edit this statistic or download as text // json
Identifier
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 2
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 3
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[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] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 4
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 4
[1,4,5,3,2] => 5
[1,5,2,3,4] => 3
[1,5,2,4,3] => 4
[1,5,3,2,4] => 4
[1,5,3,4,2] => 3
[1,5,4,2,3] => 5
[1,5,4,3,2] => 4
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 4
[2,3,1,4,5] => 2
[2,3,1,5,4] => 3
[2,3,4,1,5] => 3
[2,3,4,5,1] => 4
[2,3,5,1,4] => 4
[2,3,5,4,1] => 3
[2,4,1,3,5] => 3
[2,4,1,5,3] => 4
[2,4,3,1,5] => 4
[2,4,3,5,1] => 3
[2,4,5,1,3] => 5
[2,4,5,3,1] => 4
[2,5,1,3,4] => 4
[2,5,1,4,3] => 5
[2,5,3,1,4] => 3
[2,5,3,4,1] => 2
[2,5,4,1,3] => 4
[2,5,4,3,1] => 3
[3,1,2,4,5] => 2
[3,1,2,5,4] => 3
[3,1,4,2,5] => 3
[3,1,4,5,2] => 4
[3,1,5,2,4] => 4
[3,1,5,4,2] => 5
[3,2,1,4,5] => 3
[3,2,1,5,4] => 4
[3,2,4,1,5] => 4
[3,2,4,5,1] => 3
[3,2,5,1,4] => 5
[3,2,5,4,1] => 4
[3,4,1,2,5] => 4
[3,4,1,5,2] => 5
[3,4,2,1,5] => 5
[3,4,2,5,1] => 4
[3,4,5,1,2] => 6
[3,4,5,2,1] => 5
[3,5,1,2,4] => 5
[3,5,1,4,2] => 4
[3,5,2,1,4] => 4
>>> Load all 1200 entries. <<<
[3,5,2,4,1] => 3
[3,5,4,1,2] => 5
[3,5,4,2,1] => 4
[4,1,2,3,5] => 3
[4,1,2,5,3] => 4
[4,1,3,2,5] => 4
[4,1,3,5,2] => 3
[4,1,5,2,3] => 5
[4,1,5,3,2] => 4
[4,2,1,3,5] => 4
[4,2,1,5,3] => 5
[4,2,3,1,5] => 3
[4,2,3,5,1] => 2
[4,2,5,1,3] => 4
[4,2,5,3,1] => 3
[4,3,1,2,5] => 5
[4,3,1,5,2] => 4
[4,3,2,1,5] => 4
[4,3,2,5,1] => 3
[4,3,5,1,2] => 5
[4,3,5,2,1] => 4
[4,5,1,2,3] => 6
[4,5,1,3,2] => 5
[4,5,2,1,3] => 5
[4,5,2,3,1] => 4
[4,5,3,1,2] => 4
[4,5,3,2,1] => 5
[5,1,2,3,4] => 4
[5,1,2,4,3] => 3
[5,1,3,2,4] => 3
[5,1,3,4,2] => 2
[5,1,4,2,3] => 4
[5,1,4,3,2] => 3
[5,2,1,3,4] => 3
[5,2,1,4,3] => 4
[5,2,3,1,4] => 2
[5,2,3,4,1] => 1
[5,2,4,1,3] => 3
[5,2,4,3,1] => 2
[5,3,1,2,4] => 4
[5,3,1,4,2] => 3
[5,3,2,1,4] => 3
[5,3,2,4,1] => 2
[5,3,4,1,2] => 4
[5,3,4,2,1] => 3
[5,4,1,2,3] => 5
[5,4,1,3,2] => 4
[5,4,2,1,3] => 4
[5,4,2,3,1] => 3
[5,4,3,1,2] => 5
[5,4,3,2,1] => 4
[1,2,3,4,5,6] => 0
[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] => 3
[1,2,4,3,5,6] => 1
[1,2,4,3,6,5] => 2
[1,2,4,5,3,6] => 2
[1,2,4,5,6,3] => 3
[1,2,4,6,3,5] => 3
[1,2,4,6,5,3] => 4
[1,2,5,3,4,6] => 2
[1,2,5,3,6,4] => 3
[1,2,5,4,3,6] => 3
[1,2,5,4,6,3] => 4
[1,2,5,6,3,4] => 4
[1,2,5,6,4,3] => 5
[1,2,6,3,4,5] => 3
[1,2,6,3,5,4] => 4
[1,2,6,4,3,5] => 4
[1,2,6,4,5,3] => 5
[1,2,6,5,3,4] => 5
[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] => 3
[1,3,2,6,4,5] => 3
[1,3,2,6,5,4] => 4
[1,3,4,2,5,6] => 2
[1,3,4,2,6,5] => 3
[1,3,4,5,2,6] => 3
[1,3,4,5,6,2] => 4
[1,3,4,6,2,5] => 4
[1,3,4,6,5,2] => 5
[1,3,5,2,4,6] => 3
[1,3,5,2,6,4] => 4
[1,3,5,4,2,6] => 4
[1,3,5,4,6,2] => 5
[1,3,5,6,2,4] => 5
[1,3,5,6,4,2] => 6
[1,3,6,2,4,5] => 4
[1,3,6,2,5,4] => 5
[1,3,6,4,2,5] => 5
[1,3,6,4,5,2] => 4
[1,3,6,5,2,4] => 6
[1,3,6,5,4,2] => 5
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 3
[1,4,2,5,3,6] => 3
[1,4,2,5,6,3] => 4
[1,4,2,6,3,5] => 4
[1,4,2,6,5,3] => 5
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 4
[1,4,3,5,2,6] => 4
[1,4,3,5,6,2] => 5
[1,4,3,6,2,5] => 5
[1,4,3,6,5,2] => 6
[1,4,5,2,3,6] => 4
[1,4,5,2,6,3] => 5
[1,4,5,3,2,6] => 5
[1,4,5,3,6,2] => 6
[1,4,5,6,2,3] => 6
[1,4,5,6,3,2] => 7
[1,4,6,2,3,5] => 5
[1,4,6,2,5,3] => 6
[1,4,6,3,2,5] => 6
[1,4,6,3,5,2] => 5
[1,4,6,5,2,3] => 7
[1,4,6,5,3,2] => 6
[1,5,2,3,4,6] => 3
[1,5,2,3,6,4] => 4
[1,5,2,4,3,6] => 4
[1,5,2,4,6,3] => 5
[1,5,2,6,3,4] => 5
[1,5,2,6,4,3] => 6
[1,5,3,2,4,6] => 4
[1,5,3,2,6,4] => 5
[1,5,3,4,2,6] => 5
[1,5,3,4,6,2] => 4
[1,5,3,6,2,4] => 6
[1,5,3,6,4,2] => 5
[1,5,4,2,3,6] => 5
[1,5,4,2,6,3] => 6
[1,5,4,3,2,6] => 6
[1,5,4,3,6,2] => 5
[1,5,4,6,2,3] => 7
[1,5,4,6,3,2] => 6
[1,5,6,2,3,4] => 6
[1,5,6,2,4,3] => 7
[1,5,6,3,2,4] => 7
[1,5,6,3,4,2] => 6
[1,5,6,4,2,3] => 8
[1,5,6,4,3,2] => 7
[1,6,2,3,4,5] => 4
[1,6,2,3,5,4] => 5
[1,6,2,4,3,5] => 5
[1,6,2,4,5,3] => 4
[1,6,2,5,3,4] => 6
[1,6,2,5,4,3] => 5
[1,6,3,2,4,5] => 5
[1,6,3,2,5,4] => 6
[1,6,3,4,2,5] => 4
[1,6,3,4,5,2] => 3
[1,6,3,5,2,4] => 5
[1,6,3,5,4,2] => 4
[1,6,4,2,3,5] => 6
[1,6,4,2,5,3] => 5
[1,6,4,3,2,5] => 5
[1,6,4,3,5,2] => 4
[1,6,4,5,2,3] => 6
[1,6,4,5,3,2] => 5
[1,6,5,2,3,4] => 7
[1,6,5,2,4,3] => 6
[1,6,5,3,2,4] => 6
[1,6,5,3,4,2] => 5
[1,6,5,4,2,3] => 7
[1,6,5,4,3,2] => 6
[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] => 3
[2,1,3,6,4,5] => 3
[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] => 3
[2,1,4,5,6,3] => 4
[2,1,4,6,3,5] => 4
[2,1,4,6,5,3] => 5
[2,1,5,3,4,6] => 3
[2,1,5,3,6,4] => 4
[2,1,5,4,3,6] => 4
[2,1,5,4,6,3] => 5
[2,1,5,6,3,4] => 5
[2,1,5,6,4,3] => 6
[2,1,6,3,4,5] => 4
[2,1,6,3,5,4] => 5
[2,1,6,4,3,5] => 5
[2,1,6,4,5,3] => 6
[2,1,6,5,3,4] => 6
[2,1,6,5,4,3] => 7
[2,3,1,4,5,6] => 2
[2,3,1,4,6,5] => 3
[2,3,1,5,4,6] => 3
[2,3,1,5,6,4] => 4
[2,3,1,6,4,5] => 4
[2,3,1,6,5,4] => 5
[2,3,4,1,5,6] => 3
[2,3,4,1,6,5] => 4
[2,3,4,5,1,6] => 4
[2,3,4,5,6,1] => 5
[2,3,4,6,1,5] => 5
[2,3,4,6,5,1] => 4
[2,3,5,1,4,6] => 4
[2,3,5,1,6,4] => 5
[2,3,5,4,1,6] => 5
[2,3,5,4,6,1] => 4
[2,3,5,6,1,4] => 6
[2,3,5,6,4,1] => 5
[2,3,6,1,4,5] => 5
[2,3,6,1,5,4] => 6
[2,3,6,4,1,5] => 4
[2,3,6,4,5,1] => 3
[2,3,6,5,1,4] => 5
[2,3,6,5,4,1] => 4
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 4
[2,4,1,5,3,6] => 4
[2,4,1,5,6,3] => 5
[2,4,1,6,3,5] => 5
[2,4,1,6,5,3] => 6
[2,4,3,1,5,6] => 4
[2,4,3,1,6,5] => 5
[2,4,3,5,1,6] => 5
[2,4,3,5,6,1] => 4
[2,4,3,6,1,5] => 6
[2,4,3,6,5,1] => 5
[2,4,5,1,3,6] => 5
[2,4,5,1,6,3] => 6
[2,4,5,3,1,6] => 6
[2,4,5,3,6,1] => 5
[2,4,5,6,1,3] => 7
[2,4,5,6,3,1] => 6
[2,4,6,1,3,5] => 6
[2,4,6,1,5,3] => 7
[2,4,6,3,1,5] => 5
[2,4,6,3,5,1] => 4
[2,4,6,5,1,3] => 6
[2,4,6,5,3,1] => 5
[2,5,1,3,4,6] => 4
[2,5,1,3,6,4] => 5
[2,5,1,4,3,6] => 5
[2,5,1,4,6,3] => 6
[2,5,1,6,3,4] => 6
[2,5,1,6,4,3] => 7
[2,5,3,1,4,6] => 5
[2,5,3,1,6,4] => 6
[2,5,3,4,1,6] => 4
[2,5,3,4,6,1] => 3
[2,5,3,6,1,4] => 5
[2,5,3,6,4,1] => 4
[2,5,4,1,3,6] => 6
[2,5,4,1,6,3] => 7
[2,5,4,3,1,6] => 5
[2,5,4,3,6,1] => 4
[2,5,4,6,1,3] => 6
[2,5,4,6,3,1] => 5
[2,5,6,1,3,4] => 7
[2,5,6,1,4,3] => 8
[2,5,6,3,1,4] => 6
[2,5,6,3,4,1] => 5
[2,5,6,4,1,3] => 7
[2,5,6,4,3,1] => 6
[2,6,1,3,4,5] => 5
[2,6,1,3,5,4] => 6
[2,6,1,4,3,5] => 6
[2,6,1,4,5,3] => 5
[2,6,1,5,3,4] => 7
[2,6,1,5,4,3] => 6
[2,6,3,1,4,5] => 4
[2,6,3,1,5,4] => 5
[2,6,3,4,1,5] => 3
[2,6,3,4,5,1] => 2
[2,6,3,5,1,4] => 4
[2,6,3,5,4,1] => 3
[2,6,4,1,3,5] => 5
[2,6,4,1,5,3] => 6
[2,6,4,3,1,5] => 4
[2,6,4,3,5,1] => 3
[2,6,4,5,1,3] => 5
[2,6,4,5,3,1] => 4
[2,6,5,1,3,4] => 6
[2,6,5,1,4,3] => 7
[2,6,5,3,1,4] => 5
[2,6,5,3,4,1] => 4
[2,6,5,4,1,3] => 6
[2,6,5,4,3,1] => 5
[3,1,2,4,5,6] => 2
[3,1,2,4,6,5] => 3
[3,1,2,5,4,6] => 3
[3,1,2,5,6,4] => 4
[3,1,2,6,4,5] => 4
[3,1,2,6,5,4] => 5
[3,1,4,2,5,6] => 3
[3,1,4,2,6,5] => 4
[3,1,4,5,2,6] => 4
[3,1,4,5,6,2] => 5
[3,1,4,6,2,5] => 5
[3,1,4,6,5,2] => 6
[3,1,5,2,4,6] => 4
[3,1,5,2,6,4] => 5
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 6
[3,1,5,6,2,4] => 6
[3,1,5,6,4,2] => 7
[3,1,6,2,4,5] => 5
[3,1,6,2,5,4] => 6
[3,1,6,4,2,5] => 6
[3,1,6,4,5,2] => 5
[3,1,6,5,2,4] => 7
[3,1,6,5,4,2] => 6
[3,2,1,4,5,6] => 3
[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] => 6
[3,2,4,1,5,6] => 4
[3,2,4,1,6,5] => 5
[3,2,4,5,1,6] => 5
[3,2,4,5,6,1] => 4
[3,2,4,6,1,5] => 6
[3,2,4,6,5,1] => 5
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 6
[3,2,5,4,1,6] => 6
[3,2,5,4,6,1] => 5
[3,2,5,6,1,4] => 7
[3,2,5,6,4,1] => 6
[3,2,6,1,4,5] => 6
[3,2,6,1,5,4] => 7
[3,2,6,4,1,5] => 5
[3,2,6,4,5,1] => 4
[3,2,6,5,1,4] => 6
[3,2,6,5,4,1] => 5
[3,4,1,2,5,6] => 4
[3,4,1,2,6,5] => 5
[3,4,1,5,2,6] => 5
[3,4,1,5,6,2] => 6
[3,4,1,6,2,5] => 6
[3,4,1,6,5,2] => 7
[3,4,2,1,5,6] => 5
[3,4,2,1,6,5] => 6
[3,4,2,5,1,6] => 6
[3,4,2,5,6,1] => 5
[3,4,2,6,1,5] => 7
[3,4,2,6,5,1] => 6
[3,4,5,1,2,6] => 6
[3,4,5,1,6,2] => 7
[3,4,5,2,1,6] => 7
[3,4,5,2,6,1] => 6
[3,4,5,6,1,2] => 8
[3,4,5,6,2,1] => 7
[3,4,6,1,2,5] => 7
[3,4,6,1,5,2] => 6
[3,4,6,2,1,5] => 6
[3,4,6,2,5,1] => 5
[3,4,6,5,1,2] => 7
[3,4,6,5,2,1] => 6
[3,5,1,2,4,6] => 5
[3,5,1,2,6,4] => 6
[3,5,1,4,2,6] => 6
[3,5,1,4,6,2] => 5
[3,5,1,6,2,4] => 7
[3,5,1,6,4,2] => 6
[3,5,2,1,4,6] => 6
[3,5,2,1,6,4] => 7
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 4
[3,5,2,6,1,4] => 6
[3,5,2,6,4,1] => 5
[3,5,4,1,2,6] => 7
[3,5,4,1,6,2] => 6
[3,5,4,2,1,6] => 6
[3,5,4,2,6,1] => 5
[3,5,4,6,1,2] => 7
[3,5,4,6,2,1] => 6
[3,5,6,1,2,4] => 8
[3,5,6,1,4,2] => 7
[3,5,6,2,1,4] => 7
[3,5,6,2,4,1] => 6
[3,5,6,4,1,2] => 6
[3,5,6,4,2,1] => 7
[3,6,1,2,4,5] => 6
[3,6,1,2,5,4] => 7
[3,6,1,4,2,5] => 5
[3,6,1,4,5,2] => 4
[3,6,1,5,2,4] => 6
[3,6,1,5,4,2] => 5
[3,6,2,1,4,5] => 5
[3,6,2,1,5,4] => 6
[3,6,2,4,1,5] => 4
[3,6,2,4,5,1] => 3
[3,6,2,5,1,4] => 5
[3,6,2,5,4,1] => 4
[3,6,4,1,2,5] => 6
[3,6,4,1,5,2] => 5
[3,6,4,2,1,5] => 5
[3,6,4,2,5,1] => 4
[3,6,4,5,1,2] => 6
[3,6,4,5,2,1] => 5
[3,6,5,1,2,4] => 7
[3,6,5,1,4,2] => 6
[3,6,5,2,1,4] => 6
[3,6,5,2,4,1] => 5
[3,6,5,4,1,2] => 7
[3,6,5,4,2,1] => 6
[4,1,2,3,5,6] => 3
[4,1,2,3,6,5] => 4
[4,1,2,5,3,6] => 4
[4,1,2,5,6,3] => 5
[4,1,2,6,3,5] => 5
[4,1,2,6,5,3] => 6
[4,1,3,2,5,6] => 4
[4,1,3,2,6,5] => 5
[4,1,3,5,2,6] => 5
[4,1,3,5,6,2] => 4
[4,1,3,6,2,5] => 6
[4,1,3,6,5,2] => 5
[4,1,5,2,3,6] => 5
[4,1,5,2,6,3] => 6
[4,1,5,3,2,6] => 6
[4,1,5,3,6,2] => 5
[4,1,5,6,2,3] => 7
[4,1,5,6,3,2] => 6
[4,1,6,2,3,5] => 6
[4,1,6,2,5,3] => 7
[4,1,6,3,2,5] => 7
[4,1,6,3,5,2] => 6
[4,1,6,5,2,3] => 8
[4,1,6,5,3,2] => 7
[4,2,1,3,5,6] => 4
[4,2,1,3,6,5] => 5
[4,2,1,5,3,6] => 5
[4,2,1,5,6,3] => 6
[4,2,1,6,3,5] => 6
[4,2,1,6,5,3] => 7
[4,2,3,1,5,6] => 5
[4,2,3,1,6,5] => 6
[4,2,3,5,1,6] => 4
[4,2,3,5,6,1] => 3
[4,2,3,6,1,5] => 5
[4,2,3,6,5,1] => 4
[4,2,5,1,3,6] => 6
[4,2,5,1,6,3] => 7
[4,2,5,3,1,6] => 5
[4,2,5,3,6,1] => 4
[4,2,5,6,1,3] => 6
[4,2,5,6,3,1] => 5
[4,2,6,1,3,5] => 7
[4,2,6,1,5,3] => 8
[4,2,6,3,1,5] => 6
[4,2,6,3,5,1] => 5
[4,2,6,5,1,3] => 7
[4,2,6,5,3,1] => 6
[4,3,1,2,5,6] => 5
[4,3,1,2,6,5] => 6
[4,3,1,5,2,6] => 6
[4,3,1,5,6,2] => 5
[4,3,1,6,2,5] => 7
[4,3,1,6,5,2] => 6
[4,3,2,1,5,6] => 6
[4,3,2,1,6,5] => 7
[4,3,2,5,1,6] => 5
[4,3,2,5,6,1] => 4
[4,3,2,6,1,5] => 6
[4,3,2,6,5,1] => 5
[4,3,5,1,2,6] => 7
[4,3,5,1,6,2] => 6
[4,3,5,2,1,6] => 6
[4,3,5,2,6,1] => 5
[4,3,5,6,1,2] => 7
[4,3,5,6,2,1] => 6
[4,3,6,1,2,5] => 8
[4,3,6,1,5,2] => 7
[4,3,6,2,1,5] => 7
[4,3,6,2,5,1] => 6
[4,3,6,5,1,2] => 6
[4,3,6,5,2,1] => 7
[4,5,1,2,3,6] => 6
[4,5,1,2,6,3] => 7
[4,5,1,3,2,6] => 7
[4,5,1,3,6,2] => 6
[4,5,1,6,2,3] => 8
[4,5,1,6,3,2] => 7
[4,5,2,1,3,6] => 7
[4,5,2,1,6,3] => 8
[4,5,2,3,1,6] => 6
[4,5,2,3,6,1] => 5
[4,5,2,6,1,3] => 7
[4,5,2,6,3,1] => 6
[4,5,3,1,2,6] => 8
[4,5,3,1,6,2] => 7
[4,5,3,2,1,6] => 7
[4,5,3,2,6,1] => 6
[4,5,3,6,1,2] => 6
[4,5,3,6,2,1] => 7
[4,5,6,1,2,3] => 9
[4,5,6,1,3,2] => 8
[4,5,6,2,1,3] => 8
[4,5,6,2,3,1] => 7
[4,5,6,3,1,2] => 7
[4,5,6,3,2,1] => 8
[4,6,1,2,3,5] => 7
[4,6,1,2,5,3] => 6
[4,6,1,3,2,5] => 6
[4,6,1,3,5,2] => 5
[4,6,1,5,2,3] => 7
[4,6,1,5,3,2] => 6
[4,6,2,1,3,5] => 6
[4,6,2,1,5,3] => 7
[4,6,2,3,1,5] => 5
[4,6,2,3,5,1] => 4
[4,6,2,5,1,3] => 6
[4,6,2,5,3,1] => 5
[4,6,3,1,2,5] => 7
[4,6,3,1,5,2] => 6
[4,6,3,2,1,5] => 6
[4,6,3,2,5,1] => 5
[4,6,3,5,1,2] => 5
[4,6,3,5,2,1] => 6
[4,6,5,1,2,3] => 8
[4,6,5,1,3,2] => 7
[4,6,5,2,1,3] => 7
[4,6,5,2,3,1] => 6
[4,6,5,3,1,2] => 6
[4,6,5,3,2,1] => 7
[5,1,2,3,4,6] => 4
[5,1,2,3,6,4] => 5
[5,1,2,4,3,6] => 5
[5,1,2,4,6,3] => 4
[5,1,2,6,3,4] => 6
[5,1,2,6,4,3] => 5
[5,1,3,2,4,6] => 5
[5,1,3,2,6,4] => 6
[5,1,3,4,2,6] => 4
[5,1,3,4,6,2] => 3
[5,1,3,6,2,4] => 5
[5,1,3,6,4,2] => 4
[5,1,4,2,3,6] => 6
[5,1,4,2,6,3] => 5
[5,1,4,3,2,6] => 5
[5,1,4,3,6,2] => 4
[5,1,4,6,2,3] => 6
[5,1,4,6,3,2] => 5
[5,1,6,2,3,4] => 7
[5,1,6,2,4,3] => 6
[5,1,6,3,2,4] => 6
[5,1,6,3,4,2] => 5
[5,1,6,4,2,3] => 7
[5,1,6,4,3,2] => 6
[5,2,1,3,4,6] => 5
[5,2,1,3,6,4] => 6
[5,2,1,4,3,6] => 6
[5,2,1,4,6,3] => 5
[5,2,1,6,3,4] => 7
[5,2,1,6,4,3] => 6
[5,2,3,1,4,6] => 4
[5,2,3,1,6,4] => 5
[5,2,3,4,1,6] => 3
[5,2,3,4,6,1] => 2
[5,2,3,6,1,4] => 4
[5,2,3,6,4,1] => 3
[5,2,4,1,3,6] => 5
[5,2,4,1,6,3] => 6
[5,2,4,3,1,6] => 4
[5,2,4,3,6,1] => 3
[5,2,4,6,1,3] => 5
[5,2,4,6,3,1] => 4
[5,2,6,1,3,4] => 6
[5,2,6,1,4,3] => 7
[5,2,6,3,1,4] => 5
[5,2,6,3,4,1] => 4
[5,2,6,4,1,3] => 6
[5,2,6,4,3,1] => 5
[5,3,1,2,4,6] => 6
[5,3,1,2,6,4] => 7
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 4
[5,3,1,6,2,4] => 6
[5,3,1,6,4,2] => 5
[5,3,2,1,4,6] => 5
[5,3,2,1,6,4] => 6
[5,3,2,4,1,6] => 4
[5,3,2,4,6,1] => 3
[5,3,2,6,1,4] => 5
[5,3,2,6,4,1] => 4
[5,3,4,1,2,6] => 6
[5,3,4,1,6,2] => 5
[5,3,4,2,1,6] => 5
[5,3,4,2,6,1] => 4
[5,3,4,6,1,2] => 6
[5,3,4,6,2,1] => 5
[5,3,6,1,2,4] => 7
[5,3,6,1,4,2] => 6
[5,3,6,2,1,4] => 6
[5,3,6,2,4,1] => 5
[5,3,6,4,1,2] => 5
[5,3,6,4,2,1] => 6
[5,4,1,2,3,6] => 7
[5,4,1,2,6,3] => 6
[5,4,1,3,2,6] => 6
[5,4,1,3,6,2] => 5
[5,4,1,6,2,3] => 7
[5,4,1,6,3,2] => 6
[5,4,2,1,3,6] => 6
[5,4,2,1,6,3] => 7
[5,4,2,3,1,6] => 5
[5,4,2,3,6,1] => 4
[5,4,2,6,1,3] => 6
[5,4,2,6,3,1] => 5
[5,4,3,1,2,6] => 7
[5,4,3,1,6,2] => 6
[5,4,3,2,1,6] => 6
[5,4,3,2,6,1] => 5
[5,4,3,6,1,2] => 7
[5,4,3,6,2,1] => 6
[5,4,6,1,2,3] => 8
[5,4,6,1,3,2] => 7
[5,4,6,2,1,3] => 7
[5,4,6,2,3,1] => 6
[5,4,6,3,1,2] => 6
[5,4,6,3,2,1] => 7
[5,6,1,2,3,4] => 8
[5,6,1,2,4,3] => 7
[5,6,1,3,2,4] => 7
[5,6,1,3,4,2] => 6
[5,6,1,4,2,3] => 6
[5,6,1,4,3,2] => 7
[5,6,2,1,3,4] => 7
[5,6,2,1,4,3] => 6
[5,6,2,3,1,4] => 6
[5,6,2,3,4,1] => 5
[5,6,2,4,1,3] => 5
[5,6,2,4,3,1] => 6
[5,6,3,1,2,4] => 6
[5,6,3,1,4,2] => 5
[5,6,3,2,1,4] => 7
[5,6,3,2,4,1] => 6
[5,6,3,4,1,2] => 4
[5,6,3,4,2,1] => 5
[5,6,4,1,2,3] => 7
[5,6,4,1,3,2] => 6
[5,6,4,2,1,3] => 6
[5,6,4,2,3,1] => 7
[5,6,4,3,1,2] => 5
[5,6,4,3,2,1] => 6
[6,1,2,3,4,5] => 5
[6,1,2,3,5,4] => 4
[6,1,2,4,3,5] => 4
[6,1,2,4,5,3] => 3
[6,1,2,5,3,4] => 5
[6,1,2,5,4,3] => 4
[6,1,3,2,4,5] => 4
[6,1,3,2,5,4] => 5
[6,1,3,4,2,5] => 3
[6,1,3,4,5,2] => 2
[6,1,3,5,2,4] => 4
[6,1,3,5,4,2] => 3
[6,1,4,2,3,5] => 5
[6,1,4,2,5,3] => 4
[6,1,4,3,2,5] => 4
[6,1,4,3,5,2] => 3
[6,1,4,5,2,3] => 5
[6,1,4,5,3,2] => 4
[6,1,5,2,3,4] => 6
[6,1,5,2,4,3] => 5
[6,1,5,3,2,4] => 5
[6,1,5,3,4,2] => 4
[6,1,5,4,2,3] => 6
[6,1,5,4,3,2] => 5
[6,2,1,3,4,5] => 4
[6,2,1,3,5,4] => 5
[6,2,1,4,3,5] => 5
[6,2,1,4,5,3] => 4
[6,2,1,5,3,4] => 6
[6,2,1,5,4,3] => 5
[6,2,3,1,4,5] => 3
[6,2,3,1,5,4] => 4
[6,2,3,4,1,5] => 2
[6,2,3,4,5,1] => 1
[6,2,3,5,1,4] => 3
[6,2,3,5,4,1] => 2
[6,2,4,1,3,5] => 4
[6,2,4,1,5,3] => 5
[6,2,4,3,1,5] => 3
[6,2,4,3,5,1] => 2
[6,2,4,5,1,3] => 4
[6,2,4,5,3,1] => 3
[6,2,5,1,3,4] => 5
[6,2,5,1,4,3] => 6
[6,2,5,3,1,4] => 4
[6,2,5,3,4,1] => 3
[6,2,5,4,1,3] => 5
[6,2,5,4,3,1] => 4
[6,3,1,2,4,5] => 5
[6,3,1,2,5,4] => 6
[6,3,1,4,2,5] => 4
[6,3,1,4,5,2] => 3
[6,3,1,5,2,4] => 5
[6,3,1,5,4,2] => 4
[6,3,2,1,4,5] => 4
[6,3,2,1,5,4] => 5
[6,3,2,4,1,5] => 3
[6,3,2,4,5,1] => 2
[6,3,2,5,1,4] => 4
[6,3,2,5,4,1] => 3
[6,3,4,1,2,5] => 5
[6,3,4,1,5,2] => 4
[6,3,4,2,1,5] => 4
[6,3,4,2,5,1] => 3
[6,3,4,5,1,2] => 5
[6,3,4,5,2,1] => 4
[6,3,5,1,2,4] => 6
[6,3,5,1,4,2] => 5
[6,3,5,2,1,4] => 5
[6,3,5,2,4,1] => 4
[6,3,5,4,1,2] => 6
[6,3,5,4,2,1] => 5
[6,4,1,2,3,5] => 6
[6,4,1,2,5,3] => 5
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 4
[6,4,1,5,2,3] => 6
[6,4,1,5,3,2] => 5
[6,4,2,1,3,5] => 5
[6,4,2,1,5,3] => 6
[6,4,2,3,1,5] => 4
[6,4,2,3,5,1] => 3
[6,4,2,5,1,3] => 5
[6,4,2,5,3,1] => 4
[6,4,3,1,2,5] => 6
[6,4,3,1,5,2] => 5
[6,4,3,2,1,5] => 5
[6,4,3,2,5,1] => 4
[6,4,3,5,1,2] => 6
[6,4,3,5,2,1] => 5
[6,4,5,1,2,3] => 7
[6,4,5,1,3,2] => 6
[6,4,5,2,1,3] => 6
[6,4,5,2,3,1] => 5
[6,4,5,3,1,2] => 7
[6,4,5,3,2,1] => 6
[6,5,1,2,3,4] => 7
[6,5,1,2,4,3] => 6
[6,5,1,3,2,4] => 6
[6,5,1,3,4,2] => 5
[6,5,1,4,2,3] => 7
[6,5,1,4,3,2] => 6
[6,5,2,1,3,4] => 6
[6,5,2,1,4,3] => 7
[6,5,2,3,1,4] => 5
[6,5,2,3,4,1] => 4
[6,5,2,4,1,3] => 6
[6,5,2,4,3,1] => 5
[6,5,3,1,2,4] => 7
[6,5,3,1,4,2] => 6
[6,5,3,2,1,4] => 6
[6,5,3,2,4,1] => 5
[6,5,3,4,1,2] => 5
[6,5,3,4,2,1] => 6
[6,5,4,1,2,3] => 8
[6,5,4,1,3,2] => 7
[6,5,4,2,1,3] => 7
[6,5,4,2,3,1] => 6
[6,5,4,3,1,2] => 6
[6,5,4,3,2,1] => 7
[1,2,3,4,5,6,7] => 0
[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] => 3
[1,2,3,5,4,6,7] => 1
[1,2,3,5,4,7,6] => 2
[1,2,3,5,6,4,7] => 2
[1,2,3,5,6,7,4] => 3
[1,2,3,5,7,4,6] => 3
[1,2,3,5,7,6,4] => 4
[1,2,3,6,4,5,7] => 2
[1,2,3,6,4,7,5] => 3
[1,2,3,6,5,4,7] => 3
[1,2,3,6,5,7,4] => 4
[1,2,3,6,7,4,5] => 4
[1,2,3,6,7,5,4] => 5
[1,2,3,7,4,5,6] => 3
[1,2,3,7,4,6,5] => 4
[1,2,3,7,5,4,6] => 4
[1,2,3,7,5,6,4] => 5
[1,2,3,7,6,4,5] => 5
[1,2,3,7,6,5,4] => 6
[1,2,4,3,5,6,7] => 1
[1,2,4,3,5,7,6] => 2
[1,2,4,3,6,5,7] => 2
[1,2,4,3,6,7,5] => 3
[1,2,4,3,7,5,6] => 3
[1,2,4,3,7,6,5] => 4
[1,2,4,5,3,6,7] => 2
[1,2,4,5,3,7,6] => 3
[1,2,4,5,6,3,7] => 3
[1,2,4,5,6,7,3] => 4
[1,2,4,5,7,3,6] => 4
[1,2,4,5,7,6,3] => 5
[1,2,4,6,3,5,7] => 3
[1,2,4,6,3,7,5] => 4
[1,2,4,6,5,3,7] => 4
[1,2,4,6,5,7,3] => 5
[1,2,4,6,7,3,5] => 5
[1,2,4,6,7,5,3] => 6
[1,2,4,7,3,5,6] => 4
[1,2,4,7,3,6,5] => 5
[1,2,4,7,5,3,6] => 5
[1,2,4,7,5,6,3] => 6
[1,2,4,7,6,3,5] => 6
[1,2,4,7,6,5,3] => 7
[1,2,5,3,4,6,7] => 2
[1,2,5,3,4,7,6] => 3
[1,2,5,3,6,4,7] => 3
[1,2,5,3,6,7,4] => 4
[1,2,5,3,7,4,6] => 4
[1,2,5,3,7,6,4] => 5
[1,2,5,4,3,6,7] => 3
[1,2,5,4,3,7,6] => 4
[1,2,5,4,6,3,7] => 4
[1,2,5,4,6,7,3] => 5
[1,2,5,4,7,3,6] => 5
[1,2,5,4,7,6,3] => 6
[1,2,5,6,3,4,7] => 4
[1,2,5,6,3,7,4] => 5
[1,2,5,6,4,3,7] => 5
[1,2,5,6,4,7,3] => 6
[1,2,5,6,7,3,4] => 6
[1,2,5,6,7,4,3] => 7
[1,2,5,7,3,4,6] => 5
[1,2,5,7,3,6,4] => 6
[1,2,5,7,4,3,6] => 6
[1,2,5,7,4,6,3] => 7
[1,2,5,7,6,3,4] => 7
[1,2,5,7,6,4,3] => 8
[1,2,6,3,4,5,7] => 3
[1,2,6,3,4,7,5] => 4
[1,2,6,3,5,4,7] => 4
[1,2,6,3,5,7,4] => 5
[1,2,6,3,7,4,5] => 5
[1,2,6,3,7,5,4] => 6
[1,2,6,4,3,5,7] => 4
[1,2,6,4,3,7,5] => 5
[1,2,6,4,5,3,7] => 5
[1,2,6,4,5,7,3] => 6
[1,2,6,4,7,3,5] => 6
[1,2,6,4,7,5,3] => 7
[1,2,6,5,3,4,7] => 5
[1,2,6,5,3,7,4] => 6
[1,2,6,5,4,3,7] => 6
[1,2,6,5,4,7,3] => 7
[1,2,6,5,7,3,4] => 7
[1,2,6,5,7,4,3] => 8
[1,2,6,7,3,4,5] => 6
[1,2,6,7,3,5,4] => 7
[1,2,6,7,4,3,5] => 7
[1,2,6,7,4,5,3] => 8
[1,2,6,7,5,3,4] => 8
[1,2,6,7,5,4,3] => 9
[1,2,7,3,4,5,6] => 4
[1,2,7,3,4,6,5] => 5
[1,2,7,3,5,4,6] => 5
[1,2,7,3,5,6,4] => 6
[1,2,7,3,6,4,5] => 6
[1,2,7,3,6,5,4] => 7
[1,2,7,4,3,5,6] => 5
[1,2,7,4,3,6,5] => 6
[1,2,7,4,5,3,6] => 6
[1,2,7,4,5,6,3] => 5
[1,2,7,4,6,3,5] => 7
[1,2,7,4,6,5,3] => 6
[1,2,7,5,3,4,6] => 6
[1,2,7,5,3,6,4] => 7
[1,2,7,5,4,3,6] => 7
[1,2,7,5,4,6,3] => 6
[1,2,7,5,6,3,4] => 8
[1,2,7,5,6,4,3] => 7
[1,2,7,6,3,4,5] => 7
[1,2,7,6,3,5,4] => 8
[1,2,7,6,4,3,5] => 8
[1,2,7,6,4,5,3] => 7
[1,2,7,6,5,3,4] => 9
[1,2,7,6,5,4,3] => 8
[1,3,2,4,5,6,7] => 1
[1,3,2,4,5,7,6] => 2
[1,3,2,4,6,5,7] => 2
[1,3,2,4,6,7,5] => 3
[1,3,2,4,7,5,6] => 3
[1,3,2,4,7,6,5] => 4
[1,3,2,5,4,6,7] => 2
[1,3,2,5,4,7,6] => 3
[1,3,2,5,6,4,7] => 3
[1,3,2,5,6,7,4] => 4
[1,3,2,5,7,4,6] => 4
[1,3,2,5,7,6,4] => 5
[1,3,2,6,4,5,7] => 3
[1,3,2,6,4,7,5] => 4
[1,3,2,6,5,4,7] => 4
[1,3,2,6,5,7,4] => 5
[1,3,2,6,7,4,5] => 5
[1,3,2,6,7,5,4] => 6
[1,3,2,7,4,5,6] => 4
[1,3,2,7,4,6,5] => 5
[1,3,2,7,5,4,6] => 5
[1,3,2,7,5,6,4] => 6
[1,3,2,7,6,4,5] => 6
[1,3,2,7,6,5,4] => 7
[1,3,4,2,5,6,7] => 2
[1,3,4,2,5,7,6] => 3
[1,3,4,2,6,5,7] => 3
[1,3,4,2,6,7,5] => 4
[1,3,4,2,7,5,6] => 4
[1,3,4,2,7,6,5] => 5
[1,3,4,5,2,6,7] => 3
[1,3,4,5,2,7,6] => 4
[1,3,4,5,6,2,7] => 4
[1,3,4,5,6,7,2] => 5
[1,3,4,5,7,2,6] => 5
[1,3,4,5,7,6,2] => 6
[1,3,4,6,2,5,7] => 4
[1,3,4,6,2,7,5] => 5
[1,3,4,6,5,2,7] => 5
[1,3,4,6,5,7,2] => 6
[1,3,4,6,7,2,5] => 6
[1,3,4,6,7,5,2] => 7
[1,3,4,7,2,5,6] => 5
[1,3,4,7,2,6,5] => 6
[1,3,4,7,5,2,6] => 6
[1,3,4,7,5,6,2] => 5
[1,3,4,7,6,2,5] => 7
[1,3,4,7,6,5,2] => 6
[1,3,5,2,4,6,7] => 3
[1,3,5,2,4,7,6] => 4
[1,3,5,2,6,4,7] => 4
[1,3,5,2,6,7,4] => 5
[1,3,5,2,7,4,6] => 5
[1,3,5,2,7,6,4] => 6
[1,3,5,4,2,6,7] => 4
[1,3,5,4,2,7,6] => 5
[1,3,5,4,6,2,7] => 5
[1,3,5,4,6,7,2] => 6
[1,3,5,4,7,2,6] => 6
[1,3,5,4,7,6,2] => 7
[1,3,5,6,2,4,7] => 5
[1,3,5,6,2,7,4] => 6
[1,3,5,6,4,2,7] => 6
[1,3,5,6,4,7,2] => 7
[1,3,5,6,7,2,4] => 7
[1,3,5,6,7,4,2] => 8
[1,3,5,7,2,4,6] => 6
[1,3,5,7,2,6,4] => 7
[1,3,5,7,4,2,6] => 7
[1,3,5,7,4,6,2] => 6
[1,3,5,7,6,2,4] => 8
[1,3,5,7,6,4,2] => 7
[1,3,6,2,4,5,7] => 4
[1,3,6,2,4,7,5] => 5
[1,3,6,2,5,4,7] => 5
[1,3,6,2,5,7,4] => 6
[1,3,6,2,7,4,5] => 6
[1,3,6,2,7,5,4] => 7
[1,3,6,4,2,5,7] => 5
[1,3,6,4,2,7,5] => 6
[1,3,6,4,5,2,7] => 6
[1,3,6,4,5,7,2] => 5
[1,3,6,4,7,2,5] => 7
[1,3,6,4,7,5,2] => 6
[1,3,6,5,2,4,7] => 6
[1,3,6,5,2,7,4] => 7
[1,3,6,5,4,2,7] => 7
[1,3,6,5,4,7,2] => 6
[1,3,6,5,7,2,4] => 8
[1,3,6,5,7,4,2] => 7
[1,3,6,7,2,4,5] => 7
[1,3,6,7,2,5,4] => 8
[1,3,6,7,4,2,5] => 8
[1,3,6,7,4,5,2] => 7
[1,3,6,7,5,2,4] => 9
[1,3,6,7,5,4,2] => 8
[1,3,7,2,4,5,6] => 5
[1,3,7,2,4,6,5] => 6
[1,3,7,2,5,4,6] => 6
[1,3,7,2,5,6,4] => 7
[1,3,7,2,6,4,5] => 7
[1,3,7,2,6,5,4] => 8
[1,3,7,4,2,5,6] => 6
[1,3,7,4,2,6,5] => 7
[1,3,7,4,5,2,6] => 5
[1,3,7,4,5,6,2] => 4
[1,3,7,4,6,2,5] => 6
[1,3,7,4,6,5,2] => 5
[1,3,7,5,2,4,6] => 7
[1,3,7,5,2,6,4] => 8
[1,3,7,5,4,2,6] => 6
[1,3,7,5,4,6,2] => 5
[1,3,7,5,6,2,4] => 7
[1,3,7,5,6,4,2] => 6
[1,3,7,6,2,4,5] => 8
[1,3,7,6,2,5,4] => 9
[1,3,7,6,4,2,5] => 7
[1,3,7,6,4,5,2] => 6
[1,3,7,6,5,2,4] => 8
[1,3,7,6,5,4,2] => 7
[1,4,2,3,5,6,7] => 2
[1,4,2,3,5,7,6] => 3
[1,4,2,3,6,5,7] => 3
[1,4,2,3,6,7,5] => 4
[1,4,2,3,7,5,6] => 4
[1,4,2,3,7,6,5] => 5
[1,4,2,5,3,6,7] => 3
[1,4,2,5,3,7,6] => 4
[1,4,2,5,6,3,7] => 4
[1,4,2,5,6,7,3] => 5
[1,4,2,5,7,3,6] => 5
[1,4,2,5,7,6,3] => 6
[1,4,2,6,3,5,7] => 4
[1,4,2,6,3,7,5] => 5
[1,4,2,6,5,3,7] => 5
[1,4,2,6,5,7,3] => 6
[1,4,2,6,7,3,5] => 6
[1,4,2,6,7,5,3] => 7
[1,4,2,7,3,5,6] => 5
[1,4,2,7,3,6,5] => 6
[1,4,2,7,5,3,6] => 6
[1,4,2,7,5,6,3] => 7
[1,4,2,7,6,3,5] => 7
[1,4,2,7,6,5,3] => 8
[1,4,3,2,5,6,7] => 3
[1,4,3,2,5,7,6] => 4
[1,4,3,2,6,5,7] => 4
[1,4,3,2,6,7,5] => 5
[1,4,3,2,7,5,6] => 5
[1,4,3,2,7,6,5] => 6
[1,4,3,5,2,6,7] => 4
[1,4,3,5,2,7,6] => 5
[1,4,3,5,6,2,7] => 5
[1,4,3,5,6,7,2] => 6
[1,4,3,5,7,2,6] => 6
[1,4,3,5,7,6,2] => 7
[1,4,3,6,2,5,7] => 5
[1,4,3,6,2,7,5] => 6
[1,4,3,6,5,2,7] => 6
[1,4,3,6,5,7,2] => 7
[1,4,3,6,7,2,5] => 7
[1,4,3,6,7,5,2] => 8
[1,4,3,7,2,5,6] => 6
[1,4,3,7,2,6,5] => 7
[1,4,3,7,5,2,6] => 7
[1,4,3,7,5,6,2] => 6
[1,4,3,7,6,2,5] => 8
[1,4,3,7,6,5,2] => 7
[1,4,5,2,3,6,7] => 4
[1,4,5,2,3,7,6] => 5
[1,4,5,2,6,3,7] => 5
[1,4,5,2,6,7,3] => 6
[1,4,5,2,7,3,6] => 6
[1,4,5,2,7,6,3] => 7
[1,4,5,3,2,6,7] => 5
[1,4,5,3,2,7,6] => 6
[1,4,5,3,6,2,7] => 6
[1,4,5,3,6,7,2] => 7
[1,4,5,3,7,2,6] => 7
[1,4,5,3,7,6,2] => 8
[1,4,5,6,2,3,7] => 6
[1,4,5,6,2,7,3] => 7
[1,4,5,6,3,2,7] => 7
[1,4,5,6,3,7,2] => 8
[1,4,5,6,7,2,3] => 8
[1,4,5,6,7,3,2] => 9
[1,4,5,7,2,3,6] => 7
[1,4,5,7,2,6,3] => 8
[1,4,5,7,3,2,6] => 8
[1,4,5,7,3,6,2] => 7
[1,4,5,7,6,2,3] => 9
[1,4,5,7,6,3,2] => 8
[1,4,6,2,3,5,7] => 5
[1,4,6,2,3,7,5] => 6
[1,4,6,2,5,3,7] => 6
[1,4,6,2,5,7,3] => 7
[1,4,6,2,7,3,5] => 7
[1,4,6,2,7,5,3] => 8
[1,4,6,3,2,5,7] => 6
[1,4,6,3,2,7,5] => 7
[1,4,6,3,5,2,7] => 7
[1,4,6,3,5,7,2] => 6
[1,4,6,3,7,2,5] => 8
[1,4,6,3,7,5,2] => 7
[1,4,6,5,2,3,7] => 7
[1,4,6,5,2,7,3] => 8
[1,4,6,5,3,2,7] => 8
[1,4,6,5,3,7,2] => 7
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Description
The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$.
Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.
References
[1] Winter, M. A graph similar to the Bruhat graph, what is it called? MathOverflow:366504
Code
def statistic(pi):
    n = len(pi)
    a = Permutation([2,1])
    b = Permutation([i for i in range(2,n+1)] + [1])
    powers = [Permutation([])]
    for k in range(n-1):
        powers.append(powers[-1]*b)
    gens = [p.inverse()*a*p for p in powers]    
    G = PermutationGroup(gens)
    pi = G(pi)
    w = gap.Factorization(G._gap_(), pi._gap_())
    return w.Length().sage()
Created
Jan 08, 2018 at 17:14 by Martin Rubey
Updated
Jan 22, 2024 at 15:11 by Martin Rubey