Identifier
Identifier
Values
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 3
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[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] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 3
[1,5,2,3,4] => 2
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 3
[1,5,4,2,3] => 3
[1,5,4,3,2] => 1
[2,1,3,4,5] => 1
[2,1,3,5,4] => 1
[2,1,4,3,5] => 1
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 1
[2,3,1,4,5] => 2
[2,3,1,5,4] => 2
[2,3,4,1,5] => 2
[2,3,4,5,1] => 4
[2,3,5,1,4] => 2
[2,3,5,4,1] => 4
[2,4,1,3,5] => 2
[2,4,1,5,3] => 2
[2,4,3,1,5] => 2
[2,4,3,5,1] => 2
[2,4,5,1,3] => 4
[2,4,5,3,1] => 4
[2,5,1,3,4] => 2
[2,5,1,4,3] => 2
[2,5,3,1,4] => 4
[2,5,3,4,1] => 4
[2,5,4,1,3] => 2
[2,5,4,3,1] => 2
[3,1,2,4,5] => 2
[3,1,2,5,4] => 2
[3,1,4,2,5] => 2
[3,1,4,5,2] => 2
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 1
[3,2,1,5,4] => 1
[3,2,4,1,5] => 2
[3,2,4,5,1] => 4
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 1
[3,4,1,5,2] => 4
[3,4,2,1,5] => 3
[3,4,2,5,1] => 4
[3,4,5,1,2] => 6
[3,4,5,2,1] => 3
[3,5,1,2,4] => 4
[3,5,1,4,2] => 1
[3,5,2,1,4] => 2
[3,5,2,4,1] => 6
[3,5,4,1,2] => 2
[3,5,4,2,1] => 3
[4,1,2,3,5] => 2
[4,1,2,5,3] => 2
[4,1,3,2,5] => 2
[4,1,3,5,2] => 4
[4,1,5,2,3] => 4
[4,1,5,3,2] => 2
[4,2,1,3,5] => 2
[4,2,1,5,3] => 2
[4,2,3,1,5] => 3
[4,2,3,5,1] => 4
[4,2,5,1,3] => 1
[4,2,5,3,1] => 6
[4,3,1,2,5] => 3
[4,3,1,5,2] => 2
[4,3,2,1,5] => 1
[4,3,2,5,1] => 2
[4,3,5,1,2] => 2
[4,3,5,2,1] => 3
[4,5,1,2,3] => 6
[4,5,1,3,2] => 2
[4,5,2,1,3] => 2
[4,5,2,3,1] => 3
[4,5,3,1,2] => 3
[4,5,3,2,1] => 4
[5,1,2,3,4] => 4
[5,1,2,4,3] => 4
[5,1,3,2,4] => 2
[5,1,3,4,2] => 4
[5,1,4,2,3] => 4
[5,1,4,3,2] => 2
[5,2,1,3,4] => 4
[5,2,1,4,3] => 2
[5,2,3,1,4] => 4
[5,2,3,4,1] => 3
[5,2,4,1,3] => 6
[5,2,4,3,1] => 3
[5,3,1,2,4] => 4
[5,3,1,4,2] => 6
[5,3,2,1,4] => 2
[5,3,2,4,1] => 3
[5,3,4,1,2] => 3
[5,3,4,2,1] => 4
[5,4,1,2,3] => 3
[5,4,1,3,2] => 3
[5,4,2,1,3] => 3
[5,4,2,3,1] => 4
[5,4,3,1,2] => 4
[5,4,3,2,1] => 1
[1,2,3,4,5,6] => 1
[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] => 1
[1,2,4,3,5,6] => 1
[1,2,4,3,6,5] => 1
[1,2,4,5,3,6] => 2
[1,2,4,5,6,3] => 2
[1,2,4,6,3,5] => 2
[1,2,4,6,5,3] => 2
[1,2,5,3,4,6] => 2
[1,2,5,3,6,4] => 2
[1,2,5,4,3,6] => 1
[1,2,5,4,6,3] => 2
[1,2,5,6,3,4] => 1
[1,2,5,6,4,3] => 3
[1,2,6,3,4,5] => 2
[1,2,6,3,5,4] => 2
[1,2,6,4,3,5] => 2
[1,2,6,4,5,3] => 3
[1,2,6,5,3,4] => 3
[1,2,6,5,4,3] => 1
[1,3,2,4,5,6] => 1
[1,3,2,4,6,5] => 1
[1,3,2,5,4,6] => 1
[1,3,2,5,6,4] => 2
[1,3,2,6,4,5] => 2
[1,3,2,6,5,4] => 1
[1,3,4,2,5,6] => 2
[1,3,4,2,6,5] => 2
[1,3,4,5,2,6] => 2
[1,3,4,5,6,2] => 4
[1,3,4,6,2,5] => 2
[1,3,4,6,5,2] => 4
[1,3,5,2,4,6] => 2
[1,3,5,2,6,4] => 2
[1,3,5,4,2,6] => 2
[1,3,5,4,6,2] => 2
[1,3,5,6,2,4] => 4
[1,3,5,6,4,2] => 4
[1,3,6,2,4,5] => 2
[1,3,6,2,5,4] => 2
[1,3,6,4,2,5] => 4
[1,3,6,4,5,2] => 4
[1,3,6,5,2,4] => 2
[1,3,6,5,4,2] => 2
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 2
[1,4,2,5,3,6] => 2
[1,4,2,5,6,3] => 2
[1,4,2,6,3,5] => 2
[1,4,2,6,5,3] => 2
[1,4,3,2,5,6] => 1
[1,4,3,2,6,5] => 1
[1,4,3,5,2,6] => 2
[1,4,3,5,6,2] => 4
[1,4,3,6,2,5] => 2
[1,4,3,6,5,2] => 2
[1,4,5,2,3,6] => 1
[1,4,5,2,6,3] => 4
[1,4,5,3,2,6] => 3
[1,4,5,3,6,2] => 4
[1,4,5,6,2,3] => 6
[1,4,5,6,3,2] => 3
[1,4,6,2,3,5] => 4
[1,4,6,2,5,3] => 1
[1,4,6,3,2,5] => 2
[1,4,6,3,5,2] => 6
[1,4,6,5,2,3] => 2
[1,4,6,5,3,2] => 3
[1,5,2,3,4,6] => 2
[1,5,2,3,6,4] => 2
[1,5,2,4,3,6] => 2
[1,5,2,4,6,3] => 4
[1,5,2,6,3,4] => 4
[1,5,2,6,4,3] => 2
[1,5,3,2,4,6] => 2
[1,5,3,2,6,4] => 2
[1,5,3,4,2,6] => 3
[1,5,3,4,6,2] => 4
[1,5,3,6,2,4] => 1
[1,5,3,6,4,2] => 6
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 2
[1,5,4,3,2,6] => 1
[1,5,4,3,6,2] => 2
[1,5,4,6,2,3] => 2
[1,5,4,6,3,2] => 3
[1,5,6,2,3,4] => 6
[1,5,6,2,4,3] => 2
[1,5,6,3,2,4] => 2
[1,5,6,3,4,2] => 3
[1,5,6,4,2,3] => 3
[1,5,6,4,3,2] => 4
[1,6,2,3,4,5] => 4
[1,6,2,3,5,4] => 4
[1,6,2,4,3,5] => 2
[1,6,2,4,5,3] => 4
[1,6,2,5,3,4] => 4
[1,6,2,5,4,3] => 2
[1,6,3,2,4,5] => 4
[1,6,3,2,5,4] => 2
[1,6,3,4,2,5] => 4
[1,6,3,4,5,2] => 3
[1,6,3,5,2,4] => 6
[1,6,3,5,4,2] => 3
[1,6,4,2,3,5] => 4
[1,6,4,2,5,3] => 6
[1,6,4,3,2,5] => 2
[1,6,4,3,5,2] => 3
[1,6,4,5,2,3] => 3
[1,6,4,5,3,2] => 4
[1,6,5,2,3,4] => 3
[1,6,5,2,4,3] => 3
[1,6,5,3,2,4] => 3
[1,6,5,3,4,2] => 4
[1,6,5,4,2,3] => 4
[1,6,5,4,3,2] => 1
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 1
[2,1,3,5,4,6] => 1
[2,1,3,5,6,4] => 2
[2,1,3,6,4,5] => 2
[2,1,3,6,5,4] => 1
[2,1,4,3,5,6] => 1
[2,1,4,3,6,5] => 1
[2,1,4,5,3,6] => 2
[2,1,4,5,6,3] => 2
[2,1,4,6,3,5] => 2
[2,1,4,6,5,3] => 2
[2,1,5,3,4,6] => 2
[2,1,5,3,6,4] => 2
[2,1,5,4,3,6] => 1
[2,1,5,4,6,3] => 2
[2,1,5,6,3,4] => 1
[2,1,5,6,4,3] => 3
[2,1,6,3,4,5] => 2
[2,1,6,3,5,4] => 2
[2,1,6,4,3,5] => 2
[2,1,6,4,5,3] => 3
[2,1,6,5,3,4] => 3
[2,1,6,5,4,3] => 1
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 2
[2,3,1,5,4,6] => 2
[2,3,1,5,6,4] => 4
[2,3,1,6,4,5] => 4
[2,3,1,6,5,4] => 2
[2,3,4,1,5,6] => 2
[2,3,4,1,6,5] => 2
[2,3,4,5,1,6] => 4
[2,3,4,5,6,1] => 4
[2,3,4,6,1,5] => 4
[2,3,4,6,5,1] => 2
[2,3,5,1,4,6] => 2
[2,3,5,1,6,4] => 2
[2,3,5,4,1,6] => 4
[2,3,5,4,6,1] => 4
[2,3,5,6,1,4] => 4
[2,3,5,6,4,1] => 8
[2,3,6,1,4,5] => 2
[2,3,6,1,5,4] => 2
[2,3,6,4,1,5] => 8
[2,3,6,4,5,1] => 8
[2,3,6,5,1,4] => 4
[2,3,6,5,4,1] => 4
[2,4,1,3,5,6] => 2
[2,4,1,3,6,5] => 2
[2,4,1,5,3,6] => 2
[2,4,1,5,6,3] => 2
[2,4,1,6,3,5] => 2
[2,4,1,6,5,3] => 2
[2,4,3,1,5,6] => 2
[2,4,3,1,6,5] => 2
[2,4,3,5,1,6] => 2
[2,4,3,5,6,1] => 4
[2,4,3,6,1,5] => 2
[2,4,3,6,5,1] => 4
[2,4,5,1,3,6] => 4
[2,4,5,1,6,3] => 8
[2,4,5,3,1,6] => 4
[2,4,5,3,6,1] => 6
[2,4,5,6,1,3] => 4
[2,4,5,6,3,1] => 6
[2,4,6,1,3,5] => 8
[2,4,6,1,5,3] => 4
[2,4,6,3,1,5] => 4
[2,4,6,3,5,1] => 8
[2,4,6,5,1,3] => 2
[2,4,6,5,3,1] => 8
[2,5,1,3,4,6] => 2
[2,5,1,3,6,4] => 2
[2,5,1,4,3,6] => 2
[2,5,1,4,6,3] => 4
[2,5,1,6,3,4] => 4
[2,5,1,6,4,3] => 2
[2,5,3,1,4,6] => 4
[2,5,3,1,6,4] => 4
[2,5,3,4,1,6] => 4
[2,5,3,4,6,1] => 6
[2,5,3,6,1,4] => 4
[2,5,3,6,4,1] => 8
[2,5,4,1,3,6] => 2
[2,5,4,1,6,3] => 4
[2,5,4,3,1,6] => 2
[2,5,4,3,6,1] => 2
[2,5,4,6,1,3] => 8
[2,5,4,6,3,1] => 4
[2,5,6,1,3,4] => 8
[2,5,6,1,4,3] => 4
[2,5,6,3,1,4] => 8
[2,5,6,3,4,1] => 2
[2,5,6,4,1,3] => 8
[2,5,6,4,3,1] => 6
[2,6,1,3,4,5] => 4
[2,6,1,3,5,4] => 4
[2,6,1,4,3,5] => 2
[2,6,1,4,5,3] => 4
[2,6,1,5,3,4] => 4
[2,6,1,5,4,3] => 2
[2,6,3,1,4,5] => 8
[2,6,3,1,5,4] => 4
[2,6,3,4,1,5] => 2
[2,6,3,4,5,1] => 6
[2,6,3,5,1,4] => 4
[2,6,3,5,4,1] => 8
[2,6,4,1,3,5] => 4
[2,6,4,1,5,3] => 8
[2,6,4,3,1,5] => 6
[2,6,4,3,5,1] => 4
[2,6,4,5,1,3] => 8
[2,6,4,5,3,1] => 6
[2,6,5,1,3,4] => 2
[2,6,5,1,4,3] => 2
[2,6,5,3,1,4] => 6
[2,6,5,3,4,1] => 6
[2,6,5,4,1,3] => 2
[2,6,5,4,3,1] => 2
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 2
[3,1,2,5,4,6] => 2
[3,1,2,5,6,4] => 4
[3,1,2,6,4,5] => 4
[3,1,2,6,5,4] => 2
[3,1,4,2,5,6] => 2
[3,1,4,2,6,5] => 2
[3,1,4,5,2,6] => 2
[3,1,4,5,6,2] => 4
[3,1,4,6,2,5] => 2
[3,1,4,6,5,2] => 4
[3,1,5,2,4,6] => 2
[3,1,5,2,6,4] => 2
[3,1,5,4,2,6] => 2
[3,1,5,4,6,2] => 2
[3,1,5,6,2,4] => 4
[3,1,5,6,4,2] => 4
[3,1,6,2,4,5] => 2
[3,1,6,2,5,4] => 2
[3,1,6,4,2,5] => 4
[3,1,6,4,5,2] => 4
[3,1,6,5,2,4] => 2
[3,1,6,5,4,2] => 2
[3,2,1,4,5,6] => 1
[3,2,1,4,6,5] => 1
[3,2,1,5,4,6] => 1
[3,2,1,5,6,4] => 2
[3,2,1,6,4,5] => 2
[3,2,1,6,5,4] => 1
[3,2,4,1,5,6] => 2
[3,2,4,1,6,5] => 2
[3,2,4,5,1,6] => 4
[3,2,4,5,6,1] => 2
[3,2,4,6,1,5] => 4
[3,2,4,6,5,1] => 3
[3,2,5,1,4,6] => 2
[3,2,5,1,6,4] => 2
[3,2,5,4,1,6] => 2
[3,2,5,4,6,1] => 4
[3,2,5,6,1,4] => 4
[3,2,5,6,4,1] => 6
[3,2,6,1,4,5] => 2
[3,2,6,1,5,4] => 2
[3,2,6,4,1,5] => 4
[3,2,6,4,5,1] => 6
[3,2,6,5,1,4] => 2
[3,2,6,5,4,1] => 2
[3,4,1,2,5,6] => 1
[3,4,1,2,6,5] => 1
[3,4,1,5,2,6] => 4
[3,4,1,5,6,2] => 4
[3,4,1,6,2,5] => 4
[3,4,1,6,5,2] => 4
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 3
[3,4,2,5,1,6] => 4
[3,4,2,5,6,1] => 8
[3,4,2,6,1,5] => 4
[3,4,2,6,5,1] => 6
[3,4,5,1,2,6] => 6
[3,4,5,1,6,2] => 4
[3,4,5,2,1,6] => 3
[3,4,5,2,6,1] => 6
[3,4,5,6,1,2] => 4
[3,4,5,6,2,1] => 9
[3,4,6,1,2,5] => 8
[3,4,6,1,5,2] => 6
[3,4,6,2,1,5] => 2
[3,4,6,2,5,1] => 8
[3,4,6,5,1,2] => 3
[3,4,6,5,2,1] => 7
[3,5,1,2,4,6] => 4
[3,5,1,2,6,4] => 4
[3,5,1,4,2,6] => 1
[3,5,1,4,6,2] => 4
[3,5,1,6,2,4] => 3
[3,5,1,6,4,2] => 6
[3,5,2,1,4,6] => 2
[3,5,2,1,6,4] => 2
[3,5,2,4,1,6] => 6
[3,5,2,4,6,1] => 8
[3,5,2,6,1,4] => 6
[3,5,2,6,4,1] => 6
[3,5,4,1,2,6] => 2
[3,5,4,1,6,2] => 8
[3,5,4,2,1,6] => 3
[3,5,4,2,6,1] => 4
[3,5,4,6,1,2] => 9
[3,5,4,6,2,1] => 3
[3,5,6,1,2,4] => 6
[3,5,6,1,4,2] => 6
[3,5,6,2,1,4] => 8
[3,5,6,2,4,1] => 7
[3,5,6,4,1,2] => 8
[3,5,6,4,2,1] => 6
[3,6,1,2,4,5] => 4
[3,6,1,2,5,4] => 4
[3,6,1,4,2,5] => 4
[3,6,1,4,5,2] => 1
[3,6,1,5,2,4] => 6
[3,6,1,5,4,2] => 1
[3,6,2,1,4,5] => 4
[3,6,2,1,5,4] => 2
[3,6,2,4,1,5] => 4
[3,6,2,4,5,1] => 8
[3,6,2,5,1,4] => 6
[3,6,2,5,4,1] => 6
[3,6,4,1,2,5] => 8
[3,6,4,1,5,2] => 8
[3,6,4,2,1,5] => 6
[3,6,4,2,5,1] => 9
[3,6,4,5,1,2] => 8
[3,6,4,5,2,1] => 6
[3,6,5,1,2,4] => 8
[3,6,5,1,4,2] => 2
[3,6,5,2,1,4] => 4
[3,6,5,2,4,1] => 8
[3,6,5,4,1,2] => 2
[3,6,5,4,2,1] => 3
[4,1,2,3,5,6] => 2
[4,1,2,3,6,5] => 2
[4,1,2,5,3,6] => 2
[4,1,2,5,6,3] => 2
[4,1,2,6,3,5] => 2
[4,1,2,6,5,3] => 2
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 2
[4,1,3,5,2,6] => 4
[4,1,3,5,6,2] => 8
[4,1,3,6,2,5] => 4
[4,1,3,6,5,2] => 4
[4,1,5,2,3,6] => 4
[4,1,5,2,6,3] => 8
[4,1,5,3,2,6] => 2
[4,1,5,3,6,2] => 4
[4,1,5,6,2,3] => 8
[4,1,5,6,3,2] => 2
[4,1,6,2,3,5] => 8
[4,1,6,2,5,3] => 4
[4,1,6,3,2,5] => 4
[4,1,6,3,5,2] => 8
[4,1,6,5,2,3] => 4
[4,1,6,5,3,2] => 2
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 2
[4,2,1,5,3,6] => 2
[4,2,1,5,6,3] => 2
[4,2,1,6,3,5] => 2
[4,2,1,6,5,3] => 2
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 3
[4,2,3,5,1,6] => 4
[4,2,3,5,6,1] => 8
[4,2,3,6,1,5] => 4
[4,2,3,6,5,1] => 6
[4,2,5,1,3,6] => 1
[4,2,5,1,6,3] => 4
[4,2,5,3,1,6] => 6
[4,2,5,3,6,1] => 8
[4,2,5,6,1,3] => 6
[4,2,5,6,3,1] => 8
[4,2,6,1,3,5] => 4
[4,2,6,1,5,3] => 1
[4,2,6,3,1,5] => 8
[4,2,6,3,5,1] => 6
[4,2,6,5,1,3] => 2
[4,2,6,5,3,1] => 6
[4,3,1,2,5,6] => 3
[4,3,1,2,6,5] => 3
[4,3,1,5,2,6] => 2
[4,3,1,5,6,2] => 4
[4,3,1,6,2,5] => 2
[4,3,1,6,5,2] => 2
[4,3,2,1,5,6] => 1
[4,3,2,1,6,5] => 1
[4,3,2,5,1,6] => 2
[4,3,2,5,6,1] => 4
[4,3,2,6,1,5] => 2
[4,3,2,6,5,1] => 2
[4,3,5,1,2,6] => 2
[4,3,5,1,6,2] => 2
[4,3,5,2,1,6] => 3
[4,3,5,2,6,1] => 8
[4,3,5,6,1,2] => 3
[4,3,5,6,2,1] => 7
[4,3,6,1,2,5] => 4
[4,3,6,1,5,2] => 2
[4,3,6,2,1,5] => 2
[4,3,6,2,5,1] => 6
[4,3,6,5,1,2] => 4
[4,3,6,5,2,1] => 4
[4,5,1,2,3,6] => 6
[4,5,1,2,6,3] => 8
[4,5,1,3,2,6] => 2
[4,5,1,3,6,2] => 8
[4,5,1,6,2,3] => 6
[4,5,1,6,3,2] => 8
[4,5,2,1,3,6] => 2
[4,5,2,1,6,3] => 4
[4,5,2,3,1,6] => 3
[4,5,2,3,6,1] => 2
[4,5,2,6,1,3] => 6
[4,5,2,6,3,1] => 7
[4,5,3,1,2,6] => 3
[4,5,3,1,6,2] => 8
[4,5,3,2,1,6] => 4
[4,5,3,2,6,1] => 6
[4,5,3,6,1,2] => 8
[4,5,3,6,2,1] => 6
[4,5,6,1,2,3] => 4
[4,5,6,1,3,2] => 4
[4,5,6,2,1,3] => 4
[4,5,6,2,3,1] => 12
[4,5,6,3,1,2] => 12
[4,5,6,3,2,1] => 4
[4,6,1,2,3,5] => 4
[4,6,1,2,5,3] => 6
[4,6,1,3,2,5] => 8
[4,6,1,3,5,2] => 8
[4,6,1,5,2,3] => 6
[4,6,1,5,3,2] => 2
[4,6,2,1,3,5] => 2
[4,6,2,1,5,3] => 2
[4,6,2,3,1,5] => 8
[4,6,2,3,5,1] => 7
[4,6,2,5,1,3] => 4
[4,6,2,5,3,1] => 4
[4,6,3,1,2,5] => 8
[4,6,3,1,5,2] => 3
[4,6,3,2,1,5] => 2
[4,6,3,2,5,1] => 8
[4,6,3,5,1,2] => 6
[4,6,3,5,2,1] => 12
[4,6,5,1,2,3] => 4
[4,6,5,1,3,2] => 1
[4,6,5,2,1,3] => 6
[4,6,5,2,3,1] => 6
[4,6,5,3,1,2] => 6
[4,6,5,3,2,1] => 4
[5,1,2,3,4,6] => 4
[5,1,2,3,6,4] => 4
[5,1,2,4,3,6] => 4
[5,1,2,4,6,3] => 8
[5,1,2,6,3,4] => 4
[5,1,2,6,4,3] => 4
[5,1,3,2,4,6] => 2
[5,1,3,2,6,4] => 2
[5,1,3,4,2,6] => 4
[5,1,3,4,6,2] => 2
[5,1,3,6,2,4] => 4
[5,1,3,6,4,2] => 4
[5,1,4,2,3,6] => 4
[5,1,4,2,6,3] => 4
[5,1,4,3,2,6] => 2
[5,1,4,3,6,2] => 6
[5,1,4,6,2,3] => 8
[5,1,4,6,3,2] => 6
[5,1,6,2,3,4] => 4
[5,1,6,2,4,3] => 2
[5,1,6,3,2,4] => 8
[5,1,6,3,4,2] => 8
[5,1,6,4,2,3] => 8
[5,1,6,4,3,2] => 2
[5,2,1,3,4,6] => 4
[5,2,1,3,6,4] => 4
[5,2,1,4,3,6] => 2
[5,2,1,4,6,3] => 4
[5,2,1,6,3,4] => 4
[5,2,1,6,4,3] => 2
[5,2,3,1,4,6] => 4
[5,2,3,1,6,4] => 4
[5,2,3,4,1,6] => 3
[5,2,3,4,6,1] => 6
[5,2,3,6,1,4] => 1
[5,2,3,6,4,1] => 8
[5,2,4,1,3,6] => 6
[5,2,4,1,6,3] => 8
[5,2,4,3,1,6] => 3
[5,2,4,3,6,1] => 4
[5,2,4,6,1,3] => 8
[5,2,4,6,3,1] => 9
[5,2,6,1,3,4] => 6
[5,2,6,1,4,3] => 2
[5,2,6,3,1,4] => 8
[5,2,6,3,4,1] => 7
[5,2,6,4,1,3] => 3
[5,2,6,4,3,1] => 8
[5,3,1,2,4,6] => 4
[5,3,1,2,6,4] => 4
[5,3,1,4,2,6] => 6
[5,3,1,4,6,2] => 4
[5,3,1,6,2,4] => 6
[5,3,1,6,4,2] => 6
[5,3,2,1,4,6] => 2
[5,3,2,1,6,4] => 2
[5,3,2,4,1,6] => 3
[5,3,2,4,6,1] => 8
[5,3,2,6,1,4] => 1
[5,3,2,6,4,1] => 6
[5,3,4,1,2,6] => 3
[5,3,4,1,6,2] => 8
[5,3,4,2,1,6] => 4
[5,3,4,2,6,1] => 6
[5,3,4,6,1,2] => 8
[5,3,4,6,2,1] => 6
[5,3,6,1,2,4] => 6
[5,3,6,1,4,2] => 4
[5,3,6,2,1,4] => 2
[5,3,6,2,4,1] => 4
[5,3,6,4,1,2] => 6
[5,3,6,4,2,1] => 12
[5,4,1,2,3,6] => 3
[5,4,1,2,6,3] => 2
[5,4,1,3,2,6] => 3
[5,4,1,3,6,2] => 6
[5,4,1,6,2,3] => 8
[5,4,1,6,3,2] => 4
[5,4,2,1,3,6] => 3
[5,4,2,1,6,3] => 2
[5,4,2,3,1,6] => 4
[5,4,2,3,6,1] => 6
[5,4,2,6,1,3] => 2
[5,4,2,6,3,1] => 8
[5,4,3,1,2,6] => 4
[5,4,3,1,6,2] => 2
[5,4,3,2,1,6] => 1
[5,4,3,2,6,1] => 2
[5,4,3,6,1,2] => 2
[5,4,3,6,2,1] => 3
[5,4,6,1,2,3] => 4
[5,4,6,1,3,2] => 6
[5,4,6,2,1,3] => 1
[5,4,6,2,3,1] => 6
[5,4,6,3,1,2] => 6
[5,4,6,3,2,1] => 4
[5,6,1,2,3,4] => 4
[5,6,1,2,4,3] => 3
[5,6,1,3,2,4] => 9
[5,6,1,3,4,2] => 8
[5,6,1,4,2,3] => 8
[5,6,1,4,3,2] => 2
[5,6,2,1,3,4] => 3
[5,6,2,1,4,3] => 4
[5,6,2,3,1,4] => 8
[5,6,2,3,4,1] => 12
[5,6,2,4,1,3] => 6
[5,6,2,4,3,1] => 6
[5,6,3,1,2,4] => 8
[5,6,3,1,4,2] => 6
[5,6,3,2,1,4] => 2
[5,6,3,2,4,1] => 6
[5,6,3,4,1,2] => 1
[5,6,3,4,2,1] => 6
[5,6,4,1,2,3] => 12
[5,6,4,1,3,2] => 6
[5,6,4,2,1,3] => 6
[5,6,4,2,3,1] => 6
[5,6,4,3,1,2] => 6
[5,6,4,3,2,1] => 5
[6,1,2,3,4,5] => 4
[6,1,2,3,5,4] => 2
[6,1,2,4,3,5] => 4
[6,1,2,4,5,3] => 8
[6,1,2,5,3,4] => 8
[6,1,2,5,4,3] => 4
[6,1,3,2,4,5] => 4
[6,1,3,2,5,4] => 4
[6,1,3,4,2,5] => 6
[6,1,3,4,5,2] => 6
[6,1,3,5,2,4] => 8
[6,1,3,5,4,2] => 8
[6,1,4,2,3,5] => 6
[6,1,4,2,5,3] => 8
[6,1,4,3,2,5] => 2
[6,1,4,3,5,2] => 4
[6,1,4,5,2,3] => 2
[6,1,4,5,3,2] => 6
[6,1,5,2,3,4] => 6
[6,1,5,2,4,3] => 8
[6,1,5,3,2,4] => 4
[6,1,5,3,4,2] => 6
[6,1,5,4,2,3] => 6
[6,1,5,4,3,2] => 2
[6,2,1,3,4,5] => 2
[6,2,1,3,5,4] => 3
[6,2,1,4,3,5] => 4
[6,2,1,4,5,3] => 6
[6,2,1,5,3,4] => 6
[6,2,1,5,4,3] => 2
[6,2,3,1,4,5] => 8
[6,2,3,1,5,4] => 6
[6,2,3,4,1,5] => 6
[6,2,3,4,5,1] => 9
[6,2,3,5,1,4] => 8
[6,2,3,5,4,1] => 7
[6,2,4,1,3,5] => 8
[6,2,4,1,5,3] => 6
[6,2,4,3,1,5] => 4
[6,2,4,3,5,1] => 3
[6,2,4,5,1,3] => 7
[6,2,4,5,3,1] => 6
[6,2,5,1,3,4] => 8
[6,2,5,1,4,3] => 6
[6,2,5,3,1,4] => 9
[6,2,5,3,4,1] => 6
[6,2,5,4,1,3] => 8
[6,2,5,4,3,1] => 3
[6,3,1,2,4,5] => 8
[6,3,1,2,5,4] => 6
[6,3,1,4,2,5] => 8
[6,3,1,4,5,2] => 8
[6,3,1,5,2,4] => 6
[6,3,1,5,4,2] => 6
[6,3,2,1,4,5] => 4
[6,3,2,1,5,4] => 2
[6,3,2,4,1,5] => 8
[6,3,2,4,5,1] => 7
[6,3,2,5,1,4] => 6
[6,3,2,5,4,1] => 4
[6,3,4,1,2,5] => 2
[6,3,4,1,5,2] => 7
[6,3,4,2,1,5] => 6
[6,3,4,2,5,1] => 6
[6,3,4,5,1,2] => 12
[6,3,4,5,2,1] => 4
[6,3,5,1,2,4] => 7
[6,3,5,1,4,2] => 4
[6,3,5,2,1,4] => 8
[6,3,5,2,4,1] => 12
[6,3,5,4,1,2] => 6
[6,3,5,4,2,1] => 4
[6,4,1,2,3,5] => 6
[6,4,1,2,5,3] => 8
[6,4,1,3,2,5] => 4
[6,4,1,3,5,2] => 9
[6,4,1,5,2,3] => 7
[6,4,1,5,3,2] => 8
[6,4,2,1,3,5] => 8
[6,4,2,1,5,3] => 6
[6,4,2,3,1,5] => 6
[6,4,2,3,5,1] => 6
[6,4,2,5,1,3] => 4
[6,4,2,5,3,1] => 12
[6,4,3,1,2,5] => 6
[6,4,3,1,5,2] => 8
[6,4,3,2,1,5] => 2
[6,4,3,2,5,1] => 3
[6,4,3,5,1,2] => 6
[6,4,3,5,2,1] => 4
[6,4,5,1,2,3] => 12
[6,4,5,1,3,2] => 6
[6,4,5,2,1,3] => 6
[6,4,5,2,3,1] => 6
[6,4,5,3,1,2] => 6
[6,4,5,3,2,1] => 5
[6,5,1,2,3,4] => 9
[6,5,1,2,4,3] => 7
[6,5,1,3,2,4] => 3
[6,5,1,3,4,2] => 6
[6,5,1,4,2,3] => 6
[6,5,1,4,3,2] => 3
[6,5,2,1,3,4] => 7
[6,5,2,1,4,3] => 4
[6,5,2,3,1,4] => 6
[6,5,2,3,4,1] => 4
[6,5,2,4,1,3] => 12
[6,5,2,4,3,1] => 4
[6,5,3,1,2,4] => 6
[6,5,3,1,4,2] => 12
[6,5,3,2,1,4] => 3
[6,5,3,2,4,1] => 4
[6,5,3,4,1,2] => 6
[6,5,3,4,2,1] => 5
[6,5,4,1,2,3] => 4
[6,5,4,1,3,2] => 4
[6,5,4,2,1,3] => 4
[6,5,4,2,3,1] => 5
[6,5,4,3,1,2] => 5
[6,5,4,3,2,1] => 1
[1,2,3,4,5,6,7] => 1
[1,2,3,4,5,7,6] => 1
[1,2,3,4,6,5,7] => 1
[1,2,3,4,6,7,5] => 2
[1,2,3,4,7,5,6] => 2
[1,2,3,4,7,6,5] => 1
[1,2,3,5,4,6,7] => 1
[1,2,3,5,4,7,6] => 1
[1,2,3,5,6,4,7] => 2
[1,2,3,5,6,7,4] => 2
[1,2,3,5,7,4,6] => 2
[1,2,3,5,7,6,4] => 2
[1,2,3,6,4,5,7] => 2
[1,2,3,6,4,7,5] => 2
[1,2,3,6,5,4,7] => 1
[1,2,3,6,5,7,4] => 2
[1,2,3,6,7,4,5] => 1
[1,2,3,6,7,5,4] => 3
[1,2,3,7,4,5,6] => 2
[1,2,3,7,4,6,5] => 2
[1,2,3,7,5,4,6] => 2
[1,2,3,7,5,6,4] => 3
[1,2,3,7,6,4,5] => 3
[1,2,3,7,6,5,4] => 1
[1,2,4,3,5,6,7] => 1
[1,2,4,3,5,7,6] => 1
[1,2,4,3,6,5,7] => 1
[1,2,4,3,6,7,5] => 2
[1,2,4,3,7,5,6] => 2
[1,2,4,3,7,6,5] => 1
[1,2,4,5,3,6,7] => 2
[1,2,4,5,3,7,6] => 2
[1,2,4,5,6,3,7] => 2
[1,2,4,5,6,7,3] => 4
[1,2,4,5,7,3,6] => 2
[1,2,4,5,7,6,3] => 4
[1,2,4,6,3,5,7] => 2
[1,2,4,6,3,7,5] => 2
[1,2,4,6,5,3,7] => 2
[1,2,4,6,5,7,3] => 2
[1,2,4,6,7,3,5] => 4
[1,2,4,6,7,5,3] => 4
[1,2,4,7,3,5,6] => 2
[1,2,4,7,3,6,5] => 2
[1,2,4,7,5,3,6] => 4
[1,2,4,7,5,6,3] => 4
[1,2,4,7,6,3,5] => 2
[1,2,4,7,6,5,3] => 2
[1,2,5,3,4,6,7] => 2
[1,2,5,3,4,7,6] => 2
[1,2,5,3,6,4,7] => 2
[1,2,5,3,6,7,4] => 2
[1,2,5,3,7,4,6] => 2
[1,2,5,3,7,6,4] => 2
[1,2,5,4,3,6,7] => 1
[1,2,5,4,3,7,6] => 1
[1,2,5,4,6,3,7] => 2
[1,2,5,4,6,7,3] => 4
[1,2,5,4,7,3,6] => 2
[1,2,5,4,7,6,3] => 2
[1,2,5,6,3,4,7] => 1
[1,2,5,6,3,7,4] => 4
[1,2,5,6,4,3,7] => 3
[1,2,5,6,4,7,3] => 4
[1,2,5,6,7,3,4] => 6
[1,2,5,6,7,4,3] => 3
[1,2,5,7,3,4,6] => 4
[1,2,5,7,3,6,4] => 1
[1,2,5,7,4,3,6] => 2
[1,2,5,7,4,6,3] => 6
[1,2,5,7,6,3,4] => 2
[1,2,5,7,6,4,3] => 3
[1,2,6,3,4,5,7] => 2
[1,2,6,3,4,7,5] => 2
[1,2,6,3,5,4,7] => 2
[1,2,6,3,5,7,4] => 4
[1,2,6,3,7,4,5] => 4
[1,2,6,3,7,5,4] => 2
[1,2,6,4,3,5,7] => 2
[1,2,6,4,3,7,5] => 2
[1,2,6,4,5,3,7] => 3
[1,2,6,4,5,7,3] => 4
[1,2,6,4,7,3,5] => 1
[1,2,6,4,7,5,3] => 6
[1,2,6,5,3,4,7] => 3
[1,2,6,5,3,7,4] => 2
[1,2,6,5,4,3,7] => 1
[1,2,6,5,4,7,3] => 2
[1,2,6,5,7,3,4] => 2
[1,2,6,5,7,4,3] => 3
[1,2,6,7,3,4,5] => 6
[1,2,6,7,3,5,4] => 2
[1,2,6,7,4,3,5] => 2
[1,2,6,7,4,5,3] => 3
[1,2,6,7,5,3,4] => 3
[1,2,6,7,5,4,3] => 4
[1,2,7,3,4,5,6] => 4
[1,2,7,3,4,6,5] => 4
[1,2,7,3,5,4,6] => 2
[1,2,7,3,5,6,4] => 4
[1,2,7,3,6,4,5] => 4
[1,2,7,3,6,5,4] => 2
[1,2,7,4,3,5,6] => 4
[1,2,7,4,3,6,5] => 2
[1,2,7,4,5,3,6] => 4
[1,2,7,4,5,6,3] => 3
[1,2,7,4,6,3,5] => 6
[1,2,7,4,6,5,3] => 3
[1,2,7,5,3,4,6] => 4
[1,2,7,5,3,6,4] => 6
[1,2,7,5,4,3,6] => 2
[1,2,7,5,4,6,3] => 3
[1,2,7,5,6,3,4] => 3
[1,2,7,5,6,4,3] => 4
[1,2,7,6,3,4,5] => 3
[1,2,7,6,3,5,4] => 3
[1,2,7,6,4,3,5] => 3
[1,2,7,6,4,5,3] => 4
[1,2,7,6,5,3,4] => 4
[1,2,7,6,5,4,3] => 1
[1,3,2,4,5,6,7] => 1
[1,3,2,4,5,7,6] => 1
[1,3,2,4,6,5,7] => 1
[1,3,2,4,6,7,5] => 2
[1,3,2,4,7,5,6] => 2
[1,3,2,4,7,6,5] => 1
[1,3,2,5,4,6,7] => 1
[1,3,2,5,4,7,6] => 1
[1,3,2,5,6,4,7] => 2
[1,3,2,5,6,7,4] => 2
[1,3,2,5,7,4,6] => 2
[1,3,2,5,7,6,4] => 2
[1,3,2,6,4,5,7] => 2
[1,3,2,6,4,7,5] => 2
[1,3,2,6,5,4,7] => 1
[1,3,2,6,5,7,4] => 2
[1,3,2,6,7,4,5] => 1
[1,3,2,6,7,5,4] => 3
[1,3,2,7,4,5,6] => 2
[1,3,2,7,4,6,5] => 2
[1,3,2,7,5,4,6] => 2
[1,3,2,7,5,6,4] => 3
[1,3,2,7,6,4,5] => 3
[1,3,2,7,6,5,4] => 1
[1,3,4,2,5,6,7] => 2
[1,3,4,2,5,7,6] => 2
[1,3,4,2,6,5,7] => 2
[1,3,4,2,6,7,5] => 4
[1,3,4,2,7,5,6] => 4
[1,3,4,2,7,6,5] => 2
[1,3,4,5,2,6,7] => 2
[1,3,4,5,2,7,6] => 2
[1,3,4,5,6,2,7] => 4
[1,3,4,5,6,7,2] => 4
[1,3,4,5,7,2,6] => 4
[1,3,4,5,7,6,2] => 2
[1,3,4,6,2,5,7] => 2
[1,3,4,6,2,7,5] => 2
[1,3,4,6,5,2,7] => 4
[1,3,4,6,5,7,2] => 4
[1,3,4,6,7,2,5] => 4
[1,3,4,6,7,5,2] => 8
[1,3,4,7,2,5,6] => 2
[1,3,4,7,2,6,5] => 2
[1,3,4,7,5,2,6] => 8
[1,3,4,7,5,6,2] => 8
[1,3,4,7,6,2,5] => 4
[1,3,4,7,6,5,2] => 4
[1,3,5,2,4,6,7] => 2
[1,3,5,2,4,7,6] => 2
[1,3,5,2,6,4,7] => 2
[1,3,5,2,6,7,4] => 2
[1,3,5,2,7,4,6] => 2
[1,3,5,2,7,6,4] => 2
[1,3,5,4,2,6,7] => 2
[1,3,5,4,2,7,6] => 2
[1,3,5,4,6,2,7] => 2
[1,3,5,4,6,7,2] => 4
[1,3,5,4,7,2,6] => 2
[1,3,5,4,7,6,2] => 4
[1,3,5,6,2,4,7] => 4
[1,3,5,6,2,7,4] => 8
[1,3,5,6,4,2,7] => 4
[1,3,5,6,4,7,2] => 6
[1,3,5,6,7,2,4] => 4
[1,3,5,6,7,4,2] => 6
[1,3,5,7,2,4,6] => 8
[1,3,5,7,2,6,4] => 4
[1,3,5,7,4,2,6] => 4
[1,3,5,7,4,6,2] => 8
[1,3,5,7,6,2,4] => 2
[1,3,5,7,6,4,2] => 8
[1,3,6,2,4,5,7] => 2
[1,3,6,2,4,7,5] => 2
[1,3,6,2,5,4,7] => 2
[1,3,6,2,5,7,4] => 4
[1,3,6,2,7,4,5] => 4
[1,3,6,2,7,5,4] => 2
[1,3,6,4,2,5,7] => 4
[1,3,6,4,2,7,5] => 4
[1,3,6,4,5,2,7] => 4
[1,3,6,4,5,7,2] => 6
[1,3,6,4,7,2,5] => 4
[1,3,6,4,7,5,2] => 8
[1,3,6,5,2,4,7] => 2
[1,3,6,5,2,7,4] => 4
[1,3,6,5,4,2,7] => 2
[1,3,6,5,4,7,2] => 2
[1,3,6,5,7,2,4] => 8
[1,3,6,5,7,4,2] => 4
[1,3,6,7,2,4,5] => 8
[1,3,6,7,2,5,4] => 4
[1,3,6,7,4,2,5] => 8
[1,3,6,7,4,5,2] => 2
[1,3,6,7,5,2,4] => 8
[1,3,6,7,5,4,2] => 6
[1,3,7,2,4,5,6] => 4
[1,3,7,2,4,6,5] => 4
[1,3,7,2,5,4,6] => 2
[1,3,7,2,5,6,4] => 4
[1,3,7,2,6,4,5] => 4
[1,3,7,2,6,5,4] => 2
[1,3,7,4,2,5,6] => 8
[1,3,7,4,2,6,5] => 4
[1,3,7,4,5,2,6] => 2
[1,3,7,4,5,6,2] => 6
[1,3,7,4,6,2,5] => 4
[1,3,7,4,6,5,2] => 8
[1,3,7,5,2,4,6] => 4
[1,3,7,5,2,6,4] => 8
[1,3,7,5,4,2,6] => 6
[1,3,7,5,4,6,2] => 4
[1,3,7,5,6,2,4] => 8
[1,3,7,5,6,4,2] => 6
[1,3,7,6,2,4,5] => 2
[1,3,7,6,2,5,4] => 2
[1,3,7,6,4,2,5] => 6
[1,3,7,6,4,5,2] => 6
[1,3,7,6,5,2,4] => 2
[1,3,7,6,5,4,2] => 2
[1,4,2,3,5,6,7] => 2
[1,4,2,3,5,7,6] => 2
[1,4,2,3,6,5,7] => 2
[1,4,2,3,6,7,5] => 4
[1,4,2,3,7,5,6] => 4
[1,4,2,3,7,6,5] => 2
[1,4,2,5,3,6,7] => 2
[1,4,2,5,3,7,6] => 2
[1,4,2,5,6,3,7] => 2
[1,4,2,5,6,7,3] => 4
[1,4,2,5,7,3,6] => 2
[1,4,2,5,7,6,3] => 4
[1,4,2,6,3,5,7] => 2
[1,4,2,6,3,7,5] => 2
[1,4,2,6,5,3,7] => 2
[1,4,2,6,5,7,3] => 2
[1,4,2,6,7,3,5] => 4
[1,4,2,6,7,5,3] => 4
[1,4,2,7,3,5,6] => 2
[1,4,2,7,3,6,5] => 2
[1,4,2,7,5,3,6] => 4
[1,4,2,7,5,6,3] => 4
[1,4,2,7,6,3,5] => 2
[1,4,2,7,6,5,3] => 2
[1,4,3,2,5,6,7] => 1
[1,4,3,2,5,7,6] => 1
[1,4,3,2,6,5,7] => 1
[1,4,3,2,6,7,5] => 2
[1,4,3,2,7,5,6] => 2
[1,4,3,2,7,6,5] => 1
[1,4,3,5,2,6,7] => 2
[1,4,3,5,2,7,6] => 2
[1,4,3,5,6,2,7] => 4
[1,4,3,5,6,7,2] => 2
[1,4,3,5,7,2,6] => 4
[1,4,3,5,7,6,2] => 3
[1,4,3,6,2,5,7] => 2
[1,4,3,6,2,7,5] => 2
[1,4,3,6,5,2,7] => 2
[1,4,3,6,5,7,2] => 4
[1,4,3,6,7,2,5] => 4
[1,4,3,6,7,5,2] => 6
[1,4,3,7,2,5,6] => 2
[1,4,3,7,2,6,5] => 2
[1,4,3,7,5,2,6] => 4
[1,4,3,7,5,6,2] => 6
[1,4,3,7,6,2,5] => 2
[1,4,3,7,6,5,2] => 2
[1,4,5,2,3,6,7] => 1
[1,4,5,2,3,7,6] => 1
[1,4,5,2,6,3,7] => 4
[1,4,5,2,6,7,3] => 4
[1,4,5,2,7,3,6] => 4
[1,4,5,2,7,6,3] => 4
[1,4,5,3,2,6,7] => 3
[1,4,5,3,2,7,6] => 3
[1,4,5,3,6,2,7] => 4
[1,4,5,3,6,7,2] => 8
[1,4,5,3,7,2,6] => 4
[1,4,5,3,7,6,2] => 6
[1,4,5,6,2,3,7] => 6
[1,4,5,6,2,7,3] => 4
[1,4,5,6,3,2,7] => 3
[1,4,5,6,3,7,2] => 6
[1,4,5,6,7,2,3] => 4
[1,4,5,6,7,3,2] => 9
[1,4,5,7,2,3,6] => 8
[1,4,5,7,2,6,3] => 6
[1,4,5,7,3,2,6] => 2
[1,4,5,7,3,6,2] => 8
[1,4,5,7,6,2,3] => 3
[1,4,5,7,6,3,2] => 7
[1,4,6,2,3,5,7] => 4
[1,4,6,2,3,7,5] => 4
[1,4,6,2,5,3,7] => 1
[1,4,6,2,5,7,3] => 4
[1,4,6,2,7,3,5] => 3
[1,4,6,2,7,5,3] => 6
[1,4,6,3,2,5,7] => 2
[1,4,6,3,2,7,5] => 2
[1,4,6,3,5,2,7] => 6
[1,4,6,3,5,7,2] => 8
[1,4,6,3,7,2,5] => 6
[1,4,6,3,7,5,2] => 6
[1,4,6,5,2,3,7] => 2
[1,4,6,5,2,7,3] => 8
[1,4,6,5,3,2,7] => 3
[1,4,6,5,3,7,2] => 4
click to show generating function       
Description
The number of permutations with the same antidiagonal sums.
The X-ray of a permutation $\pi$ is the vector of the sums of the antidiagonals of the permutation matrix of $\pi$, read from left to right. For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is
$$\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right),$$
so its X-ray is $(0, 1, 1, 1, 0, 0, 0, 2, 0)$.
This statistic records the number of permutations having the same X-ray as the given permutation. In [1] this is called the degeneracy of the X-ray of the permutation.
By [prop.1, 1], the number of different X-rays of permutations of size $n$ equals the number of nondecreasing differences of permutations of size $n$, [2].
References
[1] Bebeacua, C., Mansour, T., Postnikov, A., Severini, S. On the X-rays of permutations arXiv:math/0506334
[2] Number of nondecreasing sequences that are differences of two permutations of 1,2,...,n. OEIS:A019589
Code
def X_ray(pi):
    P = Permutation(pi).to_matrix()
    n = P.nrows()
    return tuple(sum(P[k-1-j][j] for j in range(max(0, k-n), min(k,n)))
                 for k in range(1,2*n))

@cached_function
def X_rays(n):
    return sorted(X_ray(pi) for pi in Permutations(n))
    
def statistic(pi):
    return X_rays(pi.size()).count(X_ray(pi))
Created
Jul 14, 2017 at 09:26 by Martin Rubey
Updated
Jul 14, 2017 at 11:53 by Martin Rubey