,6,3,2] => 1
[5,1,6,2,3,4] => 1
[5,1,6,2,4,3] => 8
[5,1,6,3,2,4] => 9
[5,1,6,3,4,2] => 1
[5,1,6,4,2,3] => 6
[5,1,6,4,3,2] => 1
[5,2,1,3,4,6] => 1
[5,2,1,3,6,4] => 1
[5,2,1,4,3,6] => 1
[5,2,1,4,6,3] => 1
[5,2,1,6,3,4] => 6
[5,2,1,6,4,3] => 1
[5,2,3,1,4,6] => 1
[5,2,3,1,6,4] => 1
[5,2,3,4,1,6] => 1
[5,2,3,4,6,1] => 1
[5,2,3,6,1,4] => 4
[5,2,3,6,4,1] => 1
[5,2,4,1,3,6] => 1
[5,2,4,1,6,3] => 1
[5,2,4,3,1,6] => 4
[5,2,4,3,6,1] => 6
[5,2,4,6,1,3] => 6
[5,2,4,6,3,1] => 1
[5,2,6,1,3,4] => 1
[5,2,6,1,4,3] => 6
[5,2,6,3,1,4] => 6
[5,2,6,3,4,1] => 1
[5,2,6,4,1,3] => 4
[5,2,6,4,3,1] => 1
[5,3,1,2,4,6] => 1
[5,3,1,2,6,4] => 1
[5,3,1,4,2,6] => 1
[5,3,1,4,6,2] => 1
[5,3,1,6,2,4] => 8
[5,3,1,6,4,2] => 1
[5,3,2,1,4,6] => 6
[5,3,2,1,6,4] => 8
[5,3,2,4,1,6] => 4
[5,3,2,4,6,1] => 6
[5,3,2,6,1,4] => 56
[5,3,2,6,4,1] => 8
[5,3,4,1,2,6] => 1
[5,3,4,1,6,2] => 1
[5,3,4,2,1,6] => 6
[5,3,4,2,6,1] => 9
[5,3,4,6,1,2] => 8
[5,3,4,6,2,1] => 1
[5,3,6,1,2,4] => 1
[5,3,6,1,4,2] => 9
[5,3,6,2,1,4] => 8
[5,3,6,2,4,1] => 1
[5,3,6,4,1,2] => 6
[5,3,6,4,2,1] => 1
[5,4,1,2,3,6] => 6
[5,4,1,2,6,3] => 8
[5,4,1,3,2,6] => 1
[5,4,1,3,6,2] => 1
[5,4,1,6,2,3] => 1
[5,4,1,6,3,2] => 9
[5,4,2,1,3,6] => 1
[5,4,2,1,6,3] => 1
[5,4,2,3,1,6] => 6
[5,4,2,3,6,1] => 9
[5,4,2,6,1,3] => 8
[5,4,2,6,3,1] => 1
[5,4,3,1,2,6] => 1
[5,4,3,1,6,2] => 1
[5,4,3,2,1,6] => 4
[5,4,3,2,6,1] => 6
[5,4,3,6,1,2] => 6
[5,4,3,6,2,1] => 1
[5,4,6,1,2,3] => 8
[5,4,6,1,3,2] => 1
[5,4,6,2,1,3] => 56
[5,4,6,2,3,1] => 8
[5,4,6,3,1,2] => 8
[5,4,6,3,2,1] => 1
[5,6,1,2,3,4] => 9
[5,6,1,2,4,3] => 1
[5,6,1,3,2,4] => 1
[5,6,1,3,4,2] => 8
[5,6,1,4,2,3] => 1
[5,6,1,4,3,2] => 6
[5,6,2,1,3,4] => 1
[5,6,2,1,4,3] => 9
[5,6,2,3,1,4] => 8
[5,6,2,3,4,1] => 1
[5,6,2,4,1,3] => 6
[5,6,2,4,3,1] => 1
[5,6,3,1,2,4] => 1
[5,6,3,1,4,2] => 6
[5,6,3,2,1,4] => 6
[5,6,3,2,4,1] => 1
[5,6,3,4,1,2] => 4
[5,6,3,4,2,1] => 1
[5,6,4,1,2,3] => 1
[5,6,4,1,3,2] => 8
[5,6,4,2,1,3] => 8
[5,6,4,2,3,1] => 1
[5,6,4,3,1,2] => 56
[5,6,4,3,2,1] => 8
[6,1,2,3,4,5] => 1
[6,1,2,3,5,4] => 1
[6,1,2,4,3,5] => 1
[6,1,2,4,5,3] => 1
[6,1,2,5,3,4] => 1
[6,1,2,5,4,3] => 8
[6,1,3,2,4,5] => 1
[6,1,3,2,5,4] => 1
[6,1,3,4,2,5] => 1
[6,1,3,4,5,2] => 1
[6,1,3,5,2,4] => 1
[6,1,3,5,4,2] => 6
[6,1,4,2,3,5] => 1
[6,1,4,2,5,3] => 1
[6,1,4,3,2,5] => 8
[6,1,4,3,5,2] => 6
[6,1,4,5,2,3] => 1
[6,1,4,5,3,2] => 9
[6,1,5,2,3,4] => 8
[6,1,5,2,4,3] => 1
[6,1,5,3,2,4] => 1
[6,1,5,3,4,2] => 9
[6,1,5,4,2,3] => 1
[6,1,5,4,3,2] => 6
[6,2,1,3,4,5] => 1
[6,2,1,3,5,4] => 1
[6,2,1,4,3,5] => 1
[6,2,1,4,5,3] => 1
[6,2,1,5,3,4] => 1
[6,2,1,5,4,3] => 6
[6,2,3,1,4,5] => 1
[6,2,3,1,5,4] => 1
[6,2,3,4,1,5] => 1
[6,2,3,4,5,1] => 1
[6,2,3,5,1,4] => 1
[6,2,3,5,4,1] => 4
[6,2,4,1,3,5] => 1
[6,2,4,1,5,3] => 1
[6,2,4,3,1,5] => 6
[6,2,4,3,5,1] => 4
[6,2,4,5,1,3] => 1
[6,2,4,5,3,1] => 6
[6,2,5,1,3,4] => 6
[6,2,5,1,4,3] => 1
[6,2,5,3,1,4] => 1
[6,2,5,3,4,1] => 6
[6,2,5,4,1,3] => 1
[6,2,5,4,3,1] => 4
[6,3,1,2,4,5] => 1
[6,3,1,2,5,4] => 1
[6,3,1,4,2,5] => 1
[6,3,1,4,5,2] => 1
[6,3,1,5,2,4] => 1
[6,3,1,5,4,2] => 8
[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] => 8
[6,3,2,5,4,1] => 56
[6,3,4,1,2,5] => 1
[6,3,4,1,5,2] => 1
[6,3,4,2,1,5] => 9
[6,3,4,2,5,1] => 6
[6,3,4,5,1,2] => 1
[6,3,4,5,2,1] => 8
[6,3,5,1,2,4] => 9
[6,3,5,1,4,2] => 1
[6,3,5,2,1,4] => 1
[6,3,5,2,4,1] => 8
[6,3,5,4,1,2] => 1
[6,3,5,4,2,1] => 6
[6,4,1,2,3,5] => 8
[6,4,1,2,5,3] => 6
[6,4,1,3,2,5] => 1
[6,4,1,3,5,2] => 1
[6,4,1,5,2,3] => 9
[6,4,1,5,3,2] => 1
[6,4,2,1,3,5] => 1
[6,4,2,1,5,3] => 1
[6,4,2,3,1,5] => 9
[6,4,2,3,5,1] => 6
[6,4,2,5,1,3] => 1
[6,4,2,5,3,1] => 8
[6,4,3,1,2,5] => 1
[6,4,3,1,5,2] => 1
[6,4,3,2,1,5] => 6
[6,4,3,2,5,1] => 4
[6,4,3,5,1,2] => 1
[6,4,3,5,2,1] => 6
[6,4,5,1,2,3] => 1
[6,4,5,1,3,2] => 8
[6,4,5,2,1,3] => 8
[6,4,5,2,3,1] => 56
[6,4,5,3,1,2] => 1
[6,4,5,3,2,1] => 8
[6,5,1,2,3,4] => 1
[6,5,1,2,4,3] => 9
[6,5,1,3,2,4] => 8
[6,5,1,3,4,2] => 1
[6,5,1,4,2,3] => 6
[6,5,1,4,3,2] => 1
[6,5,2,1,3,4] => 9
[6,5,2,1,4,3] => 1
[6,5,2,3,1,4] => 1
[6,5,2,3,4,1] => 8
[6,5,2,4,1,3] => 1
[6,5,2,4,3,1] => 6
[6,5,3,1,2,4] => 6
[6,5,3,1,4,2] => 1
[6,5,3,2,1,4] => 1
[6,5,3,2,4,1] => 6
[6,5,3,4,1,2] => 1
[6,5,3,4,2,1] => 4
[6,5,4,1,2,3] => 8
[6,5,4,1,3,2] => 1
[6,5,4,2,1,3] => 1
[6,5,4,2,3,1] => 8
[6,5,4,3,1,2] => 8
[6,5,4,3,2,1] => 56
[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] => 24
[1,2,3,5,6,4,7] => 3
[1,2,3,5,6,7,4] => 4
[1,2,3,5,7,4,6] => 4
[1,2,3,5,7,6,4] => 3
[1,2,3,6,4,5,7] => 3
[1,2,3,6,4,7,5] => 4
[1,2,3,6,5,4,7] => 2
[1,2,3,6,5,7,4] => 3
[1,2,3,6,7,4,5] => 24
[1,2,3,6,7,5,4] => 4
[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] => 4
[1,2,3,7,6,5,4] => 24
[1,2,4,3,5,6,7] => 2
[1,2,4,3,5,7,6] => 24
[1,2,4,3,6,5,7] => 24
[1,2,4,3,6,7,5] => 42
[1,2,4,3,7,5,6] => 42
[1,2,4,3,7,6,5] => 24
[1,2,4,5,3,6,7] => 3
[1,2,4,5,3,7,6] => 42
[1,2,4,5,6,3,7] => 4
[1,2,4,5,6,7,3] => 5
[1,2,4,5,7,3,6] => 5
[1,2,4,5,7,6,3] => 4
[1,2,4,6,3,5,7] => 4
[1,2,4,6,3,7,5] => 5
[1,2,4,6,5,3,7] => 3
[1,2,4,6,5,7,3] => 4
[1,2,4,6,7,3,5] => 42
[1,2,4,6,7,5,3] => 5
[1,2,4,7,3,5,6] => 5
[1,2,4,7,3,6,5] => 4
[1,2,4,7,5,3,6] => 4
[1,2,4,7,5,6,3] => 3
[1,2,4,7,6,3,5] => 5
[1,2,4,7,6,5,3] => 42
[1,2,5,3,4,6,7] => 3
[1,2,5,3,4,7,6] => 42
[1,2,5,3,6,4,7] => 4
[1,2,5,3,6,7,4] => 5
[1,2,5,3,7,4,6] => 5
[1,2,5,3,7,6,4] => 4
[1,2,5,4,3,6,7] => 2
[1,2,5,4,3,7,6] => 24
[1,2,5,4,6,3,7] => 3
[1,2,5,4,6,7,3] => 4
[1,2,5,4,7,3,6] => 4
[1,2,5,4,7,6,3] => 3
[1,2,5,6,3,4,7] => 24
[1,2,5,6,3,7,4] => 42
[1,2,5,6,4,3,7] => 4
[1,2,5,6,4,7,3] => 5
[1,2,5,6,7,3,4] => 5
[1,2,5,6,7,4,3] => 42
[1,2,5,7,3,4,6] => 42
[1,2,5,7,3,6,4] => 24
[1,2,5,7,4,3,6] => 5
[1,2,5,7,4,6,3] => 4
[1,2,5,7,6,3,4] => 42
[1,2,5,7,6,4,3] => 5
[1,2,6,3,4,5,7] => 4
[1,2,6,3,4,7,5] => 5
[1,2,6,3,5,4,7] => 3
[1,2,6,3,5,7,4] => 4
[1,2,6,3,7,4,5] => 42
[1,2,6,3,7,5,4] => 5
[1,2,6,4,3,5,7] => 3
[1,2,6,4,3,7,5] => 4
[1,2,6,4,5,3,7] => 2
[1,2,6,4,5,7,3] => 3
[1,2,6,4,7,3,5] => 24
[1,2,6,4,7,5,3] => 4
[1,2,6,5,3,4,7] => 4
[1,2,6,5,3,7,4] => 5
[1,2,6,5,4,3,7] => 24
[1,2,6,5,4,7,3] => 42
[1,2,6,5,7,3,4] => 42
[1,2,6,5,7,4,3] => 5
[1,2,6,7,3,4,5] => 5
[1,2,6,7,3,5,4] => 42
[1,2,6,7,4,3,5] => 42
[1,2,6,7,4,5,3] => 5
[1,2,6,7,5,3,4] => 24
[1,2,6,7,5,4,3] => 4
[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] => 5
[1,2,7,3,6,5,4] => 42
[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] => 4
[1,2,7,4,6,5,3] => 24
[1,2,7,5,3,4,6] => 5
[1,2,7,5,3,6,4] => 4
[1,2,7,5,4,3,6] => 42
[1,2,7,5,4,6,3] => 24
[1,2,7,5,6,3,4] => 5
[1,2,7,5,6,4,3] => 42
[1,2,7,6,3,4,5] => 42
[1,2,7,6,3,5,4] => 5
[1,2,7,6,4,3,5] => 5
[1,2,7,6,4,5,3] => 42
[1,2,7,6,5,3,4] => 4
[1,2,7,6,5,4,3] => 24
[1,3,2,4,5,6,7] => 2
[1,3,2,4,5,7,6] => 24
[1,3,2,4,6,5,7] => 24
[1,3,2,4,6,7,5] => 42
[1,3,2,4,7,5,6] => 42
[1,3,2,4,7,6,5] => 24
[1,3,2,5,4,6,7] => 24
[1,3,2,5,4,7,6] => 576
[1,3,2,5,6,4,7] => 42
[1,3,2,5,6,7,4] => 64
[1,3,2,5,7,4,6] => 64
[1,3,2,5,7,6,4] => 42
[1,3,2,6,4,5,7] => 42
[1,3,2,6,4,7,5] => 64
[1,3,2,6,5,4,7] => 24
[1,3,2,6,5,7,4] => 42
[1,3,2,6,7,4,5] => 576
[1,3,2,6,7,5,4] => 64
[1,3,2,7,4,5,6] => 64
[1,3,2,7,4,6,5] => 42
[1,3,2,7,5,4,6] => 42
[1,3,2,7,5,6,4] => 24
[1,3,2,7,6,4,5] => 64
[1,3,2,7,6,5,4] => 576
[1,3,4,2,5,6,7] => 3
[1,3,4,2,5,7,6] => 42
[1,3,4,2,6,5,7] => 42
[1,3,4,2,6,7,5] => 72
[1,3,4,2,7,5,6] => 72
[1,3,4,2,7,6,5] => 42
[1,3,4,5,2,6,7] => 4
[1,3,4,5,2,7,6] => 64
[1,3,4,5,6,2,7] => 5
[1,3,4,5,6,7,2] => 6
[1,3,4,5,7,2,6] => 6
[1,3,4,5,7,6,2] => 5
[1,3,4,6,2,5,7] => 5
[1,3,4,6,2,7,5] => 6
[1,3,4,6,5,2,7] => 4
[1,3,4,6,5,7,2] => 5
[1,3,4,6,7,2,5] => 64
[1,3,4,6,7,5,2] => 6
[1,3,4,7,2,5,6] => 6
[1,3,4,7,2,6,5] => 5
[1,3,4,7,5,2,6] => 5
[1,3,4,7,5,6,2] => 4
[1,3,4,7,6,2,5] => 6
[1,3,4,7,6,5,2] => 64
[1,3,5,2,4,6,7] => 4
[1,3,5,2,4,7,6] => 64
[1,3,5,2,6,4,7] => 5
[1,3,5,2,6,7,4] => 6
[1,3,5,2,7,4,6] => 6
[1,3,5,2,7,6,4] => 5
[1,3,5,4,2,6,7] => 3
[1,3,5,4,2,7,6] => 42
[1,3,5,4,6,2,7] => 4
[1,3,5,4,6,7,2] => 5
[1,3,5,4,7,2,6] => 5
[1,3,5,4,7,6,2] => 4
[1,3,5,6,2,4,7] => 42
[1,3,5,6,2,7,4] => 72
[1,3,5,6,4,2,7] => 5
[1,3,5,6,4,7,2] => 6
[1,3,5,6,7,2,4] => 6
[1,3,5,6,7,4,2] => 64
[1,3,5,7,2,4,6] => 72
[1,3,5,7,2,6,4] => 42
[1,3,5,7,4,2,6] => 6
[1,3,5,7,4,6,2] => 5
[1,3,5,7,6,2,4] => 64
[1,3,5,7,6,4,2] => 6
[1,3,6,2,4,5,7] => 5
[1,3,6,2,4,7,5] => 6
[1,3,6,2,5,4,7] => 4
[1,3,6,2,5,7,4] => 5
[1,3,6,2,7,4,5] => 64
[1,3,6,2,7,5,4] => 6
[1,3,6,4,2,5,7] => 4
[1,3,6,4,2,7,5] => 5
[1,3,6,4,5,2,7] => 3
[1,3,6,4,5,7,2] => 4
[1,3,6,4,7,2,5] => 42
[1,3,6,4,7,5,2] => 5
[1,3,6,5,2,4,7] => 5
[1,3,6,5,2,7,4] => 6
[1,3,6,5,4,2,7] => 42
[1,3,6,5,4,7,2] => 64
[1,3,6,5,7,2,4] => 72
[1,3,6,5,7,4,2] => 6
[1,3,6,7,2,4,5] => 6
[1,3,6,7,2,5,4] => 64
[1,3,6,7,4,2,5] => 72
[1,3,6,7,4,5,2] => 6
[1,3,6,7,5,2,4] => 42
[1,3,6,7,5,4,2] => 5
[1,3,7,2,4,5,6] => 6
[1,3,7,2,4,6,5] => 5
[1,3,7,2,5,4,6] => 5
[1,3,7,2,5,6,4] => 4
[1,3,7,2,6,4,5] => 6
[1,3,7,2,6,5,4] => 64
[1,3,7,4,2,5,6] => 5
[1,3,7,4,2,6,5] => 4
[1,3,7,4,5,2,6] => 4
[1,3,7,4,5,6,2] => 3
[1,3,7,4,6,2,5] => 5
[1,3,7,4,6,5,2] => 42
[1,3,7,5,2,4,6] => 6
[1,3,7,5,2,6,4] => 5
[1,3,7,5,4,2,6] => 64
[1,3,7,5,4,6,2] => 42
[1,3,7,5,6,2,4] => 6
[1,3,7,5,6,4,2] => 72
[1,3,7,6,2,4,5] => 64
[1,3,7,6,2,5,4] => 6
[1,3,7,6,4,2,5] => 6
[1,3,7,6,4,5,2] => 72
[1,3,7,6,5,2,4] => 5
[1,3,7,6,5,4,2] => 42
[1,4,2,3,5,6,7] => 3
[1,4,2,3,5,7,6] => 42
[1,4,2,3,6,5,7] => 42
[1,4,2,3,6,7,5] => 72
[1,4,2,3,7,5,6] => 72
[1,4,2,3,7,6,5] => 42
[1,4,2,5,3,6,7] => 4
[1,4,2,5,3,7,6] => 64
[1,4,2,5,6,3,7] => 5
[1,4,2,5,6,7,3] => 6
[1,4,2,5,7,3,6] => 6
[1,4,2,5,7,6,3] => 5
[1,4,2,6,3,5,7] => 5
[1,4,2,6,3,7,5] => 6
[1,4,2,6,5,3,7] => 4
[1,4,2,6,5,7,3] => 5
[1,4,2,6,7,3,5] => 64
[1,4,2,6,7,5,3] => 6
[1,4,2,7,3,5,6] => 6
[1,4,2,7,3,6,5] => 5
[1,4,2,7,5,3,6] => 5
[1,4,2,7,5,6,3] => 4
[1,4,2,7,6,3,5] => 6
[1,4,2,7,6,5,3] => 64
[1,4,3,2,5,6,7] => 2
[1,4,3,2,5,7,6] => 24
[1,4,3,2,6,5,7] => 24
[1,4,3,2,6,7,5] => 42
[1,4,3,2,7,5,6] => 42
[1,4,3,2,7,6,5] => 24
[1,4,3,5,2,6,7] => 3
[1,4,3,5,2,7,6] => 42
[1,4,3,5,6,2,7] => 4
[1,4,3,5,6,7,2] => 5
[1,4,3,5,7,2,6] => 5
[1,4,3,5,7,6,2] => 4
[1,4,3,6,2,5,7] => 4
[1,4,3,6,2,7,5] => 5
[1,4,3,6,5,2,7] => 3
[1,4,3,6,5,7,2] => 4
[1,4,3,6,7,2,5] => 42
[1,4,3,6,7,5,2] => 5
[1,4,3,7,2,5,6] => 5
[1,4,3,7,2,6,5] => 4
[1,4,3,7,5,2,6] => 4
[1,4,3,7,5,6,2] => 3
[1,4,3,7,6,2,5] => 5
[1,4,3,7,6,5,2] => 42
[1,4,5,2,3,6,7] => 24
[1,4,5,2,3,7,6] => 576
[1,4,5,2,6,3,7] => 42
[1,4,5,2,6,7,3] => 64
[1,4,5,2,7,3,6] => 64
[1,4,5,2,7,6,3] => 42
[1,4,5,3,2,6,7] => 4
[1,4,5,3,2,7,6] => 64
[1,4,5,3,6,2,7] => 5
[1,4,5,3,6,7,2] => 6
[1,4,5,3,7,2,6] => 6
[1,4,5,3,7,6,2] => 5
[1,4,5,6,2,3,7] => 5
[1,4,5,6,2,7,3] => 6
[1,4,5,6,3,2,7] => 42
[1,4,5,6,3,7,2] => 64
[1,4,5,6,7,2,3] => 72
[1,4,5,6,7,3,2] => 6
[1,4,5,7,2,3,6] => 6
[1,4,5,7,2,6,3] => 5
[1,4,5,7,3,2,6] => 64
[1,4,5,7,3,6,2] => 42
[1,4,5,7,6,2,3] => 6
[1,4,5,7,6,3,2] => 72
[1,4,6,2,3,5,7] => 42
[1,4,6,2,3,7,5] => 64
[1,4,6,2,5,3,7] => 24
[1,4,6,2,5,7,3] => 42
[1,4,6,2,7,3,5] => 576
[1,4,6,2,7,5,3] => 64
[1,4,6,3,2,5,7] => 5
[1,4,6,3,2,7,5] => 6
[1,4,6,3,5,2,7] => 4
[1,4,6,3,5,7,2] => 5
[1,4,6,3,7,2,5] => 64
[1,4,6,3,7,5,2] => 6
[1,4,6,5,2,3,7] => 42
[1,4,6,5,2,7,3] => 72
[1,4,6,5,3,2,7] => 5
[1,4,6,5,3,7,2] => 6
click to show generating function       
Description
The number of minimal length factorizations of a permutation into star transpositions.
For a permutation $\pi\in\mathfrak S_n$ a minimal length factorization into star transpositions is a factorization of the form
$$\pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n,$$
where $\tau_a = (1,a)$ for $2 \leq a \leq n$ and $k$ is minimal.
[1, lem.2.1] shows that the minimal length of such a factorization is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$, see St001077The prefix exchange distance of a permutation..
[2, cor.2] shows that the number of such minimal factorizations is
$$ \frac{(n+m-2(k+1))!}{(n-k)!}\ell_1\cdots\ell_m, $$
where $\ell_1,\dots,\ell_m$ is the cycle type of $\pi$ and $k$ is the number of fixed point different from $1$.
References
[1] Irving, J., Rattan, A. Minimal factorizations of permutations into star transpositions MathSciNet:2721480
Code
def statistic(pi):
    L = pi.cycle_type()
    k = pi.number_of_fixed_points()
    if pi(1) == 1:
        k -= 1
    m = len(L)    
    n = len(pi)
    return factorial(n+m-2*(k+1))/factorial(n-k)*prod(L)

# alternative code for checking
@cached_function
def number_of_minimal_star_factorizations_dict(n):
    S = Permutations(n)
    g = [S(Permutation((1, i))) for i in range(2, n+1)]

    result = dict()
    result[S(Permutation([]))] = 1
    l = 1 # the length of the minimal factorization
    while len(result) < factorial(n):
        result_l = dict()
        for w in cartesian_product([g]*l):
            pi = prod(w)
            if pi not in result:
                result_l[pi] = result_l.get(pi, 0) + 1
        for pi, v in result_l.items():
            result[pi] = v
        l += 1
    return result

def statistic(pi):
    d = number_of_minimal_star_factorizations_dict(len(pi))
    return d[pi]
Created
Jan 09, 2018 at 21:52 by Martin Rubey
Updated
Jan 09, 2018 at 21:52 by Martin Rubey