- FindStat

edit this statistic or download as text // json

Identifier

St001461: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1]=>1
[1,2]=>2
[2,1]=>1
[1,2,3]=>3
[1,3,2]=>2
[2,1,3]=>2
[2,3,1]=>1
[3,1,2]=>1
[3,2,1]=>2
[1,2,3,4]=>4
[1,2,4,3]=>3
[1,3,2,4]=>3
[1,3,4,2]=>2
[1,4,2,3]=>2
[1,4,3,2]=>3
[2,1,3,4]=>3
[2,1,4,3]=>2
[2,3,1,4]=>2
[2,3,4,1]=>1
[2,4,1,3]=>1
[2,4,3,1]=>2
[3,1,2,4]=>2
[3,1,4,2]=>1
[3,2,1,4]=>3
[3,2,4,1]=>2
[3,4,1,2]=>1
[3,4,2,1]=>1
[4,1,2,3]=>1
[4,1,3,2]=>2
[4,2,1,3]=>2
[4,2,3,1]=>3
[4,3,1,2]=>1
[4,3,2,1]=>2
[1,2,3,4,5]=>5
[1,2,3,5,4]=>4
[1,2,4,3,5]=>4
[1,2,4,5,3]=>3
[1,2,5,3,4]=>3
[1,2,5,4,3]=>4
[1,3,2,4,5]=>4
[1,3,2,5,4]=>3
[1,3,4,2,5]=>3
[1,3,4,5,2]=>2
[1,3,5,2,4]=>2
[1,3,5,4,2]=>3
[1,4,2,3,5]=>3
[1,4,2,5,3]=>2
[1,4,3,2,5]=>4
[1,4,3,5,2]=>3
[1,4,5,2,3]=>2
[1,4,5,3,2]=>2
[1,5,2,3,4]=>2
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>4
[1,5,4,2,3]=>2
[1,5,4,3,2]=>3
[2,1,3,4,5]=>4
[2,1,3,5,4]=>3
[2,1,4,3,5]=>3
[2,1,4,5,3]=>2
[2,1,5,3,4]=>2
[2,1,5,4,3]=>3
[2,3,1,4,5]=>3
[2,3,1,5,4]=>2
[2,3,4,1,5]=>2
[2,3,4,5,1]=>1
[2,3,5,1,4]=>1
[2,3,5,4,1]=>2
[2,4,1,3,5]=>2
[2,4,1,5,3]=>1
[2,4,3,1,5]=>3
[2,4,3,5,1]=>2
[2,4,5,1,3]=>1
[2,4,5,3,1]=>1
[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]=>1
[2,5,4,3,1]=>2
[3,1,2,4,5]=>3
[3,1,2,5,4]=>2
[3,1,4,2,5]=>2
[3,1,4,5,2]=>1
[3,1,5,2,4]=>1
[3,1,5,4,2]=>2
[3,2,1,4,5]=>4
[3,2,1,5,4]=>3
[3,2,4,1,5]=>3
[3,2,4,5,1]=>2
[3,2,5,1,4]=>2
[3,2,5,4,1]=>3
[3,4,1,2,5]=>2
[3,4,1,5,2]=>1
[3,4,2,1,5]=>2
[3,4,2,5,1]=>1
[3,4,5,1,2]=>1
[3,4,5,2,1]=>1
[3,5,1,2,4]=>1
[3,5,1,4,2]=>2
[3,5,2,1,4]=>1
[3,5,2,4,1]=>2
[3,5,4,1,2]=>1
[3,5,4,2,1]=>1
[4,1,2,3,5]=>2
[4,1,2,5,3]=>1
[4,1,3,2,5]=>3
[4,1,3,5,2]=>2
[4,1,5,2,3]=>1
[4,1,5,3,2]=>1
[4,2,1,3,5]=>3
[4,2,1,5,3]=>2
[4,2,3,1,5]=>4
[4,2,3,5,1]=>3
[4,2,5,1,3]=>2
[4,2,5,3,1]=>2
[4,3,1,2,5]=>2
[4,3,1,5,2]=>1
[4,3,2,1,5]=>3
[4,3,2,5,1]=>2
[4,3,5,1,2]=>1
[4,3,5,2,1]=>1
[4,5,1,2,3]=>1
[4,5,1,3,2]=>1
[4,5,2,1,3]=>1
[4,5,2,3,1]=>1
[4,5,3,1,2]=>2
[4,5,3,2,1]=>2
[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]=>1
[5,1,4,3,2]=>2
[5,2,1,3,4]=>2
[5,2,1,4,3]=>3
[5,2,3,1,4]=>3
[5,2,3,4,1]=>4
[5,2,4,1,3]=>2
[5,2,4,3,1]=>3
[5,3,1,2,4]=>1
[5,3,1,4,2]=>2
[5,3,2,1,4]=>2
[5,3,2,4,1]=>3
[5,3,4,1,2]=>1
[5,3,4,2,1]=>2
[5,4,1,2,3]=>1
[5,4,1,3,2]=>1
[5,4,2,1,3]=>1
[5,4,2,3,1]=>2
[5,4,3,1,2]=>2
[5,4,3,2,1]=>3
[1,2,3,4,5,6]=>6
[1,2,3,4,6,5]=>5
[1,2,3,5,4,6]=>5
[1,2,3,5,6,4]=>4
[1,2,3,6,4,5]=>4
[1,2,3,6,5,4]=>5
[1,2,4,3,5,6]=>5
[1,2,4,3,6,5]=>4
[1,2,4,5,3,6]=>4
[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]=>4
[1,2,5,3,6,4]=>3
[1,2,5,4,3,6]=>5
[1,2,5,4,6,3]=>4
[1,2,5,6,3,4]=>3
[1,2,5,6,4,3]=>3
[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]=>3
[1,2,6,5,4,3]=>4
[1,3,2,4,5,6]=>5
[1,3,2,4,6,5]=>4
[1,3,2,5,4,6]=>4
[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]=>4
[1,3,4,2,6,5]=>3
[1,3,4,5,2,6]=>3
[1,3,4,5,6,2]=>2
[1,3,4,6,2,5]=>2
[1,3,4,6,5,2]=>3
[1,3,5,2,4,6]=>3
[1,3,5,2,6,4]=>2
[1,3,5,4,2,6]=>4
[1,3,5,4,6,2]=>3
[1,3,5,6,2,4]=>2
[1,3,5,6,4,2]=>2
[1,3,6,2,4,5]=>2
[1,3,6,2,5,4]=>3
[1,3,6,4,2,5]=>3
[1,3,6,4,5,2]=>4
[1,3,6,5,2,4]=>2
[1,3,6,5,4,2]=>3
[1,4,2,3,5,6]=>4
[1,4,2,3,6,5]=>3
[1,4,2,5,3,6]=>3
[1,4,2,5,6,3]=>2
[1,4,2,6,3,5]=>2
[1,4,2,6,5,3]=>3
[1,4,3,2,5,6]=>5
[1,4,3,2,6,5]=>4
[1,4,3,5,2,6]=>4
[1,4,3,5,6,2]=>3
[1,4,3,6,2,5]=>3
[1,4,3,6,5,2]=>4
[1,4,5,2,3,6]=>3
[1,4,5,2,6,3]=>2
[1,4,5,3,2,6]=>3
[1,4,5,3,6,2]=>2
[1,4,5,6,2,3]=>2
[1,4,5,6,3,2]=>2
[1,4,6,2,3,5]=>2
[1,4,6,2,5,3]=>3
[1,4,6,3,2,5]=>2
[1,4,6,3,5,2]=>3
[1,4,6,5,2,3]=>2
[1,4,6,5,3,2]=>2
[1,5,2,3,4,6]=>3
[1,5,2,3,6,4]=>2
[1,5,2,4,3,6]=>4
[1,5,2,4,6,3]=>3
[1,5,2,6,3,4]=>2
[1,5,2,6,4,3]=>2
[1,5,3,2,4,6]=>4
[1,5,3,2,6,4]=>3
[1,5,3,4,2,6]=>5
[1,5,3,4,6,2]=>4
[1,5,3,6,2,4]=>3
[1,5,3,6,4,2]=>3
[1,5,4,2,3,6]=>3
[1,5,4,2,6,3]=>2
[1,5,4,3,2,6]=>4
[1,5,4,3,6,2]=>3
[1,5,4,6,2,3]=>2
[1,5,4,6,3,2]=>2
[1,5,6,2,3,4]=>2
[1,5,6,2,4,3]=>2
[1,5,6,3,2,4]=>2
[1,5,6,3,4,2]=>2
[1,5,6,4,2,3]=>3
[1,5,6,4,3,2]=>3
[1,6,2,3,4,5]=>2
[1,6,2,3,5,4]=>3
[1,6,2,4,3,5]=>3
[1,6,2,4,5,3]=>4
[1,6,2,5,3,4]=>2
[1,6,2,5,4,3]=>3
[1,6,3,2,4,5]=>3
[1,6,3,2,5,4]=>4
[1,6,3,4,2,5]=>4
[1,6,3,4,5,2]=>5
[1,6,3,5,2,4]=>3
[1,6,3,5,4,2]=>4
[1,6,4,2,3,5]=>2
[1,6,4,2,5,3]=>3
[1,6,4,3,2,5]=>3
[1,6,4,3,5,2]=>4
[1,6,4,5,2,3]=>2
[1,6,4,5,3,2]=>3
[1,6,5,2,3,4]=>2
[1,6,5,2,4,3]=>2
[1,6,5,3,2,4]=>2
[1,6,5,3,4,2]=>3
[1,6,5,4,2,3]=>3
[1,6,5,4,3,2]=>4
[2,1,3,4,5,6]=>5
[2,1,3,4,6,5]=>4
[2,1,3,5,4,6]=>4
[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]=>4
[2,1,4,3,6,5]=>3
[2,1,4,5,3,6]=>3
[2,1,4,5,6,3]=>2
[2,1,4,6,3,5]=>2
[2,1,4,6,5,3]=>3
[2,1,5,3,4,6]=>3
[2,1,5,3,6,4]=>2
[2,1,5,4,3,6]=>4
[2,1,5,4,6,3]=>3
[2,1,5,6,3,4]=>2
[2,1,5,6,4,3]=>2
[2,1,6,3,4,5]=>2
[2,1,6,3,5,4]=>3
[2,1,6,4,3,5]=>3
[2,1,6,4,5,3]=>4
[2,1,6,5,3,4]=>2
[2,1,6,5,4,3]=>3
[2,3,1,4,5,6]=>4
[2,3,1,4,6,5]=>3
[2,3,1,5,4,6]=>3
[2,3,1,5,6,4]=>2
[2,3,1,6,4,5]=>2
[2,3,1,6,5,4]=>3
[2,3,4,1,5,6]=>3
[2,3,4,1,6,5]=>2
[2,3,4,5,1,6]=>2
[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]=>2
[2,3,5,1,6,4]=>1
[2,3,5,4,1,6]=>3
[2,3,5,4,6,1]=>2
[2,3,5,6,1,4]=>1
[2,3,5,6,4,1]=>1
[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]=>1
[2,3,6,5,4,1]=>2
[2,4,1,3,5,6]=>3
[2,4,1,3,6,5]=>2
[2,4,1,5,3,6]=>2
[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]=>4
[2,4,3,1,6,5]=>3
[2,4,3,5,1,6]=>3
[2,4,3,5,6,1]=>2
[2,4,3,6,1,5]=>2
[2,4,3,6,5,1]=>3
[2,4,5,1,3,6]=>2
[2,4,5,1,6,3]=>1
[2,4,5,3,1,6]=>2
[2,4,5,3,6,1]=>1
[2,4,5,6,1,3]=>1
[2,4,5,6,3,1]=>1
[2,4,6,1,3,5]=>1
[2,4,6,1,5,3]=>2
[2,4,6,3,1,5]=>1
[2,4,6,3,5,1]=>2
[2,4,6,5,1,3]=>1
[2,4,6,5,3,1]=>1
[2,5,1,3,4,6]=>2
[2,5,1,3,6,4]=>1
[2,5,1,4,3,6]=>3
[2,5,1,4,6,3]=>2
[2,5,1,6,3,4]=>1
[2,5,1,6,4,3]=>1
[2,5,3,1,4,6]=>3
[2,5,3,1,6,4]=>2
[2,5,3,4,1,6]=>4
[2,5,3,4,6,1]=>3
[2,5,3,6,1,4]=>2
[2,5,3,6,4,1]=>2
[2,5,4,1,3,6]=>2
[2,5,4,1,6,3]=>1
[2,5,4,3,1,6]=>3
[2,5,4,3,6,1]=>2
[2,5,4,6,1,3]=>1
[2,5,4,6,3,1]=>1
[2,5,6,1,3,4]=>1
[2,5,6,1,4,3]=>1
[2,5,6,3,1,4]=>1
[2,5,6,3,4,1]=>1
[2,5,6,4,1,3]=>2
[2,5,6,4,3,1]=>2
[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]=>1
[2,6,1,5,4,3]=>2
[2,6,3,1,4,5]=>2
[2,6,3,1,5,4]=>3
[2,6,3,4,1,5]=>3
[2,6,3,4,5,1]=>4
[2,6,3,5,1,4]=>2
[2,6,3,5,4,1]=>3
[2,6,4,1,3,5]=>1
[2,6,4,1,5,3]=>2
[2,6,4,3,1,5]=>2
[2,6,4,3,5,1]=>3
[2,6,4,5,1,3]=>1
[2,6,4,5,3,1]=>2
[2,6,5,1,3,4]=>1
[2,6,5,1,4,3]=>1
[2,6,5,3,1,4]=>1
[2,6,5,3,4,1]=>2
[2,6,5,4,1,3]=>2
[2,6,5,4,3,1]=>3
[3,1,2,4,5,6]=>4
[3,1,2,4,6,5]=>3
[3,1,2,5,4,6]=>3
[3,1,2,5,6,4]=>2
[3,1,2,6,4,5]=>2
[3,1,2,6,5,4]=>3
[3,1,4,2,5,6]=>3
[3,1,4,2,6,5]=>2
[3,1,4,5,2,6]=>2
[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]=>2
[3,1,5,2,6,4]=>1
[3,1,5,4,2,6]=>3
[3,1,5,4,6,2]=>2
[3,1,5,6,2,4]=>1
[3,1,5,6,4,2]=>1
[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]=>1
[3,1,6,5,4,2]=>2
[3,2,1,4,5,6]=>5
[3,2,1,4,6,5]=>4
[3,2,1,5,4,6]=>4
[3,2,1,5,6,4]=>3
[3,2,1,6,4,5]=>3
[3,2,1,6,5,4]=>4
[3,2,4,1,5,6]=>4
[3,2,4,1,6,5]=>3
[3,2,4,5,1,6]=>3
[3,2,4,5,6,1]=>2
[3,2,4,6,1,5]=>2
[3,2,4,6,5,1]=>3
[3,2,5,1,4,6]=>3
[3,2,5,1,6,4]=>2
[3,2,5,4,1,6]=>4
[3,2,5,4,6,1]=>3
[3,2,5,6,1,4]=>2
[3,2,5,6,4,1]=>2
[3,2,6,1,4,5]=>2
[3,2,6,1,5,4]=>3
[3,2,6,4,1,5]=>3
[3,2,6,4,5,1]=>4
[3,2,6,5,1,4]=>2
[3,2,6,5,4,1]=>3
[3,4,1,2,5,6]=>3
[3,4,1,2,6,5]=>2
[3,4,1,5,2,6]=>2
[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]=>2
[3,4,2,5,1,6]=>2
[3,4,2,5,6,1]=>1
[3,4,2,6,1,5]=>1
[3,4,2,6,5,1]=>2
[3,4,5,1,2,6]=>2
[3,4,5,1,6,2]=>1
[3,4,5,2,1,6]=>2
[3,4,5,2,6,1]=>1
[3,4,5,6,1,2]=>1
[3,4,5,6,2,1]=>1
[3,4,6,1,2,5]=>1
[3,4,6,1,5,2]=>2
[3,4,6,2,1,5]=>1
[3,4,6,2,5,1]=>2
[3,4,6,5,1,2]=>1
[3,4,6,5,2,1]=>1
[3,5,1,2,4,6]=>2
[3,5,1,2,6,4]=>1
[3,5,1,4,2,6]=>3
[3,5,1,4,6,2]=>2
[3,5,1,6,2,4]=>1
[3,5,1,6,4,2]=>1
[3,5,2,1,4,6]=>2
[3,5,2,1,6,4]=>1
[3,5,2,4,1,6]=>3
[3,5,2,4,6,1]=>2
[3,5,2,6,1,4]=>1
[3,5,2,6,4,1]=>1
[3,5,4,1,2,6]=>2
[3,5,4,1,6,2]=>1
[3,5,4,2,1,6]=>2
[3,5,4,2,6,1]=>1
[3,5,4,6,1,2]=>1
[3,5,4,6,2,1]=>1
[3,5,6,1,2,4]=>1
[3,5,6,1,4,2]=>1
[3,5,6,2,1,4]=>1
[3,5,6,2,4,1]=>1
[3,5,6,4,1,2]=>2
[3,5,6,4,2,1]=>2
[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]=>1
[3,6,1,5,4,2]=>2
[3,6,2,1,4,5]=>1
[3,6,2,1,5,4]=>2
[3,6,2,4,1,5]=>2
[3,6,2,4,5,1]=>3
[3,6,2,5,1,4]=>1
[3,6,2,5,4,1]=>2
[3,6,4,1,2,5]=>1
[3,6,4,1,5,2]=>2
[3,6,4,2,1,5]=>1
[3,6,4,2,5,1]=>2
[3,6,4,5,1,2]=>1
[3,6,4,5,2,1]=>1
[3,6,5,1,2,4]=>1
[3,6,5,1,4,2]=>1
[3,6,5,2,1,4]=>1
[3,6,5,2,4,1]=>1
[3,6,5,4,1,2]=>2
[3,6,5,4,2,1]=>2
[4,1,2,3,5,6]=>3
[4,1,2,3,6,5]=>2
[4,1,2,5,3,6]=>2
[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]=>4
[4,1,3,2,6,5]=>3
[4,1,3,5,2,6]=>3
[4,1,3,5,6,2]=>2
[4,1,3,6,2,5]=>2
[4,1,3,6,5,2]=>3
[4,1,5,2,3,6]=>2
[4,1,5,2,6,3]=>1
[4,1,5,3,2,6]=>2
[4,1,5,3,6,2]=>1
[4,1,5,6,2,3]=>1
[4,1,5,6,3,2]=>1
[4,1,6,2,3,5]=>1
[4,1,6,2,5,3]=>2
[4,1,6,3,2,5]=>1
[4,1,6,3,5,2]=>2
[4,1,6,5,2,3]=>1
[4,1,6,5,3,2]=>1
[4,2,1,3,5,6]=>4
[4,2,1,3,6,5]=>3
[4,2,1,5,3,6]=>3
[4,2,1,5,6,3]=>2
[4,2,1,6,3,5]=>2
[4,2,1,6,5,3]=>3
[4,2,3,1,5,6]=>5
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>4
[4,2,3,5,6,1]=>3
[4,2,3,6,1,5]=>3
[4,2,3,6,5,1]=>4
[4,2,5,1,3,6]=>3
[4,2,5,1,6,3]=>2
[4,2,5,3,1,6]=>3
[4,2,5,3,6,1]=>2
[4,2,5,6,1,3]=>2
[4,2,5,6,3,1]=>2
[4,2,6,1,3,5]=>2
[4,2,6,1,5,3]=>3
[4,2,6,3,1,5]=>2
[4,2,6,3,5,1]=>3
[4,2,6,5,1,3]=>2
[4,2,6,5,3,1]=>2
[4,3,1,2,5,6]=>3
[4,3,1,2,6,5]=>2
[4,3,1,5,2,6]=>2
[4,3,1,5,6,2]=>1
[4,3,1,6,2,5]=>1
[4,3,1,6,5,2]=>2
[4,3,2,1,5,6]=>4
[4,3,2,1,6,5]=>3
[4,3,2,5,1,6]=>3
[4,3,2,5,6,1]=>2
[4,3,2,6,1,5]=>2
[4,3,2,6,5,1]=>3
[4,3,5,1,2,6]=>2
[4,3,5,1,6,2]=>1
[4,3,5,2,1,6]=>2
[4,3,5,2,6,1]=>1
[4,3,5,6,1,2]=>1
[4,3,5,6,2,1]=>1
[4,3,6,1,2,5]=>1
[4,3,6,1,5,2]=>2
[4,3,6,2,1,5]=>1
[4,3,6,2,5,1]=>2
[4,3,6,5,1,2]=>1
[4,3,6,5,2,1]=>1
[4,5,1,2,3,6]=>2
[4,5,1,2,6,3]=>1
[4,5,1,3,2,6]=>2
[4,5,1,3,6,2]=>1
[4,5,1,6,2,3]=>1
[4,5,1,6,3,2]=>1
[4,5,2,1,3,6]=>2
[4,5,2,1,6,3]=>1
[4,5,2,3,1,6]=>2
[4,5,2,3,6,1]=>1
[4,5,2,6,1,3]=>1
[4,5,2,6,3,1]=>1
[4,5,3,1,2,6]=>3
[4,5,3,1,6,2]=>2
[4,5,3,2,1,6]=>3
[4,5,3,2,6,1]=>2
[4,5,3,6,1,2]=>2
[4,5,3,6,2,1]=>2
[4,5,6,1,2,3]=>1
[4,5,6,1,3,2]=>1
[4,5,6,2,1,3]=>1
[4,5,6,2,3,1]=>1
[4,5,6,3,1,2]=>1
[4,5,6,3,2,1]=>1
[4,6,1,2,3,5]=>1
[4,6,1,2,5,3]=>2
[4,6,1,3,2,5]=>1
[4,6,1,3,5,2]=>2
[4,6,1,5,2,3]=>1
[4,6,1,5,3,2]=>1
[4,6,2,1,3,5]=>1
[4,6,2,1,5,3]=>2
[4,6,2,3,1,5]=>1
[4,6,2,3,5,1]=>2
[4,6,2,5,1,3]=>1
[4,6,2,5,3,1]=>1
[4,6,3,1,2,5]=>2
[4,6,3,1,5,2]=>3
[4,6,3,2,1,5]=>2
[4,6,3,2,5,1]=>3
[4,6,3,5,1,2]=>2
[4,6,3,5,2,1]=>2
[4,6,5,1,2,3]=>1
[4,6,5,1,3,2]=>1
[4,6,5,2,1,3]=>1
[4,6,5,2,3,1]=>1
[4,6,5,3,1,2]=>1
[4,6,5,3,2,1]=>1
[5,1,2,3,4,6]=>2
[5,1,2,3,6,4]=>1
[5,1,2,4,3,6]=>3
[5,1,2,4,6,3]=>2
[5,1,2,6,3,4]=>1
[5,1,2,6,4,3]=>1
[5,1,3,2,4,6]=>3
[5,1,3,2,6,4]=>2
[5,1,3,4,2,6]=>4
[5,1,3,4,6,2]=>3
[5,1,3,6,2,4]=>2
[5,1,3,6,4,2]=>2
[5,1,4,2,3,6]=>2
[5,1,4,2,6,3]=>1
[5,1,4,3,2,6]=>3
[5,1,4,3,6,2]=>2
[5,1,4,6,2,3]=>1
[5,1,4,6,3,2]=>1
[5,1,6,2,3,4]=>1
[5,1,6,2,4,3]=>1
[5,1,6,3,2,4]=>1
[5,1,6,3,4,2]=>1
[5,1,6,4,2,3]=>2
[5,1,6,4,3,2]=>2
[5,2,1,3,4,6]=>3
[5,2,1,3,6,4]=>2
[5,2,1,4,3,6]=>4
[5,2,1,4,6,3]=>3
[5,2,1,6,3,4]=>2
[5,2,1,6,4,3]=>2
[5,2,3,1,4,6]=>4
[5,2,3,1,6,4]=>3
[5,2,3,4,1,6]=>5
[5,2,3,4,6,1]=>4
[5,2,3,6,1,4]=>3
[5,2,3,6,4,1]=>3
[5,2,4,1,3,6]=>3
[5,2,4,1,6,3]=>2
[5,2,4,3,1,6]=>4
[5,2,4,3,6,1]=>3
[5,2,4,6,1,3]=>2
[5,2,4,6,3,1]=>2
[5,2,6,1,3,4]=>2
[5,2,6,1,4,3]=>2
[5,2,6,3,1,4]=>2
[5,2,6,3,4,1]=>2
[5,2,6,4,1,3]=>3
[5,2,6,4,3,1]=>3
[5,3,1,2,4,6]=>2
[5,3,1,2,6,4]=>1
[5,3,1,4,2,6]=>3
[5,3,1,4,6,2]=>2
[5,3,1,6,2,4]=>1
[5,3,1,6,4,2]=>1
[5,3,2,1,4,6]=>3
[5,3,2,1,6,4]=>2
[5,3,2,4,1,6]=>4
[5,3,2,4,6,1]=>3
[5,3,2,6,1,4]=>2
[5,3,2,6,4,1]=>2
[5,3,4,1,2,6]=>2
[5,3,4,1,6,2]=>1
[5,3,4,2,1,6]=>3
[5,3,4,2,6,1]=>2
[5,3,4,6,1,2]=>1
[5,3,4,6,2,1]=>1
[5,3,6,1,2,4]=>1
[5,3,6,1,4,2]=>1
[5,3,6,2,1,4]=>1
[5,3,6,2,4,1]=>1
[5,3,6,4,1,2]=>2
[5,3,6,4,2,1]=>2
[5,4,1,2,3,6]=>2
[5,4,1,2,6,3]=>1
[5,4,1,3,2,6]=>2
[5,4,1,3,6,2]=>1
[5,4,1,6,2,3]=>1
[5,4,1,6,3,2]=>1
[5,4,2,1,3,6]=>2
[5,4,2,1,6,3]=>1
[5,4,2,3,1,6]=>3
[5,4,2,3,6,1]=>2
[5,4,2,6,1,3]=>1
[5,4,2,6,3,1]=>1
[5,4,3,1,2,6]=>3
[5,4,3,1,6,2]=>2
[5,4,3,2,1,6]=>4
[5,4,3,2,6,1]=>3
[5,4,3,6,1,2]=>2
[5,4,3,6,2,1]=>2
[5,4,6,1,2,3]=>1
[5,4,6,1,3,2]=>1
[5,4,6,2,1,3]=>1
[5,4,6,2,3,1]=>1
[5,4,6,3,1,2]=>1
[5,4,6,3,2,1]=>1
[5,6,1,2,3,4]=>1
[5,6,1,2,4,3]=>1
[5,6,1,3,2,4]=>1
[5,6,1,3,4,2]=>1
[5,6,1,4,2,3]=>2
[5,6,1,4,3,2]=>2
[5,6,2,1,3,4]=>1
[5,6,2,1,4,3]=>1
[5,6,2,3,1,4]=>1
[5,6,2,3,4,1]=>1
[5,6,2,4,1,3]=>2
[5,6,2,4,3,1]=>2
[5,6,3,1,2,4]=>2
[5,6,3,1,4,2]=>2
[5,6,3,2,1,4]=>2
[5,6,3,2,4,1]=>2
[5,6,3,4,1,2]=>3
[5,6,3,4,2,1]=>3
[5,6,4,1,2,3]=>1
[5,6,4,1,3,2]=>1
[5,6,4,2,1,3]=>1
[5,6,4,2,3,1]=>1
[5,6,4,3,1,2]=>2
[5,6,4,3,2,1]=>2
[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]=>1
[6,1,2,5,4,3]=>2
[6,1,3,2,4,5]=>2
[6,1,3,2,5,4]=>3
[6,1,3,4,2,5]=>3
[6,1,3,4,5,2]=>4
[6,1,3,5,2,4]=>2
[6,1,3,5,4,2]=>3
[6,1,4,2,3,5]=>1
[6,1,4,2,5,3]=>2
[6,1,4,3,2,5]=>2
[6,1,4,3,5,2]=>3
[6,1,4,5,2,3]=>1
[6,1,4,5,3,2]=>2
[6,1,5,2,3,4]=>1
[6,1,5,2,4,3]=>1
[6,1,5,3,2,4]=>1
[6,1,5,3,4,2]=>2
[6,1,5,4,2,3]=>2
[6,1,5,4,3,2]=>3
[6,2,1,3,4,5]=>2
[6,2,1,3,5,4]=>3
[6,2,1,4,3,5]=>3
[6,2,1,4,5,3]=>4
[6,2,1,5,3,4]=>2
[6,2,1,5,4,3]=>3
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>4
[6,2,3,4,1,5]=>4
[6,2,3,4,5,1]=>5
[6,2,3,5,1,4]=>3
[6,2,3,5,4,1]=>4
[6,2,4,1,3,5]=>2
[6,2,4,1,5,3]=>3
[6,2,4,3,1,5]=>3
[6,2,4,3,5,1]=>4
[6,2,4,5,1,3]=>2
[6,2,4,5,3,1]=>3
[6,2,5,1,3,4]=>2
[6,2,5,1,4,3]=>2
[6,2,5,3,1,4]=>2
[6,2,5,3,4,1]=>3
[6,2,5,4,1,3]=>3
[6,2,5,4,3,1]=>4
[6,3,1,2,4,5]=>1
[6,3,1,2,5,4]=>2
[6,3,1,4,2,5]=>2
[6,3,1,4,5,2]=>3
[6,3,1,5,2,4]=>1
[6,3,1,5,4,2]=>2
[6,3,2,1,4,5]=>2
[6,3,2,1,5,4]=>3
[6,3,2,4,1,5]=>3
[6,3,2,4,5,1]=>4
[6,3,2,5,1,4]=>2
[6,3,2,5,4,1]=>3
[6,3,4,1,2,5]=>1
[6,3,4,1,5,2]=>2
[6,3,4,2,1,5]=>2
[6,3,4,2,5,1]=>3
[6,3,4,5,1,2]=>1
[6,3,4,5,2,1]=>2
[6,3,5,1,2,4]=>1
[6,3,5,1,4,2]=>1
[6,3,5,2,1,4]=>1
[6,3,5,2,4,1]=>2
[6,3,5,4,1,2]=>2
[6,3,5,4,2,1]=>3
[6,4,1,2,3,5]=>1
[6,4,1,2,5,3]=>2
[6,4,1,3,2,5]=>1
[6,4,1,3,5,2]=>2
[6,4,1,5,2,3]=>1
[6,4,1,5,3,2]=>1
[6,4,2,1,3,5]=>1
[6,4,2,1,5,3]=>2
[6,4,2,3,1,5]=>2
[6,4,2,3,5,1]=>3
[6,4,2,5,1,3]=>1
[6,4,2,5,3,1]=>2
[6,4,3,1,2,5]=>2
[6,4,3,1,5,2]=>3
[6,4,3,2,1,5]=>3
[6,4,3,2,5,1]=>4
[6,4,3,5,1,2]=>2
[6,4,3,5,2,1]=>3
[6,4,5,1,2,3]=>1
[6,4,5,1,3,2]=>1
[6,4,5,2,1,3]=>1
[6,4,5,2,3,1]=>2
[6,4,5,3,1,2]=>1
[6,4,5,3,2,1]=>2
[6,5,1,2,3,4]=>1
[6,5,1,2,4,3]=>1
[6,5,1,3,2,4]=>1
[6,5,1,3,4,2]=>1
[6,5,1,4,2,3]=>2
[6,5,1,4,3,2]=>2
[6,5,2,1,3,4]=>1
[6,5,2,1,4,3]=>1
[6,5,2,3,1,4]=>1
[6,5,2,3,4,1]=>2
[6,5,2,4,1,3]=>2
[6,5,2,4,3,1]=>3
[6,5,3,1,2,4]=>2
[6,5,3,1,4,2]=>2
[6,5,3,2,1,4]=>2
[6,5,3,2,4,1]=>3
[6,5,3,4,1,2]=>3
[6,5,3,4,2,1]=>4
[6,5,4,1,2,3]=>1
[6,5,4,1,3,2]=>1
[6,5,4,2,1,3]=>1
[6,5,4,2,3,1]=>2
[6,5,4,3,1,2]=>2
[6,5,4,3,2,1]=>3
[1,2,3,4,5,6,7]=>7
[1,2,3,4,5,7,6]=>6
[1,2,3,4,6,5,7]=>6
[1,2,3,4,6,7,5]=>5
[1,2,3,4,7,5,6]=>5
[1,2,3,4,7,6,5]=>6
[1,2,3,5,4,6,7]=>6
[1,2,3,5,4,7,6]=>5
[1,2,3,5,6,4,7]=>5
[1,2,3,5,6,7,4]=>4
[1,2,3,5,7,4,6]=>4
[1,2,3,5,7,6,4]=>5
[1,2,3,6,4,5,7]=>5
[1,2,3,6,4,7,5]=>4
[1,2,3,6,5,4,7]=>6
[1,2,3,6,5,7,4]=>5
[1,2,3,6,7,4,5]=>4
[1,2,3,6,7,5,4]=>4
[1,2,3,7,4,5,6]=>4
[1,2,3,7,4,6,5]=>5
[1,2,3,7,5,4,6]=>5
[1,2,3,7,5,6,4]=>6
[1,2,3,7,6,4,5]=>4
[1,2,3,7,6,5,4]=>5
[1,2,4,3,5,6,7]=>6
[1,2,4,3,5,7,6]=>5
[1,2,4,3,6,5,7]=>5
[1,2,4,3,6,7,5]=>4
[1,2,4,3,7,5,6]=>4
[1,2,4,3,7,6,5]=>5
[1,2,4,5,3,6,7]=>5
[1,2,4,5,3,7,6]=>4
[1,2,4,5,6,3,7]=>4
[1,2,4,5,6,7,3]=>3
[1,2,4,5,7,3,6]=>3
[1,2,4,5,7,6,3]=>4
[1,2,4,6,3,5,7]=>4
[1,2,4,6,3,7,5]=>3
[1,2,4,6,5,3,7]=>5
[1,2,4,6,5,7,3]=>4
[1,2,4,6,7,3,5]=>3
[1,2,4,6,7,5,3]=>3
[1,2,4,7,3,5,6]=>3
[1,2,4,7,3,6,5]=>4
[1,2,4,7,5,3,6]=>4
[1,2,4,7,5,6,3]=>5
[1,2,4,7,6,3,5]=>3
[1,2,4,7,6,5,3]=>4
[1,2,5,3,4,6,7]=>5
[1,2,5,3,4,7,6]=>4
[1,2,5,3,6,4,7]=>4
[1,2,5,3,6,7,4]=>3
[1,2,5,3,7,4,6]=>3
[1,2,5,3,7,6,4]=>4
[1,2,5,4,3,6,7]=>6
[1,2,5,4,3,7,6]=>5
[1,2,5,4,6,3,7]=>5
[1,2,5,4,6,7,3]=>4
[1,2,5,4,7,3,6]=>4
[1,2,5,4,7,6,3]=>5
[1,2,5,6,3,4,7]=>4
[1,2,5,6,3,7,4]=>3
[1,2,5,6,4,3,7]=>4
[1,2,5,6,4,7,3]=>3
[1,2,5,6,7,3,4]=>3
[1,2,5,6,7,4,3]=>3
[1,2,5,7,3,4,6]=>3
[1,2,5,7,3,6,4]=>4
[1,2,5,7,4,3,6]=>3
[1,2,5,7,4,6,3]=>4
[1,2,5,7,6,3,4]=>3
[1,2,5,7,6,4,3]=>3
[1,2,6,3,4,5,7]=>4
[1,2,6,3,4,7,5]=>3
[1,2,6,3,5,4,7]=>5
[1,2,6,3,5,7,4]=>4
[1,2,6,3,7,4,5]=>3
[1,2,6,3,7,5,4]=>3
[1,2,6,4,3,5,7]=>5
[1,2,6,4,3,7,5]=>4
[1,2,6,4,5,3,7]=>6
[1,2,6,4,5,7,3]=>5
[1,2,6,4,7,3,5]=>4
[1,2,6,4,7,5,3]=>4
[1,2,6,5,3,4,7]=>4
[1,2,6,5,3,7,4]=>3
[1,2,6,5,4,3,7]=>5
[1,2,6,5,4,7,3]=>4
[1,2,6,5,7,3,4]=>3
[1,2,6,5,7,4,3]=>3
[1,2,6,7,3,4,5]=>3
[1,2,6,7,3,5,4]=>3
[1,2,6,7,4,3,5]=>3
[1,2,6,7,4,5,3]=>3
[1,2,6,7,5,3,4]=>4
[1,2,6,7,5,4,3]=>4
[1,2,7,3,4,5,6]=>3
[1,2,7,3,4,6,5]=>4
[1,2,7,3,5,4,6]=>4
[1,2,7,3,5,6,4]=>5
[1,2,7,3,6,4,5]=>3
[1,2,7,3,6,5,4]=>4
[1,2,7,4,3,5,6]=>4
[1,2,7,4,3,6,5]=>5
[1,2,7,4,5,3,6]=>5
[1,2,7,4,5,6,3]=>6
[1,2,7,4,6,3,5]=>4
[1,2,7,4,6,5,3]=>5
[1,2,7,5,3,4,6]=>3
[1,2,7,5,3,6,4]=>4
[1,2,7,5,4,3,6]=>4
[1,2,7,5,4,6,3]=>5
[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]=>5
[1,3,2,4,5,6,7]=>6
[1,3,2,4,5,7,6]=>5
[1,3,2,4,6,5,7]=>5
[1,3,2,4,6,7,5]=>4
[1,3,2,4,7,5,6]=>4
[1,3,2,4,7,6,5]=>5
[1,3,2,5,4,6,7]=>5
[1,3,2,5,4,7,6]=>4
[1,3,2,5,6,4,7]=>4
[1,3,2,5,6,7,4]=>3
[1,3,2,5,7,4,6]=>3
[1,3,2,5,7,6,4]=>4
[1,3,2,6,4,5,7]=>4
[1,3,2,6,4,7,5]=>3
[1,3,2,6,5,4,7]=>5
[1,3,2,6,5,7,4]=>4
[1,3,2,6,7,4,5]=>3
[1,3,2,6,7,5,4]=>3
[1,3,2,7,4,5,6]=>3
[1,3,2,7,4,6,5]=>4
[1,3,2,7,5,4,6]=>4
[1,3,2,7,5,6,4]=>5
[1,3,2,7,6,4,5]=>3
[1,3,2,7,6,5,4]=>4
[1,3,4,2,5,6,7]=>5
[1,3,4,2,5,7,6]=>4
[1,3,4,2,6,5,7]=>4
[1,3,4,2,6,7,5]=>3
[1,3,4,2,7,5,6]=>3
[1,3,4,2,7,6,5]=>4
[1,3,4,5,2,6,7]=>4
[1,3,4,5,2,7,6]=>3
[1,3,4,5,6,2,7]=>3
[1,3,4,5,6,7,2]=>2
[1,3,4,5,7,2,6]=>2
[1,3,4,5,7,6,2]=>3
[1,3,4,6,2,5,7]=>3
[1,3,4,6,2,7,5]=>2
[1,3,4,6,5,2,7]=>4
[1,3,4,6,5,7,2]=>3
[1,3,4,6,7,2,5]=>2
[1,3,4,6,7,5,2]=>2
[1,3,4,7,2,5,6]=>2
[1,3,4,7,2,6,5]=>3
[1,3,4,7,5,2,6]=>3
[1,3,4,7,5,6,2]=>4
[1,3,4,7,6,2,5]=>2
[1,3,4,7,6,5,2]=>3
[1,3,5,2,4,6,7]=>4
[1,3,5,2,4,7,6]=>3
[1,3,5,2,6,4,7]=>3
[1,3,5,2,6,7,4]=>2
[1,3,5,2,7,4,6]=>2
[1,3,5,2,7,6,4]=>3
[1,3,5,4,2,6,7]=>5
[1,3,5,4,2,7,6]=>4
[1,3,5,4,6,2,7]=>4
[1,3,5,4,6,7,2]=>3
[1,3,5,4,7,2,6]=>3
[1,3,5,4,7,6,2]=>4
[1,3,5,6,2,4,7]=>3
[1,3,5,6,2,7,4]=>2
[1,3,5,6,4,2,7]=>3
[1,3,5,6,4,7,2]=>2
[1,3,5,6,7,2,4]=>2
[1,3,5,6,7,4,2]=>2
[1,3,5,7,2,4,6]=>2
[1,3,5,7,2,6,4]=>3
[1,3,5,7,4,2,6]=>2
[1,3,5,7,4,6,2]=>3
[1,3,5,7,6,2,4]=>2
[1,3,5,7,6,4,2]=>2
[1,3,6,2,4,5,7]=>3
[1,3,6,2,4,7,5]=>2
[1,3,6,2,5,4,7]=>4
[1,3,6,2,5,7,4]=>3
[1,3,6,2,7,4,5]=>2
[1,3,6,2,7,5,4]=>2
[1,3,6,4,2,5,7]=>4
[1,3,6,4,2,7,5]=>3
[1,3,6,4,5,2,7]=>5
[1,3,6,4,5,7,2]=>4
[1,3,6,4,7,2,5]=>3
[1,3,6,4,7,5,2]=>3
[1,3,6,5,2,4,7]=>3
[1,3,6,5,2,7,4]=>2
[1,3,6,5,4,2,7]=>4
[1,3,6,5,4,7,2]=>3
[1,3,6,5,7,2,4]=>2
[1,3,6,5,7,4,2]=>2
[1,3,6,7,2,4,5]=>2
[1,3,6,7,2,5,4]=>2
[1,3,6,7,4,2,5]=>2
[1,3,6,7,4,5,2]=>2
[1,3,6,7,5,2,4]=>3
[1,3,6,7,5,4,2]=>3
[1,3,7,2,4,5,6]=>2
[1,3,7,2,4,6,5]=>3
[1,3,7,2,5,4,6]=>3
[1,3,7,2,5,6,4]=>4
[1,3,7,2,6,4,5]=>2
[1,3,7,2,6,5,4]=>3
[1,3,7,4,2,5,6]=>3
[1,3,7,4,2,6,5]=>4
[1,3,7,4,5,2,6]=>4
[1,3,7,4,5,6,2]=>5
[1,3,7,4,6,2,5]=>3
[1,3,7,4,6,5,2]=>4
[1,3,7,5,2,4,6]=>2
[1,3,7,5,2,6,4]=>3
[1,3,7,5,4,2,6]=>3
[1,3,7,5,4,6,2]=>4
[1,3,7,5,6,2,4]=>2
[1,3,7,5,6,4,2]=>3
[1,3,7,6,2,4,5]=>2
[1,3,7,6,2,5,4]=>2
[1,3,7,6,4,2,5]=>2
[1,3,7,6,4,5,2]=>3
[1,3,7,6,5,2,4]=>3
[1,3,7,6,5,4,2]=>4
[1,4,2,3,5,6,7]=>5
[1,4,2,3,5,7,6]=>4
[1,4,2,3,6,5,7]=>4
[1,4,2,3,6,7,5]=>3
[1,4,2,3,7,5,6]=>3
[1,4,2,3,7,6,5]=>4
[1,4,2,5,3,6,7]=>4
[1,4,2,5,3,7,6]=>3
[1,4,2,5,6,3,7]=>3
[1,4,2,5,6,7,3]=>2
[1,4,2,5,7,3,6]=>2
[1,4,2,5,7,6,3]=>3
[1,4,2,6,3,5,7]=>3
[1,4,2,6,3,7,5]=>2
[1,4,2,6,5,3,7]=>4
[1,4,2,6,5,7,3]=>3
[1,4,2,6,7,3,5]=>2
[1,4,2,6,7,5,3]=>2
[1,4,2,7,3,5,6]=>2
[1,4,2,7,3,6,5]=>3
[1,4,2,7,5,3,6]=>3
[1,4,2,7,5,6,3]=>4
[1,4,2,7,6,3,5]=>2
[1,4,2,7,6,5,3]=>3
[1,4,3,2,5,6,7]=>6
[1,4,3,2,5,7,6]=>5
[1,4,3,2,6,5,7]=>5
[1,4,3,2,6,7,5]=>4
[1,4,3,2,7,5,6]=>4
[1,4,3,2,7,6,5]=>5
[1,4,3,5,2,6,7]=>5
[1,4,3,5,2,7,6]=>4
[1,4,3,5,6,2,7]=>4
[1,4,3,5,6,7,2]=>3
[1,4,3,5,7,2,6]=>3
[1,4,3,5,7,6,2]=>4
[1,4,3,6,2,5,7]=>4
[1,4,3,6,2,7,5]=>3
[1,4,3,6,5,2,7]=>5
[1,4,3,6,5,7,2]=>4
[1,4,3,6,7,2,5]=>3
[1,4,3,6,7,5,2]=>3
[1,4,3,7,2,5,6]=>3
[1,4,3,7,2,6,5]=>4
[1,4,3,7,5,2,6]=>4
[1,4,3,7,5,6,2]=>5
[1,4,3,7,6,2,5]=>3
[1,4,3,7,6,5,2]=>4
[1,4,5,2,3,6,7]=>4
[1,4,5,2,3,7,6]=>3
[1,4,5,2,6,3,7]=>3
[1,4,5,2,6,7,3]=>2
[1,4,5,2,7,3,6]=>2
[1,4,5,2,7,6,3]=>3
[1,4,5,3,2,6,7]=>4
[1,4,5,3,2,7,6]=>3
[1,4,5,3,6,2,7]=>3
[1,4,5,3,6,7,2]=>2
[1,4,5,3,7,2,6]=>2
[1,4,5,3,7,6,2]=>3
[1,4,5,6,2,3,7]=>3
[1,4,5,6,2,7,3]=>2
[1,4,5,6,3,2,7]=>3
[1,4,5,6,3,7,2]=>2
[1,4,5,6,7,2,3]=>2
[1,4,5,6,7,3,2]=>2
[1,4,5,7,2,3,6]=>2
[1,4,5,7,2,6,3]=>3
[1,4,5,7,3,2,6]=>2
[1,4,5,7,3,6,2]=>3
[1,4,5,7,6,2,3]=>2
[1,4,5,7,6,3,2]=>2
[1,4,6,2,3,5,7]=>3
[1,4,6,2,3,7,5]=>2
[1,4,6,2,5,3,7]=>4
[1,4,6,2,5,7,3]=>3
[1,4,6,2,7,3,5]=>2
[1,4,6,2,7,5,3]=>2
[1,4,6,3,2,5,7]=>3
[1,4,6,3,2,7,5]=>2
[1,4,6,3,5,2,7]=>4
[1,4,6,3,5,7,2]=>3
[1,4,6,3,7,2,5]=>2
[1,4,6,3,7,5,2]=>2
[1,4,6,5,2,3,7]=>3
[1,4,6,5,2,7,3]=>2
[1,4,6,5,3,2,7]=>3
                    

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Description

The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.

References

[1] Callan, D. Counting Stabilized-Interval-Free Permutations arXiv:math/0310157
[2] Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n > 0. OEIS:A075834

Code

def factor(pi):
    """
    Return smallest i such that pi([1,...,i]) equals [1,...,i]
    """
    pi = list(pi)
    for i in range(1, len(pi)+1):
        A = pi[:i]
        if max(A) == i:
            return Permutation(A), Permutation([e-i for e in pi[i:]])

def rotate(pi):
    r = len(pi)
    pi1 = [x % r + 1 for x in pi]
    return Permutation(pi1[-1:] + pi1[:-1])

def orbit(pi):
    o = [pi]
    new_pi = rotate(pi)
    while pi != new_pi:
        o += [new_pi]
        new_pi = rotate(new_pi)
    return o

@cached_function
def statistic(pi):
    o = orbit(pi)
    for pi in o:
        A, B = factor(pi)
        if len(B) > 0:
            return statistic(A) + statistic(B)
    return 1

Created

Aug 09, 2019 at 21:30 by Martin Rubey

Updated

Aug 09, 2019 at 21:30 by Martin Rubey