- FindStat

edit this statistic or download as text // json

Identifier

St001078: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1,2]=>0
[2,1]=>0
[1,2,3]=>0
[1,3,2]=>1
[2,1,3]=>1
[2,3,1]=>0
[3,1,2]=>0
[3,2,1]=>1
[1,2,3,4]=>0
[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]=>0
[2,4,1,3]=>1
[2,4,3,1]=>1
[3,1,2,4]=>1
[3,1,4,2]=>1
[3,2,1,4]=>2
[3,2,4,1]=>1
[3,4,1,2]=>0
[3,4,2,1]=>1
[4,1,2,3]=>0
[4,1,3,2]=>1
[4,2,1,3]=>1
[4,2,3,1]=>1
[4,3,1,2]=>1
[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]=>1
[1,3,5,2,4]=>3
[1,3,5,4,2]=>2
[1,4,2,3,5]=>2
[1,4,2,5,3]=>3
[1,4,3,2,5]=>3
[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]=>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]=>1
[2,3,4,5,1]=>0
[2,3,5,1,4]=>2
[2,3,5,4,1]=>1
[2,4,1,3,5]=>3
[2,4,1,5,3]=>2
[2,4,3,1,5]=>2
[2,4,3,5,1]=>1
[2,4,5,1,3]=>1
[2,4,5,3,1]=>2
[2,5,1,3,4]=>2
[2,5,1,4,3]=>3
[2,5,3,1,4]=>3
[2,5,3,4,1]=>2
[2,5,4,1,3]=>2
[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]=>2
[3,1,5,2,4]=>2
[3,1,5,4,2]=>3
[3,2,1,4,5]=>3
[3,2,1,5,4]=>4
[3,2,4,1,5]=>2
[3,2,4,5,1]=>1
[3,2,5,1,4]=>3
[3,2,5,4,1]=>2
[3,4,1,2,5]=>2
[3,4,1,5,2]=>1
[3,4,2,1,5]=>3
[3,4,2,5,1]=>2
[3,4,5,1,2]=>0
[3,4,5,2,1]=>1
[3,5,1,2,4]=>1
[3,5,1,4,2]=>2
[3,5,2,1,4]=>2
[3,5,2,4,1]=>3
[3,5,4,1,2]=>1
[3,5,4,2,1]=>2
[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]=>1
[4,1,5,3,2]=>2
[4,2,1,3,5]=>2
[4,2,1,5,3]=>3
[4,2,3,1,5]=>3
[4,2,3,5,1]=>2
[4,2,5,1,3]=>2
[4,2,5,3,1]=>3
[4,3,1,2,5]=>3
[4,3,1,5,2]=>2
[4,3,2,1,5]=>4
[4,3,2,5,1]=>3
[4,3,5,1,2]=>1
[4,3,5,2,1]=>2
[4,5,1,2,3]=>0
[4,5,1,3,2]=>1
[4,5,2,1,3]=>1
[4,5,2,3,1]=>2
[4,5,3,1,2]=>2
[4,5,3,2,1]=>3
[5,1,2,3,4]=>0
[5,1,2,4,3]=>1
[5,1,3,2,4]=>1
[5,1,3,4,2]=>2
[5,1,4,2,3]=>2
[5,1,4,3,2]=>3
[5,2,1,3,4]=>1
[5,2,1,4,3]=>2
[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]=>2
[5,3,1,4,2]=>3
[5,3,2,1,4]=>3
[5,3,2,4,1]=>2
[5,3,4,1,2]=>2
[5,3,4,2,1]=>3
[5,4,1,2,3]=>1
[5,4,1,3,2]=>2
[5,4,2,1,3]=>2
[5,4,2,3,1]=>3
[5,4,3,1,2]=>3
[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]=>2
[1,2,4,6,3,5]=>3
[1,2,4,6,5,3]=>3
[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]=>3
[1,2,5,6,3,4]=>4
[1,2,5,6,4,3]=>4
[1,2,6,3,4,5]=>2
[1,2,6,3,5,4]=>3
[1,2,6,4,3,5]=>3
[1,2,6,4,5,3]=>4
[1,2,6,5,3,4]=>4
[1,2,6,5,4,3]=>5
[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]=>2
[1,3,4,5,6,2]=>1
[1,3,4,6,2,5]=>3
[1,3,4,6,5,2]=>2
[1,3,5,2,4,6]=>3
[1,3,5,2,6,4]=>4
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>2
[1,3,5,6,2,4]=>3
[1,3,5,6,4,2]=>3
[1,3,6,2,4,5]=>3
[1,3,6,2,5,4]=>4
[1,3,6,4,2,5]=>4
[1,3,6,4,5,2]=>3
[1,3,6,5,2,4]=>4
[1,3,6,5,4,2]=>4
[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]=>3
[1,4,2,6,3,5]=>4
[1,4,2,6,5,3]=>4
[1,4,3,2,5,6]=>3
[1,4,3,2,6,5]=>4
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>2
[1,4,3,6,2,5]=>4
[1,4,3,6,5,2]=>3
[1,4,5,2,3,6]=>4
[1,4,5,2,6,3]=>3
[1,4,5,3,2,6]=>4
[1,4,5,3,6,2]=>3
[1,4,5,6,2,3]=>2
[1,4,5,6,3,2]=>3
[1,4,6,2,3,5]=>3
[1,4,6,2,5,3]=>4
[1,4,6,3,2,5]=>4
[1,4,6,3,5,2]=>4
[1,4,6,5,2,3]=>3
[1,4,6,5,3,2]=>4
[1,5,2,3,4,6]=>2
[1,5,2,3,6,4]=>3
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>4
[1,5,2,6,3,4]=>3
[1,5,2,6,4,3]=>4
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>4
[1,5,3,4,2,6]=>4
[1,5,3,4,6,2]=>3
[1,5,3,6,2,4]=>4
[1,5,3,6,4,2]=>4
[1,5,4,2,3,6]=>4
[1,5,4,2,6,3]=>4
[1,5,4,3,2,6]=>5
[1,5,4,3,6,2]=>4
[1,5,4,6,2,3]=>3
[1,5,4,6,3,2]=>4
[1,5,6,2,3,4]=>2
[1,5,6,2,4,3]=>3
[1,5,6,3,2,4]=>3
[1,5,6,3,4,2]=>4
[1,5,6,4,2,3]=>4
[1,5,6,4,3,2]=>5
[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]=>4
[1,6,3,2,4,5]=>2
[1,6,3,2,5,4]=>3
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>3
[1,6,3,5,2,4]=>4
[1,6,3,5,4,2]=>4
[1,6,4,2,3,5]=>3
[1,6,4,2,5,3]=>4
[1,6,4,3,2,5]=>4
[1,6,4,3,5,2]=>4
[1,6,4,5,2,3]=>4
[1,6,4,5,3,2]=>5
[1,6,5,2,3,4]=>3
[1,6,5,2,4,3]=>4
[1,6,5,3,2,4]=>4
[1,6,5,3,4,2]=>5
[1,6,5,4,2,3]=>5
[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]=>3
[2,1,4,6,3,5]=>4
[2,1,4,6,5,3]=>4
[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]=>4
[2,1,5,6,3,4]=>4
[2,1,5,6,4,3]=>5
[2,1,6,3,4,5]=>3
[2,1,6,3,5,4]=>4
[2,1,6,4,3,5]=>4
[2,1,6,4,5,3]=>5
[2,1,6,5,3,4]=>5
[2,1,6,5,4,3]=>6
[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]=>2
[2,3,4,1,6,5]=>3
[2,3,4,5,1,6]=>1
[2,3,4,5,6,1]=>0
[2,3,4,6,1,5]=>2
[2,3,4,6,5,1]=>1
[2,3,5,1,4,6]=>3
[2,3,5,1,6,4]=>3
[2,3,5,4,1,6]=>2
[2,3,5,4,6,1]=>1
[2,3,5,6,1,4]=>2
[2,3,5,6,4,1]=>2
[2,3,6,1,4,5]=>4
[2,3,6,1,5,4]=>4
[2,3,6,4,1,5]=>3
[2,3,6,4,5,1]=>2
[2,3,6,5,1,4]=>3
[2,3,6,5,4,1]=>3
[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]=>3
[2,4,1,6,3,5]=>4
[2,4,1,6,5,3]=>4
[2,4,3,1,5,6]=>3
[2,4,3,1,6,5]=>4
[2,4,3,5,1,6]=>2
[2,4,3,5,6,1]=>1
[2,4,3,6,1,5]=>3
[2,4,3,6,5,1]=>2
[2,4,5,1,3,6]=>3
[2,4,5,1,6,3]=>2
[2,4,5,3,1,6]=>3
[2,4,5,3,6,1]=>2
[2,4,5,6,1,3]=>1
[2,4,5,6,3,1]=>2
[2,4,6,1,3,5]=>3
[2,4,6,1,5,3]=>3
[2,4,6,3,1,5]=>4
[2,4,6,3,5,1]=>3
[2,4,6,5,1,3]=>2
[2,4,6,5,3,1]=>3
[2,5,1,3,4,6]=>3
[2,5,1,3,6,4]=>4
[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]=>4
[2,5,3,1,4,6]=>4
[2,5,3,1,6,4]=>4
[2,5,3,4,1,6]=>3
[2,5,3,4,6,1]=>2
[2,5,3,6,1,4]=>3
[2,5,3,6,4,1]=>3
[2,5,4,1,3,6]=>4
[2,5,4,1,6,3]=>3
[2,5,4,3,1,6]=>4
[2,5,4,3,6,1]=>3
[2,5,4,6,1,3]=>2
[2,5,4,6,3,1]=>3
[2,5,6,1,3,4]=>2
[2,5,6,1,4,3]=>3
[2,5,6,3,1,4]=>3
[2,5,6,3,4,1]=>4
[2,5,6,4,1,3]=>3
[2,5,6,4,3,1]=>4
[2,6,1,3,4,5]=>2
[2,6,1,3,5,4]=>3
[2,6,1,4,3,5]=>3
[2,6,1,4,5,3]=>4
[2,6,1,5,3,4]=>4
[2,6,1,5,4,3]=>5
[2,6,3,1,4,5]=>3
[2,6,3,1,5,4]=>4
[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]=>4
[2,6,4,1,5,3]=>4
[2,6,4,3,1,5]=>4
[2,6,4,3,5,1]=>3
[2,6,4,5,1,3]=>3
[2,6,4,5,3,1]=>4
[2,6,5,1,3,4]=>3
[2,6,5,1,4,3]=>4
[2,6,5,3,1,4]=>4
[2,6,5,3,4,1]=>4
[2,6,5,4,1,3]=>4
[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]=>3
[3,1,4,5,6,2]=>2
[3,1,4,6,2,5]=>4
[3,1,4,6,5,2]=>3
[3,1,5,2,4,6]=>4
[3,1,5,2,6,4]=>4
[3,1,5,4,2,6]=>4
[3,1,5,4,6,2]=>3
[3,1,5,6,2,4]=>3
[3,1,5,6,4,2]=>4
[3,1,6,2,4,5]=>3
[3,1,6,2,5,4]=>4
[3,1,6,4,2,5]=>4
[3,1,6,4,5,2]=>4
[3,1,6,5,2,4]=>4
[3,1,6,5,4,2]=>5
[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]=>3
[3,2,4,1,6,5]=>4
[3,2,4,5,1,6]=>2
[3,2,4,5,6,1]=>1
[3,2,4,6,1,5]=>3
[3,2,4,6,5,1]=>2
[3,2,5,1,4,6]=>4
[3,2,5,1,6,4]=>4
[3,2,5,4,1,6]=>3
[3,2,5,4,6,1]=>2
[3,2,5,6,1,4]=>3
[3,2,5,6,4,1]=>3
[3,2,6,1,4,5]=>4
[3,2,6,1,5,4]=>5
[3,2,6,4,1,5]=>4
[3,2,6,4,5,1]=>3
[3,2,6,5,1,4]=>4
[3,2,6,5,4,1]=>4
[3,4,1,2,5,6]=>4
[3,4,1,2,6,5]=>4
[3,4,1,5,2,6]=>3
[3,4,1,5,6,2]=>2
[3,4,1,6,2,5]=>3
[3,4,1,6,5,2]=>3
[3,4,2,1,5,6]=>4
[3,4,2,1,6,5]=>5
[3,4,2,5,1,6]=>3
[3,4,2,5,6,1]=>2
[3,4,2,6,1,5]=>4
[3,4,2,6,5,1]=>3
[3,4,5,1,2,6]=>2
[3,4,5,1,6,2]=>1
[3,4,5,2,1,6]=>3
[3,4,5,2,6,1]=>2
[3,4,5,6,1,2]=>0
[3,4,5,6,2,1]=>1
[3,4,6,1,2,5]=>2
[3,4,6,1,5,2]=>2
[3,4,6,2,1,5]=>3
[3,4,6,2,5,1]=>3
[3,4,6,5,1,2]=>1
[3,4,6,5,2,1]=>2
[3,5,1,2,4,6]=>3
[3,5,1,2,6,4]=>3
[3,5,1,4,2,6]=>4
[3,5,1,4,6,2]=>3
[3,5,1,6,2,4]=>2
[3,5,1,6,4,2]=>3
[3,5,2,1,4,6]=>4
[3,5,2,1,6,4]=>4
[3,5,2,4,1,6]=>4
[3,5,2,4,6,1]=>3
[3,5,2,6,1,4]=>3
[3,5,2,6,4,1]=>4
[3,5,4,1,2,6]=>3
[3,5,4,1,6,2]=>2
[3,5,4,2,1,6]=>4
[3,5,4,2,6,1]=>3
[3,5,4,6,1,2]=>1
[3,5,4,6,2,1]=>2
[3,5,6,1,2,4]=>1
[3,5,6,1,4,2]=>2
[3,5,6,2,1,4]=>2
[3,5,6,2,4,1]=>3
[3,5,6,4,1,2]=>2
[3,5,6,4,2,1]=>3
[3,6,1,2,4,5]=>2
[3,6,1,2,5,4]=>3
[3,6,1,4,2,5]=>3
[3,6,1,4,5,2]=>4
[3,6,1,5,2,4]=>3
[3,6,1,5,4,2]=>4
[3,6,2,1,4,5]=>3
[3,6,2,1,5,4]=>4
[3,6,2,4,1,5]=>4
[3,6,2,4,5,1]=>3
[3,6,2,5,1,4]=>4
[3,6,2,5,4,1]=>4
[3,6,4,1,2,5]=>3
[3,6,4,1,5,2]=>3
[3,6,4,2,1,5]=>4
[3,6,4,2,5,1]=>4
[3,6,4,5,1,2]=>2
[3,6,4,5,2,1]=>3
[3,6,5,1,2,4]=>2
[3,6,5,1,4,2]=>3
[3,6,5,2,1,4]=>3
[3,6,5,2,4,1]=>4
[3,6,5,4,1,2]=>3
[3,6,5,4,2,1]=>4
[4,1,2,3,5,6]=>2
[4,1,2,3,6,5]=>3
[4,1,2,5,3,6]=>3
[4,1,2,5,6,3]=>4
[4,1,2,6,3,5]=>3
[4,1,2,6,5,3]=>4
[4,1,3,2,5,6]=>3
[4,1,3,2,6,5]=>4
[4,1,3,5,2,6]=>4
[4,1,3,5,6,2]=>3
[4,1,3,6,2,5]=>4
[4,1,3,6,5,2]=>4
[4,1,5,2,3,6]=>3
[4,1,5,2,6,3]=>3
[4,1,5,3,2,6]=>4
[4,1,5,3,6,2]=>4
[4,1,5,6,2,3]=>2
[4,1,5,6,3,2]=>3
[4,1,6,2,3,5]=>2
[4,1,6,2,5,3]=>3
[4,1,6,3,2,5]=>3
[4,1,6,3,5,2]=>4
[4,1,6,5,2,3]=>3
[4,1,6,5,3,2]=>4
[4,2,1,3,5,6]=>3
[4,2,1,3,6,5]=>4
[4,2,1,5,3,6]=>4
[4,2,1,5,6,3]=>4
[4,2,1,6,3,5]=>4
[4,2,1,6,5,3]=>5
[4,2,3,1,5,6]=>4
[4,2,3,1,6,5]=>5
[4,2,3,5,1,6]=>3
[4,2,3,5,6,1]=>2
[4,2,3,6,1,5]=>4
[4,2,3,6,5,1]=>3
[4,2,5,1,3,6]=>4
[4,2,5,1,6,3]=>3
[4,2,5,3,1,6]=>4
[4,2,5,3,6,1]=>3
[4,2,5,6,1,3]=>2
[4,2,5,6,3,1]=>3
[4,2,6,1,3,5]=>3
[4,2,6,1,5,3]=>4
[4,2,6,3,1,5]=>4
[4,2,6,3,5,1]=>4
[4,2,6,5,1,3]=>3
[4,2,6,5,3,1]=>4
[4,3,1,2,5,6]=>4
[4,3,1,2,6,5]=>5
[4,3,1,5,2,6]=>4
[4,3,1,5,6,2]=>3
[4,3,1,6,2,5]=>4
[4,3,1,6,5,2]=>4
[4,3,2,1,5,6]=>5
[4,3,2,1,6,5]=>6
[4,3,2,5,1,6]=>4
[4,3,2,5,6,1]=>3
[4,3,2,6,1,5]=>5
[4,3,2,6,5,1]=>4
[4,3,5,1,2,6]=>3
[4,3,5,1,6,2]=>2
[4,3,5,2,1,6]=>4
[4,3,5,2,6,1]=>3
[4,3,5,6,1,2]=>1
[4,3,5,6,2,1]=>2
[4,3,6,1,2,5]=>3
[4,3,6,1,5,2]=>3
[4,3,6,2,1,5]=>4
[4,3,6,2,5,1]=>4
[4,3,6,5,1,2]=>2
[4,3,6,5,2,1]=>3
[4,5,1,2,3,6]=>2
[4,5,1,2,6,3]=>2
[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]=>2
[4,5,2,1,3,6]=>3
[4,5,2,1,6,3]=>3
[4,5,2,3,1,6]=>4
[4,5,2,3,6,1]=>4
[4,5,2,6,1,3]=>2
[4,5,2,6,3,1]=>3
[4,5,3,1,2,6]=>4
[4,5,3,1,6,2]=>3
[4,5,3,2,1,6]=>5
[4,5,3,2,6,1]=>4
[4,5,3,6,1,2]=>2
[4,5,3,6,2,1]=>3
[4,5,6,1,2,3]=>0
[4,5,6,1,3,2]=>1
[4,5,6,2,1,3]=>1
[4,5,6,2,3,1]=>2
[4,5,6,3,1,2]=>2
[4,5,6,3,2,1]=>3
[4,6,1,2,3,5]=>1
[4,6,1,2,5,3]=>2
[4,6,1,3,2,5]=>2
[4,6,1,3,5,2]=>3
[4,6,1,5,2,3]=>2
[4,6,1,5,3,2]=>3
[4,6,2,1,3,5]=>2
[4,6,2,1,5,3]=>3
[4,6,2,3,1,5]=>3
[4,6,2,3,5,1]=>3
[4,6,2,5,1,3]=>3
[4,6,2,5,3,1]=>4
[4,6,3,1,2,5]=>3
[4,6,3,1,5,2]=>4
[4,6,3,2,1,5]=>4
[4,6,3,2,5,1]=>4
[4,6,3,5,1,2]=>3
[4,6,3,5,2,1]=>4
[4,6,5,1,2,3]=>1
[4,6,5,1,3,2]=>2
[4,6,5,2,1,3]=>2
[4,6,5,2,3,1]=>3
[4,6,5,3,1,2]=>3
[4,6,5,3,2,1]=>4
[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]=>2
[5,1,2,6,4,3]=>3
[5,1,3,2,4,6]=>2
[5,1,3,2,6,4]=>3
[5,1,3,4,2,6]=>3
[5,1,3,4,6,2]=>3
[5,1,3,6,2,4]=>3
[5,1,3,6,4,2]=>4
[5,1,4,2,3,6]=>3
[5,1,4,2,6,3]=>4
[5,1,4,3,2,6]=>4
[5,1,4,3,6,2]=>4
[5,1,4,6,2,3]=>3
[5,1,4,6,3,2]=>4
[5,1,6,2,3,4]=>1
[5,1,6,2,4,3]=>2
[5,1,6,3,2,4]=>2
[5,1,6,3,4,2]=>3
[5,1,6,4,2,3]=>3
[5,1,6,4,3,2]=>4
[5,2,1,3,4,6]=>2
[5,2,1,3,6,4]=>3
[5,2,1,4,3,6]=>3
[5,2,1,4,6,3]=>4
[5,2,1,6,3,4]=>3
[5,2,1,6,4,3]=>4
[5,2,3,1,4,6]=>3
[5,2,3,1,6,4]=>4
[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]=>4
[5,2,4,1,6,3]=>4
[5,2,4,3,1,6]=>4
[5,2,4,3,6,1]=>3
[5,2,4,6,1,3]=>3
[5,2,4,6,3,1]=>4
[5,2,6,1,3,4]=>2
[5,2,6,1,4,3]=>3
[5,2,6,3,1,4]=>3
[5,2,6,3,4,1]=>3
[5,2,6,4,1,3]=>4
[5,2,6,4,3,1]=>4
[5,3,1,2,4,6]=>3
[5,3,1,2,6,4]=>4
[5,3,1,4,2,6]=>4
[5,3,1,4,6,2]=>4
[5,3,1,6,2,4]=>3
[5,3,1,6,4,2]=>4
[5,3,2,1,4,6]=>4
[5,3,2,1,6,4]=>5
[5,3,2,4,1,6]=>4
[5,3,2,4,6,1]=>3
[5,3,2,6,1,4]=>4
[5,3,2,6,4,1]=>4
[5,3,4,1,2,6]=>4
[5,3,4,1,6,2]=>3
[5,3,4,2,1,6]=>5
[5,3,4,2,6,1]=>4
[5,3,4,6,1,2]=>2
[5,3,4,6,2,1]=>3
[5,3,6,1,2,4]=>2
[5,3,6,1,4,2]=>3
[5,3,6,2,1,4]=>3
[5,3,6,2,4,1]=>4
[5,3,6,4,1,2]=>3
[5,3,6,4,2,1]=>4
[5,4,1,2,3,6]=>3
[5,4,1,2,6,3]=>3
[5,4,1,3,2,6]=>4
[5,4,1,3,6,2]=>4
[5,4,1,6,2,3]=>2
[5,4,1,6,3,2]=>3
[5,4,2,1,3,6]=>4
[5,4,2,1,6,3]=>4
[5,4,2,3,1,6]=>5
[5,4,2,3,6,1]=>4
[5,4,2,6,1,3]=>3
[5,4,2,6,3,1]=>4
[5,4,3,1,2,6]=>5
[5,4,3,1,6,2]=>4
[5,4,3,2,1,6]=>6
[5,4,3,2,6,1]=>5
[5,4,3,6,1,2]=>3
[5,4,3,6,2,1]=>4
[5,4,6,1,2,3]=>1
[5,4,6,1,3,2]=>2
[5,4,6,2,1,3]=>2
[5,4,6,2,3,1]=>3
[5,4,6,3,1,2]=>3
[5,4,6,3,2,1]=>4
[5,6,1,2,3,4]=>0
[5,6,1,2,4,3]=>1
[5,6,1,3,2,4]=>1
[5,6,1,3,4,2]=>2
[5,6,1,4,2,3]=>2
[5,6,1,4,3,2]=>3
[5,6,2,1,3,4]=>1
[5,6,2,1,4,3]=>2
[5,6,2,3,1,4]=>2
[5,6,2,3,4,1]=>2
[5,6,2,4,1,3]=>3
[5,6,2,4,3,1]=>3
[5,6,3,1,2,4]=>2
[5,6,3,1,4,2]=>3
[5,6,3,2,1,4]=>3
[5,6,3,2,4,1]=>3
[5,6,3,4,1,2]=>4
[5,6,3,4,2,1]=>4
[5,6,4,1,2,3]=>2
[5,6,4,1,3,2]=>3
[5,6,4,2,1,3]=>3
[5,6,4,2,3,1]=>4
[5,6,4,3,1,2]=>4
[5,6,4,3,2,1]=>5
[6,1,2,3,4,5]=>0
[6,1,2,3,5,4]=>1
[6,1,2,4,3,5]=>1
[6,1,2,4,5,3]=>2
[6,1,2,5,3,4]=>2
[6,1,2,5,4,3]=>3
[6,1,3,2,4,5]=>1
[6,1,3,2,5,4]=>2
[6,1,3,4,2,5]=>2
[6,1,3,4,5,2]=>2
[6,1,3,5,2,4]=>3
[6,1,3,5,4,2]=>3
[6,1,4,2,3,5]=>2
[6,1,4,2,5,3]=>3
[6,1,4,3,2,5]=>3
[6,1,4,3,5,2]=>3
[6,1,4,5,2,3]=>4
[6,1,4,5,3,2]=>4
[6,1,5,2,3,4]=>2
[6,1,5,2,4,3]=>3
[6,1,5,3,2,4]=>3
[6,1,5,3,4,2]=>4
[6,1,5,4,2,3]=>4
[6,1,5,4,3,2]=>5
[6,2,1,3,4,5]=>1
[6,2,1,3,5,4]=>2
[6,2,1,4,3,5]=>2
[6,2,1,4,5,3]=>3
[6,2,1,5,3,4]=>3
[6,2,1,5,4,3]=>4
[6,2,3,1,4,5]=>2
[6,2,3,1,5,4]=>3
[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]=>3
[6,2,4,1,5,3]=>4
[6,2,4,3,1,5]=>3
[6,2,4,3,5,1]=>2
[6,2,4,5,1,3]=>3
[6,2,4,5,3,1]=>3
[6,2,5,1,3,4]=>3
[6,2,5,1,4,3]=>4
[6,2,5,3,1,4]=>4
[6,2,5,3,4,1]=>3
[6,2,5,4,1,3]=>4
[6,2,5,4,3,1]=>4
[6,3,1,2,4,5]=>2
[6,3,1,2,5,4]=>3
[6,3,1,4,2,5]=>3
[6,3,1,4,5,2]=>3
[6,3,1,5,2,4]=>4
[6,3,1,5,4,2]=>4
[6,3,2,1,4,5]=>3
[6,3,2,1,5,4]=>4
[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]=>4
[6,3,4,1,5,2]=>3
[6,3,4,2,1,5]=>4
[6,3,4,2,5,1]=>3
[6,3,4,5,1,2]=>2
[6,3,4,5,2,1]=>3
[6,3,5,1,2,4]=>3
[6,3,5,1,4,2]=>4
[6,3,5,2,1,4]=>4
[6,3,5,2,4,1]=>4
[6,3,5,4,1,2]=>3
[6,3,5,4,2,1]=>4
[6,4,1,2,3,5]=>2
[6,4,1,2,5,3]=>3
[6,4,1,3,2,5]=>3
[6,4,1,3,5,2]=>4
[6,4,1,5,2,3]=>3
[6,4,1,5,3,2]=>4
[6,4,2,1,3,5]=>3
[6,4,2,1,5,3]=>4
[6,4,2,3,1,5]=>4
[6,4,2,3,5,1]=>3
[6,4,2,5,1,3]=>4
[6,4,2,5,3,1]=>4
[6,4,3,1,2,5]=>4
[6,4,3,1,5,2]=>4
[6,4,3,2,1,5]=>5
[6,4,3,2,5,1]=>4
[6,4,3,5,1,2]=>3
[6,4,3,5,2,1]=>4
[6,4,5,1,2,3]=>2
[6,4,5,1,3,2]=>3
[6,4,5,2,1,3]=>3
[6,4,5,2,3,1]=>4
[6,4,5,3,1,2]=>4
[6,4,5,3,2,1]=>5
[6,5,1,2,3,4]=>1
[6,5,1,2,4,3]=>2
[6,5,1,3,2,4]=>2
[6,5,1,3,4,2]=>3
[6,5,1,4,2,3]=>3
[6,5,1,4,3,2]=>4
[6,5,2,1,3,4]=>2
[6,5,2,1,4,3]=>3
[6,5,2,3,1,4]=>3
[6,5,2,3,4,1]=>3
[6,5,2,4,1,3]=>4
[6,5,2,4,3,1]=>4
[6,5,3,1,2,4]=>3
[6,5,3,1,4,2]=>4
[6,5,3,2,1,4]=>4
[6,5,3,2,4,1]=>4
[6,5,3,4,1,2]=>4
[6,5,3,4,2,1]=>5
[6,5,4,1,2,3]=>3
[6,5,4,1,3,2]=>4
[6,5,4,2,1,3]=>4
[6,5,4,2,3,1]=>5
[6,5,4,3,1,2]=>5
[6,5,4,3,2,1]=>6
[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]=>2
[1,2,4,5,7,3,6]=>4
[1,2,4,5,7,6,3]=>3
[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]=>3
[1,2,4,6,7,3,5]=>5
[1,2,4,6,7,5,3]=>4
[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]=>4
[1,2,4,7,6,3,5]=>6
[1,2,4,7,6,5,3]=>5
[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]=>3
[1,2,5,4,7,3,6]=>5
[1,2,5,4,7,6,3]=>4
[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]=>4
[1,2,5,6,7,3,4]=>4
[1,2,5,6,7,4,3]=>5
[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]=>5
[1,2,5,7,6,3,4]=>5
[1,2,5,7,6,4,3]=>6
[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]=>4
[1,2,6,4,7,3,5]=>6
[1,2,6,4,7,5,3]=>5
[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]=>5
[1,2,6,5,7,3,4]=>5
[1,2,6,5,7,4,3]=>6
[1,2,6,7,3,4,5]=>4
[1,2,6,7,3,5,4]=>5
[1,2,6,7,4,3,5]=>5
[1,2,6,7,4,5,3]=>6
[1,2,6,7,5,3,4]=>6
[1,2,6,7,5,4,3]=>7
[1,2,7,3,4,5,6]=>2
[1,2,7,3,4,6,5]=>3
[1,2,7,3,5,4,6]=>3
[1,2,7,3,5,6,4]=>4
[1,2,7,3,6,4,5]=>4
[1,2,7,3,6,5,4]=>5
[1,2,7,4,3,5,6]=>3
[1,2,7,4,3,6,5]=>4
[1,2,7,4,5,3,6]=>4
[1,2,7,4,5,6,3]=>5
[1,2,7,4,6,3,5]=>5
[1,2,7,4,6,5,3]=>6
[1,2,7,5,3,4,6]=>4
[1,2,7,5,3,6,4]=>5
[1,2,7,5,4,3,6]=>5
[1,2,7,5,4,6,3]=>6
[1,2,7,5,6,3,4]=>6
[1,2,7,5,6,4,3]=>7
[1,2,7,6,3,4,5]=>5
[1,2,7,6,3,5,4]=>6
[1,2,7,6,4,3,5]=>6
[1,2,7,6,4,5,3]=>7
[1,2,7,6,5,3,4]=>7
[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]=>2
[1,3,4,5,6,7,2]=>1
[1,3,4,5,7,2,6]=>3
[1,3,4,5,7,6,2]=>2
[1,3,4,6,2,5,7]=>4
[1,3,4,6,2,7,5]=>5
[1,3,4,6,5,2,7]=>3
[1,3,4,6,5,7,2]=>2
[1,3,4,6,7,2,5]=>4
[1,3,4,6,7,5,2]=>3
[1,3,4,7,2,5,6]=>5
[1,3,4,7,2,6,5]=>6
[1,3,4,7,5,2,6]=>4
[1,3,4,7,5,6,2]=>3
[1,3,4,7,6,2,5]=>5
[1,3,4,7,6,5,2]=>4
[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]=>3
[1,3,5,4,6,7,2]=>2
[1,3,5,4,7,2,6]=>4
[1,3,5,4,7,6,2]=>3
[1,3,5,6,2,4,7]=>5
[1,3,5,6,2,7,4]=>4
[1,3,5,6,4,2,7]=>4
[1,3,5,6,4,7,2]=>3
[1,3,5,6,7,2,4]=>3
[1,3,5,6,7,4,2]=>4
[1,3,5,7,2,4,6]=>6
[1,3,5,7,2,6,4]=>5
[1,3,5,7,4,2,6]=>5
[1,3,5,7,4,6,2]=>4
[1,3,5,7,6,2,4]=>4
[1,3,5,7,6,4,2]=>5
[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]=>4
[1,3,6,4,5,7,2]=>3
[1,3,6,4,7,2,5]=>5
[1,3,6,4,7,5,2]=>4
[1,3,6,5,2,4,7]=>6
[1,3,6,5,2,7,4]=>5
[1,3,6,5,4,2,7]=>5
[1,3,6,5,4,7,2]=>4
[1,3,6,5,7,2,4]=>4
[1,3,6,5,7,4,2]=>5
[1,3,6,7,2,4,5]=>5
[1,3,6,7,2,5,4]=>6
[1,3,6,7,4,2,5]=>6
[1,3,6,7,4,5,2]=>5
[1,3,6,7,5,2,4]=>5
[1,3,6,7,5,4,2]=>6
[1,3,7,2,4,5,6]=>3
[1,3,7,2,4,6,5]=>4
[1,3,7,2,5,4,6]=>4
[1,3,7,2,5,6,4]=>5
[1,3,7,2,6,4,5]=>5
[1,3,7,2,6,5,4]=>6
[1,3,7,4,2,5,6]=>4
[1,3,7,4,2,6,5]=>5
[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]=>5
[1,3,7,5,2,6,4]=>6
[1,3,7,5,4,2,6]=>6
[1,3,7,5,4,6,2]=>5
[1,3,7,5,6,2,4]=>5
[1,3,7,5,6,4,2]=>6
[1,3,7,6,2,4,5]=>6
[1,3,7,6,2,5,4]=>7
[1,3,7,6,4,2,5]=>7
[1,3,7,6,4,5,2]=>6
[1,3,7,6,5,2,4]=>6
[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]=>3
[1,4,2,5,7,3,6]=>5
[1,4,2,5,7,6,3]=>4
[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]=>4
[1,4,2,6,7,3,5]=>6
[1,4,2,6,7,5,3]=>5
[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]=>5
[1,4,2,7,6,3,5]=>7
[1,4,2,7,6,5,3]=>6
[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]=>3
[1,4,3,5,6,7,2]=>2
[1,4,3,5,7,2,6]=>4
[1,4,3,5,7,6,2]=>3
[1,4,3,6,2,5,7]=>5
[1,4,3,6,2,7,5]=>6
[1,4,3,6,5,2,7]=>4
[1,4,3,6,5,7,2]=>3
[1,4,3,6,7,2,5]=>5
[1,4,3,6,7,5,2]=>4
[1,4,3,7,2,5,6]=>6
[1,4,3,7,2,6,5]=>7
[1,4,3,7,5,2,6]=>5
[1,4,3,7,5,6,2]=>4
[1,4,3,7,6,2,5]=>6
[1,4,3,7,6,5,2]=>5
[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]=>4
[1,4,5,2,7,3,6]=>6
[1,4,5,2,7,6,3]=>5
[1,4,5,3,2,6,7]=>5
[1,4,5,3,2,7,6]=>6
[1,4,5,3,6,2,7]=>4
[1,4,5,3,6,7,2]=>3
[1,4,5,3,7,2,6]=>5
[1,4,5,3,7,6,2]=>4
[1,4,5,6,2,3,7]=>4
[1,4,5,6,2,7,3]=>3
[1,4,5,6,3,2,7]=>5
[1,4,5,6,3,7,2]=>4
[1,4,5,6,7,2,3]=>2
[1,4,5,6,7,3,2]=>3
[1,4,5,7,2,3,6]=>5
[1,4,5,7,2,6,3]=>4
[1,4,5,7,3,2,6]=>6
[1,4,5,7,3,6,2]=>5
[1,4,5,7,6,2,3]=>3
[1,4,5,7,6,3,2]=>4
[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]=>5
[1,4,6,2,7,3,5]=>5
[1,4,6,2,7,5,3]=>6
[1,4,6,3,2,5,7]=>6
[1,4,6,3,2,7,5]=>7
[1,4,6,3,5,2,7]=>5
[1,4,6,3,5,7,2]=>4
[1,4,6,3,7,2,5]=>6
[1,4,6,3,7,5,2]=>5
[1,4,6,5,2,3,7]=>5
[1,4,6,5,2,7,3]=>4
[1,4,6,5,3,2,7]=>6
[1,4,6,5,3,7,2]=>5
                    

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Description

The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,...,n).
Let $\rho=(1,\dots,n)$ and $\sigma=(1,2)$. Then, for a permutation $\pi\in\mathfrak S_n$, this statistic is
$$\min\{k\mid \pi=\rho^{i_0}\sigma\rho^{i_1}\sigma\dots\rho^{i_{k-1}}\sigma\rho^{i_k}, 0\leq i_0,\dots,i_k < n\}.$$
Put differently, it is the minimal length of a factorization into cyclic shifts of the transposition $(1,2)$ (see St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).) of any of the permutations $\rho^k\pi$ for $0\leq k < n$.

References

[1] Ryazanov, A. Generating $S_n$ with a fundamental transposition and a big cycle MathOverflow:290208

Code

@cached_function
def minimal_length(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()

def statistic(pi):
    n = len(pi)
    b = Permutation([i for i in range(2,n+1)] + [1])
    powers = [Permutation([])]
    for k in range(n-1):
        powers.append(powers[-1]*b)    
    return min([minimal_length(p*pi) for p in powers])

Created

Jan 08, 2018 at 19:49 by Martin Rubey

Updated

Feb 09, 2021 at 15:28 by Martin Rubey