- FindStat

edit this statistic or download as text // json

Identifier

St001402: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

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

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Description

The number of separators in a permutation.
Let $\sigma$ be a permutation. Then $\sigma_i$ is a vertical separator if $|\sigma_{i-1} - \sigma_{i+1}| = 1$, and $i$ is a horizontal separator, if it is a vertical separator of $\sigma^{-1}$. The set of separators is the union of the set of the vertical and the set of the horizontal separators.

References

[1] Bagno, E., Eisenberg, E., Reches, S., Sigron, M. Separators - a new statistic for permutations arXiv:1905.12364

Code

def statistic(pi):
    """
    sage: oeis([sum(1 for pi in Permutations(n) if not statistic(pi)) for n in range(1,7)])
    0: A137774: Number of ways to place n nonattacking empresses on an n X n board.

    """
    return len(separators(pi))

def vertical_separators(sigma):
    """
    sage: list(vertical_separators(sigma))
    [3, 2, 6, 7]
    """
    n = len(sigma)
    for i in range(n):
        if 0 < i < n-1 and abs(sigma[i-1]-sigma[i+1]) == 1:
            yield sigma[i]

def horizontal_separators(sigma):
    """
    sage: list(horizontal_separators(sigma))
    [2, 3, 5, 8]
    """
    sigma_inv = sigma.inverse()
    for i in vertical_separators(sigma_inv):
        yield sigma_inv(i)
            
def separators(sigma):
    """
    sage: separators(sigma)
    [2, 3, 5, 6, 7, 8]
    """
    return list(set(horizontal_separators(sigma)).union(set(vertical_separators(sigma))))

Created

May 30, 2019 at 14:29 by Martin Rubey

Updated

May 30, 2019 at 14:29 by Martin Rubey