Identifier
Identifier
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 3
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 8
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 3
[1,5,2,3,4] => 1
[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] => 8
[2,1,3,4,5] => 1
[2,1,3,5,4] => 1
[2,1,4,3,5] => 1
[2,1,4,5,3] => 1
[2,1,5,3,4] => 1
[2,1,5,4,3] => 2
[2,3,1,4,5] => 1
[2,3,1,5,4] => 1
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 1
[2,3,5,4,1] => 2
[2,4,1,3,5] => 1
[2,4,1,5,3] => 1
[2,4,3,1,5] => 2
[2,4,3,5,1] => 2
[2,4,5,1,3] => 1
[2,4,5,3,1] => 3
[2,5,1,3,4] => 1
[2,5,1,4,3] => 2
[2,5,3,1,4] => 2
[2,5,3,4,1] => 3
[2,5,4,1,3] => 3
[2,5,4,3,1] => 8
[3,1,2,4,5] => 1
[3,1,2,5,4] => 1
[3,1,4,2,5] => 1
[3,1,4,5,2] => 1
[3,1,5,2,4] => 1
[3,1,5,4,2] => 2
[3,2,1,4,5] => 2
[3,2,1,5,4] => 2
[3,2,4,1,5] => 2
[3,2,4,5,1] => 2
[3,2,5,1,4] => 2
[3,2,5,4,1] => 4
[3,4,1,2,5] => 1
[3,4,1,5,2] => 1
[3,4,2,1,5] => 3
[3,4,2,5,1] => 3
[3,4,5,1,2] => 1
[3,4,5,2,1] => 4
[3,5,1,2,4] => 1
[3,5,1,4,2] => 2
[3,5,2,1,4] => 3
[3,5,2,4,1] => 5
[3,5,4,1,2] => 3
[3,5,4,2,1] => 11
[4,1,2,3,5] => 1
[4,1,2,5,3] => 1
[4,1,3,2,5] => 2
[4,1,3,5,2] => 2
[4,1,5,2,3] => 1
[4,1,5,3,2] => 3
[4,2,1,3,5] => 2
[4,2,1,5,3] => 2
[4,2,3,1,5] => 3
[4,2,3,5,1] => 3
[4,2,5,1,3] => 2
[4,2,5,3,1] => 5
[4,3,1,2,5] => 3
[4,3,1,5,2] => 3
[4,3,2,1,5] => 8
[4,3,2,5,1] => 8
[4,3,5,1,2] => 3
[4,3,5,2,1] => 11
[4,5,1,2,3] => 1
[4,5,1,3,2] => 3
[4,5,2,1,3] => 3
[4,5,2,3,1] => 6
[4,5,3,1,2] => 6
[4,5,3,2,1] => 20
[5,1,2,3,4] => 1
[5,1,2,4,3] => 2
[5,1,3,2,4] => 2
[5,1,3,4,2] => 3
[5,1,4,2,3] => 3
[5,1,4,3,2] => 8
[5,2,1,3,4] => 2
[5,2,1,4,3] => 4
[5,2,3,1,4] => 3
[5,2,3,4,1] => 4
[5,2,4,1,3] => 5
[5,2,4,3,1] => 11
[5,3,1,2,4] => 3
[5,3,1,4,2] => 5
[5,3,2,1,4] => 8
[5,3,2,4,1] => 11
[5,3,4,1,2] => 6
[5,3,4,2,1] => 20
[5,4,1,2,3] => 4
[5,4,1,3,2] => 11
[5,4,2,1,3] => 11
[5,4,2,3,1] => 20
[5,4,3,1,2] => 20
[5,4,3,2,1] => 62
[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] => 1
[1,2,3,6,4,5] => 1
[1,2,3,6,5,4] => 2
[1,2,4,3,5,6] => 1
[1,2,4,3,6,5] => 1
[1,2,4,5,3,6] => 1
[1,2,4,5,6,3] => 1
[1,2,4,6,3,5] => 1
[1,2,4,6,5,3] => 2
[1,2,5,3,4,6] => 1
[1,2,5,3,6,4] => 1
[1,2,5,4,3,6] => 2
[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] => 1
[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] => 8
[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] => 1
[1,3,2,6,4,5] => 1
[1,3,2,6,5,4] => 2
[1,3,4,2,5,6] => 1
[1,3,4,2,6,5] => 1
[1,3,4,5,2,6] => 1
[1,3,4,5,6,2] => 1
[1,3,4,6,2,5] => 1
[1,3,4,6,5,2] => 2
[1,3,5,2,4,6] => 1
[1,3,5,2,6,4] => 1
[1,3,5,4,2,6] => 2
[1,3,5,4,6,2] => 2
[1,3,5,6,2,4] => 1
[1,3,5,6,4,2] => 3
[1,3,6,2,4,5] => 1
[1,3,6,2,5,4] => 2
[1,3,6,4,2,5] => 2
[1,3,6,4,5,2] => 3
[1,3,6,5,2,4] => 3
[1,3,6,5,4,2] => 8
[1,4,2,3,5,6] => 1
[1,4,2,3,6,5] => 1
[1,4,2,5,3,6] => 1
[1,4,2,5,6,3] => 1
[1,4,2,6,3,5] => 1
[1,4,2,6,5,3] => 2
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 2
[1,4,3,5,2,6] => 2
[1,4,3,5,6,2] => 2
[1,4,3,6,2,5] => 2
[1,4,3,6,5,2] => 4
[1,4,5,2,3,6] => 1
[1,4,5,2,6,3] => 1
[1,4,5,3,2,6] => 3
[1,4,5,3,6,2] => 3
[1,4,5,6,2,3] => 1
[1,4,5,6,3,2] => 4
[1,4,6,2,3,5] => 1
[1,4,6,2,5,3] => 2
[1,4,6,3,2,5] => 3
[1,4,6,3,5,2] => 5
[1,4,6,5,2,3] => 3
[1,4,6,5,3,2] => 11
[1,5,2,3,4,6] => 1
[1,5,2,3,6,4] => 1
[1,5,2,4,3,6] => 2
[1,5,2,4,6,3] => 2
[1,5,2,6,3,4] => 1
[1,5,2,6,4,3] => 3
[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] => 3
[1,5,3,6,2,4] => 2
[1,5,3,6,4,2] => 5
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 3
[1,5,4,3,2,6] => 8
[1,5,4,3,6,2] => 8
[1,5,4,6,2,3] => 3
[1,5,4,6,3,2] => 11
[1,5,6,2,3,4] => 1
[1,5,6,2,4,3] => 3
[1,5,6,3,2,4] => 3
[1,5,6,3,4,2] => 6
[1,5,6,4,2,3] => 6
[1,5,6,4,3,2] => 20
[1,6,2,3,4,5] => 1
[1,6,2,3,5,4] => 2
[1,6,2,4,3,5] => 2
[1,6,2,4,5,3] => 3
[1,6,2,5,3,4] => 3
[1,6,2,5,4,3] => 8
[1,6,3,2,4,5] => 2
[1,6,3,2,5,4] => 4
[1,6,3,4,2,5] => 3
[1,6,3,4,5,2] => 4
[1,6,3,5,2,4] => 5
[1,6,3,5,4,2] => 11
[1,6,4,2,3,5] => 3
[1,6,4,2,5,3] => 5
[1,6,4,3,2,5] => 8
[1,6,4,3,5,2] => 11
[1,6,4,5,2,3] => 6
[1,6,4,5,3,2] => 20
[1,6,5,2,3,4] => 4
[1,6,5,2,4,3] => 11
[1,6,5,3,2,4] => 11
[1,6,5,3,4,2] => 20
[1,6,5,4,2,3] => 20
[1,6,5,4,3,2] => 62
[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] => 1
[2,1,3,6,4,5] => 1
[2,1,3,6,5,4] => 2
[2,1,4,3,5,6] => 1
[2,1,4,3,6,5] => 1
[2,1,4,5,3,6] => 1
[2,1,4,5,6,3] => 1
[2,1,4,6,3,5] => 1
[2,1,4,6,5,3] => 2
[2,1,5,3,4,6] => 1
[2,1,5,3,6,4] => 1
[2,1,5,4,3,6] => 2
[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] => 1
[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] => 8
[2,3,1,4,5,6] => 1
[2,3,1,4,6,5] => 1
[2,3,1,5,4,6] => 1
[2,3,1,5,6,4] => 1
[2,3,1,6,4,5] => 1
[2,3,1,6,5,4] => 2
[2,3,4,1,5,6] => 1
[2,3,4,1,6,5] => 1
[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] => 2
[2,3,5,1,4,6] => 1
[2,3,5,1,6,4] => 1
[2,3,5,4,1,6] => 2
[2,3,5,4,6,1] => 2
[2,3,5,6,1,4] => 1
[2,3,5,6,4,1] => 3
[2,3,6,1,4,5] => 1
[2,3,6,1,5,4] => 2
[2,3,6,4,1,5] => 2
[2,3,6,4,5,1] => 3
[2,3,6,5,1,4] => 3
[2,3,6,5,4,1] => 8
[2,4,1,3,5,6] => 1
[2,4,1,3,6,5] => 1
[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] => 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] => 2
[2,4,3,6,1,5] => 2
[2,4,3,6,5,1] => 4
[2,4,5,1,3,6] => 1
[2,4,5,1,6,3] => 1
[2,4,5,3,1,6] => 3
[2,4,5,3,6,1] => 3
[2,4,5,6,1,3] => 1
[2,4,5,6,3,1] => 4
[2,4,6,1,3,5] => 1
[2,4,6,1,5,3] => 2
[2,4,6,3,1,5] => 3
[2,4,6,3,5,1] => 5
[2,4,6,5,1,3] => 3
[2,4,6,5,3,1] => 11
[2,5,1,3,4,6] => 1
[2,5,1,3,6,4] => 1
[2,5,1,4,3,6] => 2
[2,5,1,4,6,3] => 2
[2,5,1,6,3,4] => 1
[2,5,1,6,4,3] => 3
[2,5,3,1,4,6] => 2
[2,5,3,1,6,4] => 2
[2,5,3,4,1,6] => 3
[2,5,3,4,6,1] => 3
[2,5,3,6,1,4] => 2
[2,5,3,6,4,1] => 5
[2,5,4,1,3,6] => 3
[2,5,4,1,6,3] => 3
[2,5,4,3,1,6] => 8
[2,5,4,3,6,1] => 8
[2,5,4,6,1,3] => 3
[2,5,4,6,3,1] => 11
[2,5,6,1,3,4] => 1
[2,5,6,1,4,3] => 3
[2,5,6,3,1,4] => 3
[2,5,6,3,4,1] => 6
[2,5,6,4,1,3] => 6
[2,5,6,4,3,1] => 20
[2,6,1,3,4,5] => 1
[2,6,1,3,5,4] => 2
[2,6,1,4,3,5] => 2
[2,6,1,4,5,3] => 3
[2,6,1,5,3,4] => 3
[2,6,1,5,4,3] => 8
[2,6,3,1,4,5] => 2
[2,6,3,1,5,4] => 4
[2,6,3,4,1,5] => 3
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 5
[2,6,3,5,4,1] => 11
[2,6,4,1,3,5] => 3
[2,6,4,1,5,3] => 5
[2,6,4,3,1,5] => 8
[2,6,4,3,5,1] => 11
[2,6,4,5,1,3] => 6
[2,6,4,5,3,1] => 20
[2,6,5,1,3,4] => 4
[2,6,5,1,4,3] => 11
[2,6,5,3,1,4] => 11
[2,6,5,3,4,1] => 20
[2,6,5,4,1,3] => 20
[2,6,5,4,3,1] => 62
[3,1,2,4,5,6] => 1
[3,1,2,4,6,5] => 1
[3,1,2,5,4,6] => 1
[3,1,2,5,6,4] => 1
[3,1,2,6,4,5] => 1
[3,1,2,6,5,4] => 2
[3,1,4,2,5,6] => 1
[3,1,4,2,6,5] => 1
[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] => 2
[3,1,5,2,4,6] => 1
[3,1,5,2,6,4] => 1
[3,1,5,4,2,6] => 2
[3,1,5,4,6,2] => 2
[3,1,5,6,2,4] => 1
[3,1,5,6,4,2] => 3
[3,1,6,2,4,5] => 1
[3,1,6,2,5,4] => 2
[3,1,6,4,2,5] => 2
[3,1,6,4,5,2] => 3
[3,1,6,5,2,4] => 3
[3,1,6,5,4,2] => 8
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 2
[3,2,1,5,4,6] => 2
[3,2,1,5,6,4] => 2
[3,2,1,6,4,5] => 2
[3,2,1,6,5,4] => 4
[3,2,4,1,5,6] => 2
[3,2,4,1,6,5] => 2
[3,2,4,5,1,6] => 2
[3,2,4,5,6,1] => 2
[3,2,4,6,1,5] => 2
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 2
[3,2,5,1,6,4] => 2
[3,2,5,4,1,6] => 4
[3,2,5,4,6,1] => 4
[3,2,5,6,1,4] => 2
[3,2,5,6,4,1] => 6
[3,2,6,1,4,5] => 2
[3,2,6,1,5,4] => 4
[3,2,6,4,1,5] => 4
[3,2,6,4,5,1] => 6
[3,2,6,5,1,4] => 6
[3,2,6,5,4,1] => 16
[3,4,1,2,5,6] => 1
[3,4,1,2,6,5] => 1
[3,4,1,5,2,6] => 1
[3,4,1,5,6,2] => 1
[3,4,1,6,2,5] => 1
[3,4,1,6,5,2] => 2
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 3
[3,4,2,5,1,6] => 3
[3,4,2,5,6,1] => 3
[3,4,2,6,1,5] => 3
[3,4,2,6,5,1] => 6
[3,4,5,1,2,6] => 1
[3,4,5,1,6,2] => 1
[3,4,5,2,1,6] => 4
[3,4,5,2,6,1] => 4
[3,4,5,6,1,2] => 1
[3,4,5,6,2,1] => 5
[3,4,6,1,2,5] => 1
[3,4,6,1,5,2] => 2
[3,4,6,2,1,5] => 4
[3,4,6,2,5,1] => 7
[3,4,6,5,1,2] => 3
[3,4,6,5,2,1] => 14
[3,5,1,2,4,6] => 1
[3,5,1,2,6,4] => 1
[3,5,1,4,2,6] => 2
[3,5,1,4,6,2] => 2
[3,5,1,6,2,4] => 1
[3,5,1,6,4,2] => 3
[3,5,2,1,4,6] => 3
[3,5,2,1,6,4] => 3
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 5
[3,5,2,6,1,4] => 3
[3,5,2,6,4,1] => 8
[3,5,4,1,2,6] => 3
[3,5,4,1,6,2] => 3
[3,5,4,2,1,6] => 11
[3,5,4,2,6,1] => 11
[3,5,4,6,1,2] => 3
[3,5,4,6,2,1] => 14
[3,5,6,1,2,4] => 1
[3,5,6,1,4,2] => 3
[3,5,6,2,1,4] => 4
[3,5,6,2,4,1] => 9
[3,5,6,4,1,2] => 6
[3,5,6,4,2,1] => 26
[3,6,1,2,4,5] => 1
[3,6,1,2,5,4] => 2
[3,6,1,4,2,5] => 2
[3,6,1,4,5,2] => 3
[3,6,1,5,2,4] => 3
[3,6,1,5,4,2] => 8
[3,6,2,1,4,5] => 3
[3,6,2,1,5,4] => 6
[3,6,2,4,1,5] => 5
[3,6,2,4,5,1] => 7
[3,6,2,5,1,4] => 8
[3,6,2,5,4,1] => 19
[3,6,4,1,2,5] => 3
[3,6,4,1,5,2] => 5
[3,6,4,2,1,5] => 11
[3,6,4,2,5,1] => 16
[3,6,4,5,1,2] => 6
[3,6,4,5,2,1] => 26
[3,6,5,1,2,4] => 4
[3,6,5,1,4,2] => 11
[3,6,5,2,1,4] => 15
[3,6,5,2,4,1] => 31
[3,6,5,4,1,2] => 20
[3,6,5,4,2,1] => 82
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 1
[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] => 2
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 2
[4,1,3,5,2,6] => 2
[4,1,3,5,6,2] => 2
[4,1,3,6,2,5] => 2
[4,1,3,6,5,2] => 4
[4,1,5,2,3,6] => 1
[4,1,5,2,6,3] => 1
[4,1,5,3,2,6] => 3
[4,1,5,3,6,2] => 3
[4,1,5,6,2,3] => 1
[4,1,5,6,3,2] => 4
[4,1,6,2,3,5] => 1
[4,1,6,2,5,3] => 2
[4,1,6,3,2,5] => 3
[4,1,6,3,5,2] => 5
[4,1,6,5,2,3] => 3
[4,1,6,5,3,2] => 11
[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] => 4
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 3
[4,2,3,5,1,6] => 3
[4,2,3,5,6,1] => 3
[4,2,3,6,1,5] => 3
[4,2,3,6,5,1] => 6
[4,2,5,1,3,6] => 2
[4,2,5,1,6,3] => 2
[4,2,5,3,1,6] => 5
[4,2,5,3,6,1] => 5
[4,2,5,6,1,3] => 2
[4,2,5,6,3,1] => 7
[4,2,6,1,3,5] => 2
[4,2,6,1,5,3] => 4
[4,2,6,3,1,5] => 5
[4,2,6,3,5,1] => 8
[4,2,6,5,1,3] => 6
[4,2,6,5,3,1] => 19
[4,3,1,2,5,6] => 3
[4,3,1,2,6,5] => 3
[4,3,1,5,2,6] => 3
[4,3,1,5,6,2] => 3
[4,3,1,6,2,5] => 3
[4,3,1,6,5,2] => 6
[4,3,2,1,5,6] => 8
[4,3,2,1,6,5] => 8
[4,3,2,5,1,6] => 8
[4,3,2,5,6,1] => 8
[4,3,2,6,1,5] => 8
[4,3,2,6,5,1] => 16
[4,3,5,1,2,6] => 3
[4,3,5,1,6,2] => 3
[4,3,5,2,1,6] => 11
[4,3,5,2,6,1] => 11
[4,3,5,6,1,2] => 3
[4,3,5,6,2,1] => 14
[4,3,6,1,2,5] => 3
[4,3,6,1,5,2] => 6
[4,3,6,2,1,5] => 11
[4,3,6,2,5,1] => 19
[4,3,6,5,1,2] => 9
[4,3,6,5,2,1] => 39
[4,5,1,2,3,6] => 1
[4,5,1,2,6,3] => 1
[4,5,1,3,2,6] => 3
[4,5,1,3,6,2] => 3
[4,5,1,6,2,3] => 1
[4,5,1,6,3,2] => 4
[4,5,2,1,3,6] => 3
[4,5,2,1,6,3] => 3
[4,5,2,3,1,6] => 6
[4,5,2,3,6,1] => 6
[4,5,2,6,1,3] => 3
[4,5,2,6,3,1] => 9
[4,5,3,1,2,6] => 6
[4,5,3,1,6,2] => 6
[4,5,3,2,1,6] => 20
[4,5,3,2,6,1] => 20
[4,5,3,6,1,2] => 6
[4,5,3,6,2,1] => 26
[4,5,6,1,2,3] => 1
[4,5,6,1,3,2] => 4
[4,5,6,2,1,3] => 4
[4,5,6,2,3,1] => 10
[4,5,6,3,1,2] => 10
[4,5,6,3,2,1] => 40
[4,6,1,2,3,5] => 1
[4,6,1,2,5,3] => 2
[4,6,1,3,5,2] => 5
[4,6,1,5,2,3] => 3
[4,6,1,5,3,2] => 11
[4,6,2,1,3,5] => 3
[4,6,2,1,5,3] => 6
[4,6,2,3,1,5] => 6
[4,6,2,3,5,1] => 9
[4,6,2,5,1,3] => 8
[4,6,2,5,3,1] => 22
[4,6,3,1,2,5] => 6
[4,6,3,1,5,2] => 11
[4,6,3,2,1,5] => 20
[4,6,3,2,5,1] => 31
[6,2,1,4,3,5] => 4
[6,4,2,1,3,5] => 11
click to show generating function       
Description
The number of connected components of short braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for
$$(1243) = (12)(34)(23) = (34)(12)(23)$$
share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$.
This statistic counts the number connected components of such short braid moves among all reduced words.
Code
def short_braid_move_graph(pi):
    V = [ tuple(w) for w in pi.reduced_words() ]
    is_edge = lambda w1,w2: w1 != w2 and any( w1[:i] == w2[:i] and w1[i+2:] == w2[i+2:] for i in range(len(w1)-1) )
    return Graph([V,is_edge])

def statistic(pi):
    return len( short_braid_move_graph(pi).connected_components() )
Created
Jul 03, 2017 at 16:58 by Christian Stump
Updated
Dec 25, 2017 at 17:32 by Martin Rubey