Identifier
Identifier
Values
[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] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[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] => 6
[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] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[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] => 6
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 4
[2,3,1,4,5] => 1
[2,3,1,5,4] => 2
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[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] => 2
[2,4,5,3,1] => 3
[2,5,1,3,4] => 2
[2,5,1,4,3] => 4
[2,5,3,1,4] => 3
[2,5,3,4,1] => 3
[2,5,4,1,3] => 4
[2,5,4,3,1] => 6
[3,1,2,4,5] => 1
[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] => 4
[3,2,1,4,5] => 2
[3,2,1,5,4] => 4
[3,2,4,1,5] => 2
[3,2,4,5,1] => 2
[3,2,5,1,4] => 4
[3,2,5,4,1] => 4
[3,4,1,2,5] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 3
[3,4,2,5,1] => 3
[3,4,5,1,2] => 2
[3,4,5,2,1] => 4
[3,5,1,2,4] => 2
[3,5,1,4,2] => 4
[3,5,2,1,4] => 4
[3,5,2,4,1] => 5
[3,5,4,1,2] => 4
[3,5,4,2,1] => 8
[4,1,2,3,5] => 1
[4,1,2,5,3] => 2
[4,1,3,2,5] => 2
[4,1,3,5,2] => 3
[4,1,5,2,3] => 2
[4,1,5,3,2] => 4
[4,2,1,3,5] => 2
[4,2,1,5,3] => 4
[4,2,3,1,5] => 3
[4,2,3,5,1] => 3
[4,2,5,1,3] => 4
[4,2,5,3,1] => 5
[4,3,1,2,5] => 3
[4,3,1,5,2] => 4
[4,3,2,1,5] => 6
[4,3,2,5,1] => 6
[4,3,5,1,2] => 4
[4,3,5,2,1] => 8
[4,5,1,2,3] => 2
[4,5,1,3,2] => 4
[4,5,2,1,3] => 4
[4,5,2,3,1] => 6
[4,5,3,1,2] => 6
[4,5,3,2,1] => 11
[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] => 6
[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] => 8
[5,3,1,2,4] => 3
[5,3,1,4,2] => 5
[5,3,2,1,4] => 6
[5,3,2,4,1] => 8
[5,3,4,1,2] => 6
[5,3,4,2,1] => 11
[5,4,1,2,3] => 4
[5,4,1,3,2] => 8
[5,4,2,1,3] => 8
[5,4,2,3,1] => 11
[5,4,3,1,2] => 11
[5,4,3,2,1] => 22
[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] => 2
[1,2,4,5,3,6] => 1
[1,2,4,5,6,3] => 1
[1,2,4,6,3,5] => 2
[1,2,4,6,5,3] => 2
[1,2,5,3,4,6] => 1
[1,2,5,3,6,4] => 2
[1,2,5,4,3,6] => 2
[1,2,5,4,6,3] => 2
[1,2,5,6,3,4] => 2
[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] => 6
[1,3,2,4,5,6] => 1
[1,3,2,4,6,5] => 2
[1,3,2,5,4,6] => 2
[1,3,2,5,6,4] => 2
[1,3,2,6,4,5] => 2
[1,3,2,6,5,4] => 4
[1,3,4,2,5,6] => 1
[1,3,4,2,6,5] => 2
[1,3,4,5,2,6] => 1
[1,3,4,5,6,2] => 1
[1,3,4,6,2,5] => 2
[1,3,4,6,5,2] => 2
[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] => 2
[1,3,5,6,4,2] => 3
[1,3,6,2,4,5] => 2
[1,3,6,2,5,4] => 4
[1,3,6,4,2,5] => 3
[1,3,6,4,5,2] => 3
[1,3,6,5,2,4] => 4
[1,3,6,5,4,2] => 6
[1,4,2,3,5,6] => 1
[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] => 4
[1,4,3,2,5,6] => 2
[1,4,3,2,6,5] => 4
[1,4,3,5,2,6] => 2
[1,4,3,5,6,2] => 2
[1,4,3,6,2,5] => 4
[1,4,3,6,5,2] => 4
[1,4,5,2,3,6] => 2
[1,4,5,2,6,3] => 2
[1,4,5,3,2,6] => 3
[1,4,5,3,6,2] => 3
[1,4,5,6,2,3] => 2
[1,4,5,6,3,2] => 4
[1,4,6,2,3,5] => 2
[1,4,6,2,5,3] => 4
[1,4,6,3,2,5] => 4
[1,4,6,3,5,2] => 5
[1,4,6,5,2,3] => 4
[1,4,6,5,3,2] => 8
[1,5,2,3,4,6] => 1
[1,5,2,3,6,4] => 2
[1,5,2,4,3,6] => 2
[1,5,2,4,6,3] => 3
[1,5,2,6,3,4] => 2
[1,5,2,6,4,3] => 4
[1,5,3,2,4,6] => 2
[1,5,3,2,6,4] => 4
[1,5,3,4,2,6] => 3
[1,5,3,4,6,2] => 3
[1,5,3,6,2,4] => 4
[1,5,3,6,4,2] => 5
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 4
[1,5,4,3,2,6] => 6
[1,5,4,3,6,2] => 6
[1,5,4,6,2,3] => 4
[1,5,4,6,3,2] => 8
[1,5,6,2,3,4] => 2
[1,5,6,2,4,3] => 4
[1,5,6,3,2,4] => 4
[1,5,6,3,4,2] => 6
[1,5,6,4,2,3] => 6
[1,5,6,4,3,2] => 11
[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] => 6
[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] => 8
[1,6,4,2,3,5] => 3
[1,6,4,2,5,3] => 5
[1,6,4,3,2,5] => 6
[1,6,4,3,5,2] => 8
[1,6,4,5,2,3] => 6
[1,6,4,5,3,2] => 11
[1,6,5,2,3,4] => 4
[1,6,5,2,4,3] => 8
[1,6,5,3,2,4] => 8
[1,6,5,3,4,2] => 11
[1,6,5,4,2,3] => 11
[1,6,5,4,3,2] => 22
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 2
[2,1,3,5,4,6] => 2
[2,1,3,5,6,4] => 2
[2,1,3,6,4,5] => 2
[2,1,3,6,5,4] => 4
[2,1,4,3,5,6] => 2
[2,1,4,3,6,5] => 3
[2,1,4,5,3,6] => 2
[2,1,4,5,6,3] => 2
[2,1,4,6,3,5] => 3
[2,1,4,6,5,3] => 4
[2,1,5,3,4,6] => 2
[2,1,5,3,6,4] => 3
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 4
[2,1,5,6,3,4] => 3
[2,1,5,6,4,3] => 6
[2,1,6,3,4,5] => 2
[2,1,6,3,5,4] => 4
[2,1,6,4,3,5] => 4
[2,1,6,4,5,3] => 6
[2,1,6,5,3,4] => 6
[2,1,6,5,4,3] => 11
[2,3,1,4,5,6] => 1
[2,3,1,4,6,5] => 2
[2,3,1,5,4,6] => 2
[2,3,1,5,6,4] => 3
[2,3,1,6,4,5] => 3
[2,3,1,6,5,4] => 5
[2,3,4,1,5,6] => 1
[2,3,4,1,6,5] => 2
[2,3,4,5,1,6] => 1
[2,3,4,5,6,1] => 1
[2,3,4,6,1,5] => 2
[2,3,4,6,5,1] => 2
[2,3,5,1,4,6] => 2
[2,3,5,1,6,4] => 3
[2,3,5,4,1,6] => 2
[2,3,5,4,6,1] => 2
[2,3,5,6,1,4] => 3
[2,3,5,6,4,1] => 3
[2,3,6,1,4,5] => 3
[2,3,6,1,5,4] => 5
[2,3,6,4,1,5] => 3
[2,3,6,4,5,1] => 3
[2,3,6,5,1,4] => 5
[2,3,6,5,4,1] => 6
[2,4,1,3,5,6] => 2
[2,4,1,3,6,5] => 3
[2,4,1,5,3,6] => 2
[2,4,1,5,6,3] => 3
[2,4,1,6,3,5] => 3
[2,4,1,6,5,3] => 5
[2,4,3,1,5,6] => 2
[2,4,3,1,6,5] => 4
[2,4,3,5,1,6] => 2
[2,4,3,5,6,1] => 2
[2,4,3,6,1,5] => 4
[2,4,3,6,5,1] => 4
[2,4,5,1,3,6] => 2
[2,4,5,1,6,3] => 3
[2,4,5,3,1,6] => 3
[2,4,5,3,6,1] => 3
[2,4,5,6,1,3] => 3
[2,4,5,6,3,1] => 4
[2,4,6,1,3,5] => 3
[2,4,6,1,5,3] => 5
[2,4,6,3,1,5] => 5
[2,4,6,3,5,1] => 5
[2,4,6,5,1,3] => 5
[2,4,6,5,3,1] => 8
[2,5,1,3,4,6] => 2
[2,5,1,3,6,4] => 3
[2,5,1,4,3,6] => 4
[2,5,1,4,6,3] => 4
[2,5,1,6,3,4] => 3
[2,5,1,6,4,3] => 6
[2,5,3,1,4,6] => 3
[2,5,3,1,6,4] => 4
[2,5,3,4,1,6] => 3
[2,5,3,4,6,1] => 3
[2,5,3,6,1,4] => 4
[2,5,3,6,4,1] => 5
[2,5,4,1,3,6] => 4
[2,5,4,1,6,3] => 5
[2,5,4,3,1,6] => 6
[2,5,4,3,6,1] => 6
[2,5,4,6,1,3] => 5
[2,5,4,6,3,1] => 8
[2,5,6,1,3,4] => 3
[2,5,6,1,4,3] => 6
[2,5,6,3,1,4] => 5
[2,5,6,3,4,1] => 6
[2,5,6,4,1,3] => 7
[2,5,6,4,3,1] => 12
[2,6,1,3,4,5] => 2
[2,6,1,3,5,4] => 4
[2,6,1,4,3,5] => 4
[2,6,1,4,5,3] => 6
[2,6,1,5,3,4] => 6
[2,6,1,5,4,3] => 11
[2,6,3,1,4,5] => 3
[2,6,3,1,5,4] => 6
[2,6,3,4,1,5] => 4
[2,6,3,4,5,1] => 4
[2,6,3,5,1,4] => 6
[2,6,3,5,4,1] => 8
[2,6,4,1,3,5] => 5
[2,6,4,1,5,3] => 7
[2,6,4,3,1,5] => 7
[2,6,4,3,5,1] => 8
[2,6,4,5,1,3] => 7
[2,6,4,5,3,1] => 12
[2,6,5,1,3,4] => 6
[2,6,5,1,4,3] => 12
[2,6,5,3,1,4] => 9
[2,6,5,3,4,1] => 12
[2,6,5,4,1,3] => 14
[2,6,5,4,3,1] => 23
[3,1,2,4,5,6] => 1
[3,1,2,4,6,5] => 2
[3,1,2,5,4,6] => 2
[3,1,2,5,6,4] => 3
[3,1,2,6,4,5] => 3
[3,1,2,6,5,4] => 5
[3,1,4,2,5,6] => 2
[3,1,4,2,6,5] => 3
[3,1,4,5,2,6] => 2
[3,1,4,5,6,2] => 2
[3,1,4,6,2,5] => 3
[3,1,4,6,5,2] => 4
[3,1,5,2,4,6] => 2
[3,1,5,2,6,4] => 3
[3,1,5,4,2,6] => 4
[3,1,5,4,6,2] => 4
[3,1,5,6,2,4] => 3
[3,1,5,6,4,2] => 6
[3,1,6,2,4,5] => 3
[3,1,6,2,5,4] => 5
[3,1,6,4,2,5] => 4
[3,1,6,4,5,2] => 6
[3,1,6,5,2,4] => 6
[3,1,6,5,4,2] => 11
[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] => 5
[3,2,1,6,4,5] => 5
[3,2,1,6,5,4] => 10
[3,2,4,1,5,6] => 2
[3,2,4,1,6,5] => 4
[3,2,4,5,1,6] => 2
[3,2,4,5,6,1] => 2
[3,2,4,6,1,5] => 4
[3,2,4,6,5,1] => 4
[3,2,5,1,4,6] => 4
[3,2,5,1,6,4] => 5
[3,2,5,4,1,6] => 4
[3,2,5,4,6,1] => 4
[3,2,5,6,1,4] => 5
[3,2,5,6,4,1] => 6
[3,2,6,1,4,5] => 5
[3,2,6,1,5,4] => 10
[3,2,6,4,1,5] => 6
[3,2,6,4,5,1] => 6
[3,2,6,5,1,4] => 10
[3,2,6,5,4,1] => 12
[3,4,1,2,5,6] => 2
[3,4,1,2,6,5] => 3
[3,4,1,5,2,6] => 2
[3,4,1,5,6,2] => 3
[3,4,1,6,2,5] => 3
[3,4,1,6,5,2] => 5
[3,4,2,1,5,6] => 3
[3,4,2,1,6,5] => 6
[3,4,2,5,1,6] => 3
[3,4,2,5,6,1] => 3
[3,4,2,6,1,5] => 6
[3,4,2,6,5,1] => 6
[3,4,5,1,2,6] => 2
[3,4,5,1,6,2] => 3
[3,4,5,2,1,6] => 4
[3,4,5,2,6,1] => 4
[3,4,5,6,1,2] => 3
[3,4,5,6,2,1] => 5
[3,4,6,1,2,5] => 3
[3,4,6,1,5,2] => 5
[3,4,6,2,1,5] => 6
[3,4,6,2,5,1] => 7
[3,4,6,5,1,2] => 5
[3,4,6,5,2,1] => 10
[3,5,1,2,4,6] => 2
[3,5,1,2,6,4] => 3
[3,5,1,4,2,6] => 4
[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] => 4
[3,5,2,1,6,4] => 6
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 5
[3,5,2,6,1,4] => 6
[3,5,2,6,4,1] => 8
[3,5,4,1,2,6] => 4
[3,5,4,1,6,2] => 5
[3,5,4,2,1,6] => 8
[3,5,4,2,6,1] => 8
[3,5,4,6,1,2] => 5
[3,5,4,6,2,1] => 10
[3,5,6,1,2,4] => 3
[3,5,6,1,4,2] => 6
[3,5,6,2,1,4] => 6
[3,5,6,2,4,1] => 9
[3,5,6,4,1,2] => 7
[3,5,6,4,2,1] => 14
[3,6,1,2,4,5] => 3
[3,6,1,2,5,4] => 5
[3,6,1,4,2,5] => 4
[3,6,1,4,5,2] => 6
[3,6,1,5,2,4] => 6
[3,6,1,5,4,2] => 11
[3,6,2,1,4,5] => 5
[3,6,2,1,5,4] => 10
[3,6,2,4,1,5] => 6
[3,6,2,4,5,1] => 7
[3,6,2,5,1,4] => 10
[3,6,2,5,4,1] => 14
[3,6,4,1,2,5] => 5
[3,6,4,1,5,2] => 7
[3,6,4,2,1,5] => 9
[3,6,4,2,5,1] => 11
[3,6,4,5,1,2] => 7
[3,6,4,5,2,1] => 14
[3,6,5,1,2,4] => 6
[3,6,5,1,4,2] => 12
[3,6,5,2,1,4] => 12
[3,6,5,2,4,1] => 17
[3,6,5,4,1,2] => 14
[3,6,5,4,2,1] => 28
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 2
[4,1,2,5,3,6] => 2
[4,1,2,5,6,3] => 3
[4,1,2,6,3,5] => 3
[4,1,2,6,5,3] => 5
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 4
[4,1,3,5,2,6] => 3
[4,1,3,5,6,2] => 3
[4,1,3,6,2,5] => 4
[4,1,3,6,5,2] => 6
[4,1,5,2,3,6] => 2
[4,1,5,2,6,3] => 3
[4,1,5,3,2,6] => 4
[4,1,5,3,6,2] => 5
[4,1,5,6,2,3] => 3
[4,1,5,6,3,2] => 6
[4,1,6,2,3,5] => 3
[4,1,6,2,5,3] => 5
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 7
[4,1,6,5,2,3] => 6
[4,1,6,5,3,2] => 12
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 4
[4,2,1,5,3,6] => 4
[4,2,1,5,6,3] => 5
[4,2,1,6,3,5] => 5
[4,2,1,6,5,3] => 10
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 6
[4,2,3,5,1,6] => 3
[4,2,3,5,6,1] => 3
[4,2,3,6,1,5] => 6
[4,2,3,6,5,1] => 6
[4,2,5,1,3,6] => 4
[4,2,5,1,6,3] => 5
[4,2,5,3,1,6] => 5
[4,2,5,3,6,1] => 5
[4,2,5,6,1,3] => 5
[4,2,5,6,3,1] => 7
[4,2,6,1,3,5] => 5
[4,2,6,1,5,3] => 10
[4,2,6,3,1,5] => 7
[4,2,6,3,5,1] => 8
[4,2,6,5,1,3] => 10
[4,2,6,5,3,1] => 14
[4,3,1,2,5,6] => 3
[4,3,1,2,6,5] => 6
[4,3,1,5,2,6] => 4
[4,3,1,5,6,2] => 5
[4,3,1,6,2,5] => 6
[4,3,1,6,5,2] => 10
[4,3,2,1,5,6] => 6
[4,3,2,1,6,5] => 11
[4,3,2,5,1,6] => 6
[4,3,2,5,6,1] => 6
[4,3,2,6,1,5] => 11
[4,3,2,6,5,1] => 12
[4,3,5,1,2,6] => 4
[4,3,5,1,6,2] => 5
[4,3,5,2,1,6] => 8
[4,3,5,2,6,1] => 8
[4,3,5,6,1,2] => 5
[4,3,5,6,2,1] => 10
[4,3,6,1,2,5] => 6
[4,3,6,1,5,2] => 10
[4,3,6,2,1,5] => 12
[4,3,6,2,5,1] => 14
[4,3,6,5,1,2] => 10
[4,3,6,5,2,1] => 19
[4,5,1,2,3,6] => 2
[4,5,1,2,6,3] => 3
[4,5,1,3,2,6] => 4
[4,5,1,3,6,2] => 5
[4,5,1,6,2,3] => 3
[4,5,1,6,3,2] => 6
[4,5,2,1,3,6] => 4
[4,5,2,1,6,3] => 6
[4,5,2,3,1,6] => 6
[4,5,2,3,6,1] => 6
[4,5,2,6,1,3] => 6
[4,5,2,6,3,1] => 9
[4,5,3,1,2,6] => 6
[4,5,3,1,6,2] => 7
[4,5,3,2,1,6] => 11
[4,5,3,2,6,1] => 12
[4,5,3,6,1,2] => 7
[4,5,3,6,2,1] => 14
[4,5,6,1,2,3] => 3
[4,5,6,1,3,2] => 6
[4,5,6,2,1,3] => 6
[4,5,6,2,3,1] => 9
[4,5,6,3,1,2] => 9
[4,5,6,3,2,1] => 18
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 5
[4,6,1,3,2,5] => 5
[4,6,1,3,5,2] => 7
[4,6,1,5,2,3] => 6
[4,6,1,5,3,2] => 12
[4,6,2,1,3,5] => 5
[4,6,2,1,5,3] => 10
[4,6,2,3,1,5] => 7
[4,6,2,3,5,1] => 9
[4,6,2,5,1,3] => 10
[4,6,2,5,3,1] => 15
[4,6,3,1,2,5] => 7
[4,6,3,1,5,2] => 12
[4,6,3,2,1,5] => 14
[4,6,3,2,5,1] => 17
[4,6,3,5,1,2] => 12
[4,6,3,5,2,1] => 23
[4,6,5,1,2,3] => 6
[4,6,5,1,3,2] => 12
[4,6,5,2,1,3] => 12
[4,6,5,2,3,1] => 18
[4,6,5,3,1,2] => 18
[4,6,5,3,2,1] => 35
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 2
[5,1,2,4,3,6] => 2
[5,1,2,4,6,3] => 3
[5,1,2,6,3,4] => 3
[5,1,2,6,4,3] => 5
[5,1,3,2,4,6] => 2
[5,1,3,2,6,4] => 4
[5,1,3,4,2,6] => 3
[5,1,3,4,6,2] => 4
[5,1,3,6,2,4] => 4
[5,1,3,6,4,2] => 6
[5,1,4,2,3,6] => 3
[5,1,4,2,6,3] => 5
[5,1,4,3,2,6] => 6
[5,1,4,3,6,2] => 7
[5,1,4,6,2,3] => 5
[5,1,4,6,3,2] => 9
[5,1,6,2,3,4] => 3
[5,1,6,2,4,3] => 5
[5,1,6,3,2,4] => 5
[5,1,6,3,4,2] => 7
[5,1,6,4,2,3] => 7
[5,1,6,4,3,2] => 14
[5,2,1,3,4,6] => 2
[5,2,1,3,6,4] => 4
[5,2,1,4,3,6] => 4
[5,2,1,4,6,3] => 6
[5,2,1,6,3,4] => 5
[5,2,1,6,4,3] => 10
[5,2,3,1,4,6] => 3
[5,2,3,1,6,4] => 6
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 4
[5,2,3,6,1,4] => 6
[5,2,3,6,4,1] => 7
[5,2,4,1,3,6] => 5
[5,2,4,1,6,3] => 7
[5,2,4,3,1,6] => 8
[5,2,4,3,6,1] => 8
[5,2,4,6,1,3] => 7
[5,2,4,6,3,1] => 11
[5,2,6,1,3,4] => 5
[5,2,6,1,4,3] => 10
[5,2,6,3,1,4] => 7
[5,2,6,3,4,1] => 9
[5,2,6,4,1,3] => 12
[5,2,6,4,3,1] => 17
[5,3,1,2,4,6] => 3
[5,3,1,2,6,4] => 6
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 6
[5,3,1,6,2,4] => 6
[5,3,1,6,4,2] => 10
[5,3,2,1,4,6] => 6
[5,3,2,1,6,4] => 11
[5,3,2,4,1,6] => 8
[5,3,2,4,6,1] => 8
[5,3,2,6,1,4] => 11
[5,3,2,6,4,1] => 14
[5,3,4,1,2,6] => 6
[5,3,4,1,6,2] => 7
[5,3,4,2,1,6] => 11
[5,3,4,2,6,1] => 12
[5,3,4,6,1,2] => 7
[5,3,4,6,2,1] => 14
[5,3,6,1,2,4] => 6
[5,3,6,1,4,2] => 10
[5,3,6,2,1,4] => 12
[5,3,6,2,4,1] => 15
[5,3,6,4,1,2] => 12
[5,3,6,4,2,1] => 23
[5,4,1,2,3,6] => 4
[5,4,1,2,6,3] => 6
[5,4,1,3,2,6] => 8
[5,4,1,3,6,2] => 9
[5,4,1,6,2,3] => 6
[5,4,1,6,3,2] => 12
[5,4,2,1,3,6] => 8
[5,4,2,1,6,3] => 12
[5,4,2,3,1,6] => 11
[5,4,2,3,6,1] => 12
[5,4,2,6,1,3] => 12
[5,4,2,6,3,1] => 17
[5,4,3,1,2,6] => 11
[5,4,3,1,6,2] => 14
[5,4,3,2,1,6] => 22
[5,4,3,2,6,1] => 23
[5,4,3,6,1,2] => 14
[5,4,3,6,2,1] => 28
[5,4,6,1,2,3] => 6
[5,4,6,1,3,2] => 12
[5,4,6,2,1,3] => 12
[5,4,6,2,3,1] => 18
[5,4,6,3,1,2] => 18
[5,4,6,3,2,1] => 35
[5,6,1,2,3,4] => 3
[5,6,1,2,4,3] => 5
[5,6,1,3,2,4] => 5
[5,6,1,3,4,2] => 7
[5,6,1,4,2,3] => 7
[5,6,1,4,3,2] => 14
[5,6,2,1,3,4] => 5
[5,6,2,1,4,3] => 10
[5,6,2,3,1,4] => 7
[5,6,2,3,4,1] => 9
[5,6,2,4,1,3] => 12
[5,6,2,4,3,1] => 18
[5,6,3,1,2,4] => 7
[5,6,3,1,4,2] => 12
[5,6,3,2,1,4] => 14
[5,6,3,2,4,1] => 18
[5,6,3,4,1,2] => 14
[5,6,3,4,2,1] => 26
[5,6,4,1,2,3] => 9
[5,6,4,1,3,2] => 18
[5,6,4,2,1,3] => 18
[5,6,4,2,3,1] => 26
[5,6,4,3,1,2] => 26
[5,6,4,3,2,1] => 52
[6,1,2,3,4,5] => 1
[6,1,2,3,5,4] => 2
[6,1,2,4,3,5] => 2
[6,1,2,4,5,3] => 3
[6,1,2,5,3,4] => 3
[6,1,2,5,4,3] => 6
[6,1,3,2,4,5] => 2
[6,1,3,2,5,4] => 4
[6,1,3,4,2,5] => 3
[6,1,3,4,5,2] => 4
[6,1,3,5,2,4] => 5
[6,1,3,5,4,2] => 8
[6,1,4,2,3,5] => 3
[6,1,4,2,5,3] => 5
[6,1,4,3,2,5] => 6
[6,1,4,3,5,2] => 8
[6,1,4,5,2,3] => 6
[6,1,4,5,3,2] => 12
[6,1,5,2,3,4] => 4
[6,1,5,2,4,3] => 8
[6,1,5,3,2,4] => 8
[6,1,5,3,4,2] => 12
[6,1,5,4,2,3] => 12
[6,1,5,4,3,2] => 23
[6,2,1,3,4,5] => 2
[6,2,1,3,5,4] => 4
[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] => 12
[6,2,3,1,4,5] => 3
[6,2,3,1,5,4] => 6
[6,2,3,4,1,5] => 4
[6,2,3,4,5,1] => 5
[6,2,3,5,1,4] => 7
[6,2,3,5,4,1] => 10
[6,2,4,1,3,5] => 5
[6,2,4,1,5,3] => 8
[6,2,4,3,1,5] => 8
[6,2,4,3,5,1] => 10
[6,2,4,5,1,3] => 9
[6,2,4,5,3,1] => 14
[6,2,5,1,3,4] => 7
[6,2,5,1,4,3] => 14
[6,2,5,3,1,4] => 11
[6,2,5,3,4,1] => 14
[6,2,5,4,1,3] => 17
[6,2,5,4,3,1] => 28
[6,3,1,2,4,5] => 3
[6,3,1,2,5,4] => 6
[6,3,1,4,2,5] => 5
[6,3,1,4,5,2] => 7
[6,3,1,5,2,4] => 8
[6,3,1,5,4,2] => 14
[6,3,2,1,4,5] => 6
[6,3,2,1,5,4] => 12
[6,3,2,4,1,5] => 8
[6,3,2,4,5,1] => 10
[6,3,2,5,1,4] => 14
[6,3,2,5,4,1] => 19
[6,3,4,1,2,5] => 6
[6,3,4,1,5,2] => 9
[6,3,4,2,1,5] => 12
[6,3,4,2,5,1] => 14
[6,3,4,5,1,2] => 9
[6,3,4,5,2,1] => 18
[6,3,5,1,2,4] => 9
[6,3,5,1,4,2] => 15
[6,3,5,2,1,4] => 17
[6,3,5,2,4,1] => 23
[6,3,5,4,1,2] => 18
[6,3,5,4,2,1] => 35
[6,4,1,2,3,5] => 4
[6,4,1,2,5,3] => 7
[6,4,1,3,2,5] => 8
[6,4,1,3,5,2] => 11
[6,4,1,5,2,3] => 9
[6,4,1,5,3,2] => 17
[6,4,2,1,3,5] => 8
[6,4,2,1,5,3] => 14
[6,4,2,3,1,5] => 12
[6,4,2,3,5,1] => 14
[6,4,2,5,1,3] => 15
[6,4,2,5,3,1] => 23
[6,4,3,1,2,5] => 12
[6,4,3,1,5,2] => 17
[6,4,3,2,1,5] => 23
[6,4,3,2,5,1] => 28
[6,4,3,5,1,2] => 18
[6,4,3,5,2,1] => 35
[6,4,5,1,2,3] => 9
[6,4,5,1,3,2] => 18
[6,4,5,2,1,3] => 18
[6,4,5,2,3,1] => 26
[6,4,5,3,1,2] => 26
[6,4,5,3,2,1] => 52
[6,5,1,2,3,4] => 5
[6,5,1,2,4,3] => 10
[6,5,1,3,2,4] => 10
[6,5,1,3,4,2] => 14
[6,5,1,4,2,3] => 14
[6,5,1,4,3,2] => 28
[6,5,2,1,3,4] => 10
[6,5,2,1,4,3] => 19
[6,5,2,3,1,4] => 14
[6,5,2,3,4,1] => 18
[6,5,2,4,1,3] => 23
[6,5,2,4,3,1] => 35
[6,5,3,1,2,4] => 14
[6,5,3,1,4,2] => 23
[6,5,3,2,1,4] => 28
[6,5,3,2,4,1] => 35
[6,5,3,4,1,2] => 26
[6,5,3,4,2,1] => 52
[6,5,4,1,2,3] => 18
[6,5,4,1,3,2] => 35
[6,5,4,2,1,3] => 35
[6,5,4,2,3,1] => 52
[6,5,4,3,1,2] => 52
[6,5,4,3,2,1] => 101
click to show generating function       
Description
The width of a permutation.
Let $w$ be a permutation. The interval $[e,w]$ in the weak order is ranked, and we define $r_i=r_i(w)$ to be the number of elements at rank $i$ in $[e,w]$, where $i \in \{0, \dots, \ell(w)\}$. The width of
$w$ is the maximum of $\{r_0,r_1,\ldots,r_{\ell(w)}\}$. See [1].
References
[1] Fishel, S., Milićević, E., Patrias, R., Tenner, B. E. Enumerations relating braid and commutation classes arXiv:1708.04372
Code
def statistic(pi):
    S = SymmetricGroup(len(pi))
    pi = S(pi)
    P = S.weak_poset()
    return P.subposet(P.principal_order_ideal(pi)).width()

Created
Jun 26, 2018 at 12:54 by Susanna Fishel
Updated
Jun 26, 2018 at 13:13 by Christian Stump