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] => 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] => 4
[4,3,1,2] => 3
[4,3,2,1] => 5
[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] => 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] => 4
[1,5,4,2,3] => 3
[1,5,4,3,2] => 5
[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] => 2
[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] => 4
[2,5,4,1,3] => 3
[2,5,4,3,1] => 5
[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] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 3
[3,4,2,5,1] => 3
[3,4,5,1,2] => 3
[3,4,5,2,1] => 5
[3,5,1,2,4] => 2
[3,5,1,4,2] => 4
>>> Load all 873 entries. <<<[3,5,2,1,4] => 3
[3,5,2,4,1] => 6
[3,5,4,1,2] => 5
[3,5,4,2,1] => 8
[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] => 2
[4,1,5,3,2] => 3
[4,2,1,3,5] => 2
[4,2,1,5,3] => 2
[4,2,3,1,5] => 4
[4,2,3,5,1] => 4
[4,2,5,1,3] => 4
[4,2,5,3,1] => 6
[4,3,1,2,5] => 3
[4,3,1,5,2] => 3
[4,3,2,1,5] => 5
[4,3,2,5,1] => 5
[4,3,5,1,2] => 5
[4,3,5,2,1] => 8
[4,5,1,2,3] => 3
[4,5,1,3,2] => 5
[4,5,2,1,3] => 5
[4,5,2,3,1] => 8
[4,5,3,1,2] => 9
[4,5,3,2,1] => 14
[5,1,2,3,4] => 1
[5,1,2,4,3] => 2
[5,1,3,2,4] => 2
[5,1,3,4,2] => 4
[5,1,4,2,3] => 3
[5,1,4,3,2] => 5
[5,2,1,3,4] => 2
[5,2,1,4,3] => 4
[5,2,3,1,4] => 4
[5,2,3,4,1] => 8
[5,2,4,1,3] => 6
[5,2,4,3,1] => 10
[5,3,1,2,4] => 3
[5,3,1,4,2] => 6
[5,3,2,1,4] => 5
[5,3,2,4,1] => 10
[5,3,4,1,2] => 8
[5,3,4,2,1] => 13
[5,4,1,2,3] => 5
[5,4,1,3,2] => 8
[5,4,2,1,3] => 8
[5,4,2,3,1] => 13
[5,4,3,1,2] => 14
[5,4,3,2,1] => 23
[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] => 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] => 4
[1,2,6,5,3,4] => 3
[1,2,6,5,4,3] => 5
[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] => 2
[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] => 4
[1,3,6,5,2,4] => 3
[1,3,6,5,4,2] => 5
[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] => 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] => 3
[1,4,5,6,3,2] => 5
[1,4,6,2,3,5] => 2
[1,4,6,2,5,3] => 4
[1,4,6,3,2,5] => 3
[1,4,6,3,5,2] => 6
[1,4,6,5,2,3] => 5
[1,4,6,5,3,2] => 8
[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] => 2
[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] => 4
[1,5,3,4,6,2] => 4
[1,5,3,6,2,4] => 4
[1,5,3,6,4,2] => 6
[1,5,4,2,3,6] => 3
[1,5,4,2,6,3] => 3
[1,5,4,3,2,6] => 5
[1,5,4,3,6,2] => 5
[1,5,4,6,2,3] => 5
[1,5,4,6,3,2] => 8
[1,5,6,2,3,4] => 3
[1,5,6,2,4,3] => 5
[1,5,6,3,2,4] => 5
[1,5,6,3,4,2] => 8
[1,5,6,4,2,3] => 9
[1,5,6,4,3,2] => 14
[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] => 4
[1,6,2,5,3,4] => 3
[1,6,2,5,4,3] => 5
[1,6,3,2,4,5] => 2
[1,6,3,2,5,4] => 4
[1,6,3,4,2,5] => 4
[1,6,3,4,5,2] => 8
[1,6,3,5,2,4] => 6
[1,6,3,5,4,2] => 10
[1,6,4,2,3,5] => 3
[1,6,4,2,5,3] => 6
[1,6,4,3,2,5] => 5
[1,6,4,3,5,2] => 10
[1,6,4,5,2,3] => 8
[1,6,4,5,3,2] => 13
[1,6,5,2,3,4] => 5
[1,6,5,2,4,3] => 8
[1,6,5,3,2,4] => 8
[1,6,5,3,4,2] => 13
[1,6,5,4,2,3] => 14
[1,6,5,4,3,2] => 23
[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] => 2
[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] => 4
[2,1,6,5,3,4] => 3
[2,1,6,5,4,3] => 5
[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] => 2
[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] => 4
[2,3,6,5,1,4] => 3
[2,3,6,5,4,1] => 5
[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] => 2
[2,4,5,1,6,3] => 2
[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] => 5
[2,4,6,1,3,5] => 2
[2,4,6,1,5,3] => 4
[2,4,6,3,1,5] => 3
[2,4,6,3,5,1] => 6
[2,4,6,5,1,3] => 5
[2,4,6,5,3,1] => 8
[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] => 2
[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] => 4
[2,5,3,4,6,1] => 4
[2,5,3,6,1,4] => 4
[2,5,3,6,4,1] => 6
[2,5,4,1,3,6] => 3
[2,5,4,1,6,3] => 3
[2,5,4,3,1,6] => 5
[2,5,4,3,6,1] => 5
[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] => 5
[2,5,6,3,1,4] => 5
[2,5,6,3,4,1] => 8
[2,5,6,4,1,3] => 9
[2,5,6,4,3,1] => 14
[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] => 4
[2,6,1,5,3,4] => 3
[2,6,1,5,4,3] => 5
[2,6,3,1,4,5] => 2
[2,6,3,1,5,4] => 4
[2,6,3,4,1,5] => 4
[2,6,3,4,5,1] => 8
[2,6,3,5,1,4] => 6
[2,6,3,5,4,1] => 10
[2,6,4,1,3,5] => 3
[2,6,4,1,5,3] => 6
[2,6,4,3,1,5] => 5
[2,6,4,3,5,1] => 10
[2,6,4,5,1,3] => 8
[2,6,4,5,3,1] => 13
[2,6,5,1,3,4] => 5
[2,6,5,1,4,3] => 8
[2,6,5,3,1,4] => 8
[2,6,5,3,4,1] => 13
[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] => 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] => 2
[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] => 4
[3,1,6,5,2,4] => 3
[3,1,6,5,4,2] => 5
[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] => 4
[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] => 8
[3,2,6,5,1,4] => 6
[3,2,6,5,4,1] => 10
[3,4,1,2,5,6] => 2
[3,4,1,2,6,5] => 2
[3,4,1,5,2,6] => 2
[3,4,1,5,6,2] => 2
[3,4,1,6,2,5] => 2
[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] => 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] => 3
[3,4,5,1,6,2] => 3
[3,4,5,2,1,6] => 5
[3,4,5,2,6,1] => 5
[3,4,5,6,1,2] => 5
[3,4,5,6,2,1] => 8
[3,4,6,1,2,5] => 3
[3,4,6,1,5,2] => 6
[3,4,6,2,1,5] => 5
[3,4,6,2,5,1] => 10
[3,4,6,5,1,2] => 8
[3,4,6,5,2,1] => 13
[3,5,1,2,4,6] => 2
[3,5,1,2,6,4] => 2
[3,5,1,4,2,6] => 4
[3,5,1,4,6,2] => 4
[3,5,1,6,2,4] => 4
[3,5,1,6,4,2] => 6
[3,5,2,1,4,6] => 3
[3,5,2,1,6,4] => 3
[3,5,2,4,1,6] => 6
[3,5,2,4,6,1] => 6
[3,5,2,6,1,4] => 6
[3,5,2,6,4,1] => 9
[3,5,4,1,2,6] => 5
[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] => 8
[3,5,4,6,2,1] => 13
[3,5,6,1,2,4] => 5
[3,5,6,1,4,2] => 8
[3,5,6,2,1,4] => 8
[3,5,6,2,4,1] => 13
[3,5,6,4,1,2] => 14
[3,5,6,4,2,1] => 23
[3,6,1,2,4,5] => 2
[3,6,1,2,5,4] => 4
[3,6,1,4,2,5] => 4
[3,6,1,4,5,2] => 8
[3,6,1,5,2,4] => 6
[3,6,1,5,4,2] => 10
[3,6,2,1,4,5] => 3
[3,6,2,1,5,4] => 6
[3,6,2,4,1,5] => 6
[3,6,2,4,5,1] => 12
[3,6,2,5,1,4] => 9
[3,6,2,5,4,1] => 15
[3,6,4,1,2,5] => 5
[3,6,4,1,5,2] => 10
[3,6,4,2,1,5] => 8
[3,6,4,2,5,1] => 16
[3,6,4,5,1,2] => 13
[3,6,4,5,2,1] => 21
[3,6,5,1,2,4] => 8
[3,6,5,1,4,2] => 13
[3,6,5,2,1,4] => 13
[3,6,5,2,4,1] => 21
[3,6,5,4,1,2] => 23
[3,6,5,4,2,1] => 37
[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] => 2
[4,1,5,2,6,3] => 2
[4,1,5,3,2,6] => 3
[4,1,5,3,6,2] => 3
[4,1,5,6,2,3] => 3
[4,1,5,6,3,2] => 5
[4,1,6,2,3,5] => 2
[4,1,6,2,5,3] => 4
[4,1,6,3,2,5] => 3
[4,1,6,3,5,2] => 6
[4,1,6,5,2,3] => 5
[4,1,6,5,3,2] => 8
[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] => 4
[4,2,3,1,6,5] => 4
[4,2,3,5,1,6] => 4
[4,2,3,5,6,1] => 4
[4,2,3,6,1,5] => 4
[4,2,3,6,5,1] => 8
[4,2,5,1,3,6] => 4
[4,2,5,1,6,3] => 4
[4,2,5,3,1,6] => 6
[4,2,5,3,6,1] => 6
[4,2,5,6,1,3] => 6
[4,2,5,6,3,1] => 10
[4,2,6,1,3,5] => 4
[4,2,6,1,5,3] => 8
[4,2,6,3,1,5] => 6
[4,2,6,3,5,1] => 12
[4,2,6,5,1,3] => 10
[4,2,6,5,3,1] => 16
[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] => 5
[4,3,2,1,6,5] => 5
[4,3,2,5,1,6] => 5
[4,3,2,5,6,1] => 5
[4,3,2,6,1,5] => 5
[4,3,2,6,5,1] => 10
[4,3,5,1,2,6] => 5
[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] => 8
[4,3,5,6,2,1] => 13
[4,3,6,1,2,5] => 5
[4,3,6,1,5,2] => 10
[4,3,6,2,1,5] => 8
[4,3,6,2,5,1] => 16
[4,3,6,5,1,2] => 13
[4,3,6,5,2,1] => 21
[4,5,1,2,3,6] => 3
[4,5,1,2,6,3] => 3
[4,5,1,3,2,6] => 5
[4,5,1,3,6,2] => 5
[4,5,1,6,2,3] => 5
[4,5,1,6,3,2] => 8
[4,5,2,1,3,6] => 5
[4,5,2,1,6,3] => 5
[4,5,2,3,1,6] => 8
[4,5,2,3,6,1] => 8
[4,5,2,6,1,3] => 8
[4,5,2,6,3,1] => 13
[4,5,3,1,2,6] => 9
[4,5,3,1,6,2] => 9
[4,5,3,2,1,6] => 14
[4,5,3,2,6,1] => 14
[4,5,3,6,1,2] => 14
[4,5,3,6,2,1] => 23
[4,5,6,1,2,3] => 9
[4,5,6,1,3,2] => 14
[4,5,6,2,1,3] => 14
[4,5,6,2,3,1] => 22
[4,5,6,3,1,2] => 22
[4,5,6,3,2,1] => 36
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 6
[4,6,1,3,2,5] => 5
[4,6,1,3,5,2] => 10
[4,6,1,5,2,3] => 8
[4,6,1,5,3,2] => 13
[4,6,2,1,3,5] => 5
[4,6,2,1,5,3] => 10
[4,6,2,3,1,5] => 8
[4,6,2,3,5,1] => 16
[4,6,2,5,1,3] => 13
[4,6,2,5,3,1] => 21
[4,6,3,1,2,5] => 9
[4,6,3,1,5,2] => 18
[4,6,3,2,1,5] => 14
[4,6,3,2,5,1] => 28
[4,6,3,5,1,2] => 23
[4,6,3,5,2,1] => 37
[4,6,5,1,2,3] => 14
[4,6,5,1,3,2] => 23
[4,6,5,2,1,3] => 22
[4,6,5,2,3,1] => 36
[4,6,5,3,1,2] => 36
[4,6,5,3,2,1] => 58
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 1
[5,1,2,4,3,6] => 2
[5,1,2,4,6,3] => 2
[5,1,2,6,3,4] => 2
[5,1,2,6,4,3] => 3
[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] => 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] => 3
[5,1,4,3,2,6] => 5
[5,1,4,3,6,2] => 5
[5,1,4,6,2,3] => 5
[5,1,4,6,3,2] => 8
[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] => 8
[5,1,6,4,2,3] => 9
[5,1,6,4,3,2] => 14
[5,2,1,3,4,6] => 2
[5,2,1,3,6,4] => 2
[5,2,1,4,3,6] => 4
[5,2,1,4,6,3] => 4
[5,2,1,6,3,4] => 4
[5,2,1,6,4,3] => 6
[5,2,3,1,4,6] => 4
[5,2,3,1,6,4] => 4
[5,2,3,4,1,6] => 8
[5,2,3,4,6,1] => 8
[5,2,3,6,1,4] => 8
[5,2,3,6,4,1] => 12
[5,2,4,1,3,6] => 6
[5,2,4,1,6,3] => 6
[5,2,4,3,1,6] => 10
[5,2,4,3,6,1] => 10
[5,2,4,6,1,3] => 10
[5,2,4,6,3,1] => 16
[5,2,6,1,3,4] => 6
[5,2,6,1,4,3] => 10
[5,2,6,3,1,4] => 10
[5,2,6,3,4,1] => 16
[5,2,6,4,1,3] => 18
[5,2,6,4,3,1] => 28
[5,3,1,2,4,6] => 3
[5,3,1,2,6,4] => 3
[5,3,1,4,2,6] => 6
[5,3,1,4,6,2] => 6
[5,3,1,6,2,4] => 6
[5,3,1,6,4,2] => 9
[5,3,2,1,4,6] => 5
[5,3,2,1,6,4] => 5
[5,3,2,4,1,6] => 10
[5,3,2,4,6,1] => 10
[5,3,2,6,1,4] => 10
[5,3,2,6,4,1] => 15
[5,3,4,1,2,6] => 8
[5,3,4,1,6,2] => 8
[5,3,4,2,1,6] => 13
[5,3,4,2,6,1] => 13
[5,3,4,6,1,2] => 13
[5,3,4,6,2,1] => 21
[5,3,6,1,2,4] => 8
[5,3,6,1,4,2] => 13
[5,3,6,2,1,4] => 13
[5,3,6,2,4,1] => 21
[5,3,6,4,1,2] => 23
[5,3,6,4,2,1] => 37
[5,4,1,2,3,6] => 5
[5,4,1,2,6,3] => 5
[5,4,1,3,2,6] => 8
[5,4,1,3,6,2] => 8
[5,4,1,6,2,3] => 8
[5,4,1,6,3,2] => 13
[5,4,2,1,3,6] => 8
[5,4,2,1,6,3] => 8
[5,4,2,3,1,6] => 13
[5,4,2,3,6,1] => 13
[5,4,2,6,1,3] => 13
[5,4,2,6,3,1] => 21
[5,4,3,1,2,6] => 14
[5,4,3,1,6,2] => 14
[5,4,3,2,1,6] => 23
[5,4,3,2,6,1] => 23
[5,4,3,6,1,2] => 23
[5,4,3,6,2,1] => 37
[5,4,6,1,2,3] => 14
[5,4,6,1,3,2] => 22
[5,4,6,2,1,3] => 23
[5,4,6,2,3,1] => 36
[5,4,6,3,1,2] => 36
[5,4,6,3,2,1] => 58
[5,6,1,2,3,4] => 5
[5,6,1,2,4,3] => 8
[5,6,1,3,2,4] => 8
[5,6,1,3,4,2] => 13
[5,6,1,4,2,3] => 14
[5,6,1,4,3,2] => 23
[5,6,2,1,3,4] => 8
[5,6,2,1,4,3] => 13
[5,6,2,3,1,4] => 13
[5,6,2,3,4,1] => 21
[5,6,2,4,1,3] => 23
[5,6,2,4,3,1] => 37
[5,6,3,1,2,4] => 14
[5,6,3,1,4,2] => 23
[5,6,3,2,1,4] => 23
[5,6,3,2,4,1] => 37
[5,6,3,4,1,2] => 41
[5,6,3,4,2,1] => 65
[5,6,4,1,2,3] => 22
[5,6,4,1,3,2] => 36
[5,6,4,2,1,3] => 36
[5,6,4,2,3,1] => 58
[5,6,4,3,1,2] => 59
[5,6,4,3,2,1] => 94
[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] => 4
[6,1,2,5,3,4] => 3
[6,1,2,5,4,3] => 5
[6,1,3,2,4,5] => 2
[6,1,3,2,5,4] => 4
[6,1,3,4,2,5] => 4
[6,1,3,4,5,2] => 8
[6,1,3,5,2,4] => 6
[6,1,3,5,4,2] => 10
[6,1,4,2,3,5] => 3
[6,1,4,2,5,3] => 6
[6,1,4,3,2,5] => 5
[6,1,4,3,5,2] => 10
[6,1,4,5,2,3] => 8
[6,1,4,5,3,2] => 13
[6,1,5,2,3,4] => 5
[6,1,5,2,4,3] => 8
[6,1,5,3,2,4] => 8
[6,1,5,3,4,2] => 13
[6,1,5,4,2,3] => 14
[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] => 8
[6,2,1,5,3,4] => 6
[6,2,1,5,4,3] => 10
[6,2,3,1,4,5] => 4
[6,2,3,1,5,4] => 8
[6,2,3,4,1,5] => 8
[6,2,3,4,5,1] => 16
[6,2,3,5,1,4] => 12
[6,2,3,5,4,1] => 20
[6,2,4,1,3,5] => 6
[6,2,4,1,5,3] => 12
[6,2,4,3,1,5] => 10
[6,2,4,3,5,1] => 20
[6,2,4,5,1,3] => 16
[6,2,4,5,3,1] => 26
[6,2,5,1,3,4] => 10
[6,2,5,1,4,3] => 16
[6,2,5,3,1,4] => 16
[6,2,5,3,4,1] => 26
[6,2,5,4,1,3] => 28
[6,2,5,4,3,1] => 46
[6,3,1,2,4,5] => 3
[6,3,1,2,5,4] => 6
[6,3,1,4,2,5] => 6
[6,3,1,4,5,2] => 12
[6,3,1,5,2,4] => 9
[6,3,1,5,4,2] => 15
[6,3,2,1,4,5] => 5
[6,3,2,1,5,4] => 10
[6,3,2,4,1,5] => 10
[6,3,2,4,5,1] => 20
[6,3,2,5,1,4] => 15
[6,3,2,5,4,1] => 25
[6,3,4,1,2,5] => 8
[6,3,4,1,5,2] => 16
[6,3,4,2,1,5] => 13
[6,3,4,2,5,1] => 26
[6,3,4,5,1,2] => 21
[6,3,4,5,2,1] => 34
[6,3,5,1,2,4] => 13
[6,3,5,1,4,2] => 21
[6,3,5,2,1,4] => 21
[6,3,5,2,4,1] => 34
[6,3,5,4,1,2] => 37
[6,3,5,4,2,1] => 60
[6,4,1,2,3,5] => 5
[6,4,1,2,5,3] => 10
[6,4,1,3,2,5] => 8
[6,4,1,3,5,2] => 16
[6,4,1,5,2,3] => 13
[6,4,1,5,3,2] => 21
[6,4,2,1,3,5] => 8
[6,4,2,1,5,3] => 16
[6,4,2,3,1,5] => 13
[6,4,2,3,5,1] => 26
[6,4,2,5,1,3] => 21
[6,4,2,5,3,1] => 34
[6,4,3,1,2,5] => 14
[6,4,3,1,5,2] => 28
[6,4,3,2,1,5] => 23
[6,4,3,2,5,1] => 46
[6,4,3,5,1,2] => 37
[6,4,3,5,2,1] => 60
[6,4,5,1,2,3] => 22
[6,4,5,1,3,2] => 36
[6,4,5,2,1,3] => 36
[6,4,5,2,3,1] => 59
[6,4,5,3,1,2] => 58
[6,4,5,3,2,1] => 94
[6,5,1,2,3,4] => 8
[6,5,1,2,4,3] => 13
[6,5,1,3,2,4] => 13
[6,5,1,3,4,2] => 21
[6,5,1,4,2,3] => 23
[6,5,1,4,3,2] => 37
[6,5,2,1,3,4] => 13
[6,5,2,1,4,3] => 21
[6,5,2,3,1,4] => 21
[6,5,2,3,4,1] => 34
[6,5,2,4,1,3] => 37
[6,5,2,4,3,1] => 60
[6,5,3,1,2,4] => 23
[6,5,3,1,4,2] => 37
[6,5,3,2,1,4] => 37
[6,5,3,2,4,1] => 60
[6,5,3,4,1,2] => 65
[6,5,3,4,2,1] => 106
[6,5,4,1,2,3] => 36
[6,5,4,1,3,2] => 58
[6,5,4,2,1,3] => 58
[6,5,4,2,3,1] => 94
[6,5,4,3,1,2] => 94
[6,5,4,3,2,1] => 153
click to show known generating functions
Search the OEIS for these generating functions
Search the Online Encyclopedia of Integer
Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
5,1 13,7,2,1,1 34,31,14,10,12,4,0,7,1,2,0,0,2,2,0,0,0,0,0,0,0,0,1
$F_{1} = q$
$F_{2} = 2\ q$
$F_{3} = 5\ q + q^{2}$
$F_{4} = 13\ q + 7\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$
$F_{5} = 34\ q + 31\ q^{2} + 14\ q^{3} + 10\ q^{4} + 12\ q^{5} + 4\ q^{6} + 7\ q^{8} + q^{9} + 2\ q^{10} + 2\ q^{13} + 2\ q^{14} + q^{23}$
$F_{6} = 89\ q + 115\ q^{2} + 62\ q^{3} + 59\ q^{4} + 65\ q^{5} + 40\ q^{6} + 65\ q^{8} + 11\ q^{9} + 30\ q^{10} + 6\ q^{12} + 36\ q^{13} + 20\ q^{14} + 4\ q^{15} + 13\ q^{16} + 2\ q^{18} + 3\ q^{20} + 16\ q^{21} + 6\ q^{22} + 20\ q^{23} + q^{25} + 4\ q^{26} + 4\ q^{28} + 4\ q^{34} + 10\ q^{36} + 12\ q^{37} + q^{41} + 2\ q^{46} + 6\ q^{58} + 2\ q^{59} + 4\ q^{60} + 2\ q^{65} + 4\ q^{94} + q^{106} + q^{153}$
Description
The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5.3]) with parameters given by the identity and the permutation.
Also the number of paths in the Bruhat order from the identity to the permutation that are increasing with respect to a given reflection ordering as defined in Björner and Brenti [1, Section 5.2].
Code
def statistic(w):
n = len(w)
R.=QQ[]
W = SymmetricGroup(n)
KL = KazhdanLusztigPolynomial(W,q)
return KL.R_tilde(W.one(),W(w)).subs(q=1)
Created
Jan 31, 2024 at 16:32 by Joel Lewis, Alejandro Morales
Updated
Nov 06, 2024 at 16:35 by Nupur Jain