- FindStat

edit this statistic or download as text // json

Identifier

St001077: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1,2]=>0
[2,1]=>1
[1,2,3]=>0
[1,3,2]=>3
[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]=>3
[1,3,2,4]=>3
[1,3,4,2]=>4
[1,4,2,3]=>4
[1,4,3,2]=>3
[2,1,3,4]=>1
[2,1,4,3]=>4
[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]=>1
[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]=>4
[1,2,3,4,5]=>0
[1,2,3,5,4]=>3
[1,2,4,3,5]=>3
[1,2,4,5,3]=>4
[1,2,5,3,4]=>4
[1,2,5,4,3]=>3
[1,3,2,4,5]=>3
[1,3,2,5,4]=>6
[1,3,4,2,5]=>4
[1,3,4,5,2]=>5
[1,3,5,2,4]=>5
[1,3,5,4,2]=>4
[1,4,2,3,5]=>4
[1,4,2,5,3]=>5
[1,4,3,2,5]=>3
[1,4,3,5,2]=>4
[1,4,5,2,3]=>6
[1,4,5,3,2]=>5
[1,5,2,3,4]=>5
[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]=>6
[2,1,3,4,5]=>1
[2,1,3,5,4]=>4
[2,1,4,3,5]=>4
[2,1,4,5,3]=>5
[2,1,5,3,4]=>5
[2,1,5,4,3]=>4
[2,3,1,4,5]=>2
[2,3,1,5,4]=>5
[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]=>2
[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]=>3
[2,5,3,1,4]=>3
[2,5,3,4,1]=>2
[2,5,4,1,3]=>4
[2,5,4,3,1]=>5
[3,1,2,4,5]=>2
[3,1,2,5,4]=>5
[3,1,4,2,5]=>3
[3,1,4,5,2]=>4
[3,1,5,2,4]=>4
[3,1,5,4,2]=>3
[3,2,1,4,5]=>1
[3,2,1,5,4]=>4
[3,2,4,1,5]=>2
[3,2,4,5,1]=>3
[3,2,5,1,4]=>3
[3,2,5,4,1]=>2
[3,4,1,2,5]=>4
[3,4,1,5,2]=>5
[3,4,2,1,5]=>3
[3,4,2,5,1]=>4
[3,4,5,1,2]=>4
[3,4,5,2,1]=>5
[3,5,1,2,4]=>5
[3,5,1,4,2]=>4
[3,5,2,1,4]=>4
[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]=>2
[4,1,3,5,2]=>3
[4,1,5,2,3]=>5
[4,1,5,3,2]=>4
[4,2,1,3,5]=>2
[4,2,1,5,3]=>3
[4,2,3,1,5]=>1
[4,2,3,5,1]=>2
[4,2,5,1,3]=>4
[4,2,5,3,1]=>3
[4,3,1,2,5]=>3
[4,3,1,5,2]=>4
[4,3,2,1,5]=>4
[4,3,2,5,1]=>5
[4,3,5,1,2]=>5
[4,3,5,2,1]=>4
[4,5,1,2,3]=>4
[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]=>3
[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]=>5
[5,2,1,3,4]=>3
[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]=>4
[5,3,1,2,4]=>4
[5,3,1,4,2]=>3
[5,3,2,1,4]=>5
[5,3,2,4,1]=>4
[5,3,4,1,2]=>4
[5,3,4,2,1]=>5
[5,4,1,2,3]=>5
[5,4,1,3,2]=>4
[5,4,2,1,3]=>4
[5,4,2,3,1]=>5
[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]=>3
[1,2,3,5,4,6]=>3
[1,2,3,5,6,4]=>4
[1,2,3,6,4,5]=>4
[1,2,3,6,5,4]=>3
[1,2,4,3,5,6]=>3
[1,2,4,3,6,5]=>6
[1,2,4,5,3,6]=>4
[1,2,4,5,6,3]=>5
[1,2,4,6,3,5]=>5
[1,2,4,6,5,3]=>4
[1,2,5,3,4,6]=>4
[1,2,5,3,6,4]=>5
[1,2,5,4,3,6]=>3
[1,2,5,4,6,3]=>4
[1,2,5,6,3,4]=>6
[1,2,5,6,4,3]=>5
[1,2,6,3,4,5]=>5
[1,2,6,3,5,4]=>4
[1,2,6,4,3,5]=>4
[1,2,6,4,5,3]=>3
[1,2,6,5,3,4]=>5
[1,2,6,5,4,3]=>6
[1,3,2,4,5,6]=>3
[1,3,2,4,6,5]=>6
[1,3,2,5,4,6]=>6
[1,3,2,5,6,4]=>7
[1,3,2,6,4,5]=>7
[1,3,2,6,5,4]=>6
[1,3,4,2,5,6]=>4
[1,3,4,2,6,5]=>7
[1,3,4,5,2,6]=>5
[1,3,4,5,6,2]=>6
[1,3,4,6,2,5]=>6
[1,3,4,6,5,2]=>5
[1,3,5,2,4,6]=>5
[1,3,5,2,6,4]=>6
[1,3,5,4,2,6]=>4
[1,3,5,4,6,2]=>5
[1,3,5,6,2,4]=>7
[1,3,5,6,4,2]=>6
[1,3,6,2,4,5]=>6
[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]=>7
[1,4,2,3,5,6]=>4
[1,4,2,3,6,5]=>7
[1,4,2,5,3,6]=>5
[1,4,2,5,6,3]=>6
[1,4,2,6,3,5]=>6
[1,4,2,6,5,3]=>5
[1,4,3,2,5,6]=>3
[1,4,3,2,6,5]=>6
[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]=>4
[1,4,5,2,3,6]=>6
[1,4,5,2,6,3]=>7
[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]=>7
[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]=>5
[1,5,2,3,6,4]=>6
[1,5,2,4,3,6]=>4
[1,5,2,4,6,3]=>5
[1,5,2,6,3,4]=>7
[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]=>3
[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]=>7
[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]=>6
[1,5,6,4,3,2]=>5
[1,6,2,3,4,5]=>6
[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]=>7
[1,6,3,2,4,5]=>5
[1,6,3,2,5,4]=>4
[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]=>6
[1,6,4,2,3,5]=>6
[1,6,4,2,5,3]=>5
[1,6,4,3,2,5]=>7
[1,6,4,3,5,2]=>6
[1,6,4,5,2,3]=>6
[1,6,4,5,3,2]=>7
[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]=>7
[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]=>4
[2,1,3,5,4,6]=>4
[2,1,3,5,6,4]=>5
[2,1,3,6,4,5]=>5
[2,1,3,6,5,4]=>4
[2,1,4,3,5,6]=>4
[2,1,4,3,6,5]=>7
[2,1,4,5,3,6]=>5
[2,1,4,5,6,3]=>6
[2,1,4,6,3,5]=>6
[2,1,4,6,5,3]=>5
[2,1,5,3,4,6]=>5
[2,1,5,3,6,4]=>6
[2,1,5,4,3,6]=>4
[2,1,5,4,6,3]=>5
[2,1,5,6,3,4]=>7
[2,1,5,6,4,3]=>6
[2,1,6,3,4,5]=>6
[2,1,6,3,5,4]=>5
[2,1,6,4,3,5]=>5
[2,1,6,4,5,3]=>4
[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]=>5
[2,3,1,5,4,6]=>5
[2,3,1,5,6,4]=>6
[2,3,1,6,4,5]=>6
[2,3,1,6,5,4]=>5
[2,3,4,1,5,6]=>3
[2,3,4,1,6,5]=>6
[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]=>3
[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]=>4
[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]=>6
[2,4,1,3,5,6]=>3
[2,4,1,3,6,5]=>6
[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]=>4
[2,4,3,1,5,6]=>2
[2,4,3,1,6,5]=>5
[2,4,3,5,1,6]=>3
[2,4,3,5,6,1]=>4
[2,4,3,6,1,5]=>4
[2,4,3,6,5,1]=>3
[2,4,5,1,3,6]=>5
[2,4,5,1,6,3]=>6
[2,4,5,3,1,6]=>4
[2,4,5,3,6,1]=>5
[2,4,5,6,1,3]=>5
[2,4,5,6,3,1]=>6
[2,4,6,1,3,5]=>6
[2,4,6,1,5,3]=>5
[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]=>3
[2,5,1,4,6,3]=>4
[2,5,1,6,3,4]=>6
[2,5,1,6,4,3]=>5
[2,5,3,1,4,6]=>3
[2,5,3,1,6,4]=>4
[2,5,3,4,1,6]=>2
[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]=>4
[2,5,4,1,6,3]=>5
[2,5,4,3,1,6]=>5
[2,5,4,3,6,1]=>6
[2,5,4,6,1,3]=>6
[2,5,4,6,3,1]=>5
[2,5,6,1,3,4]=>5
[2,5,6,1,4,3]=>6
[2,5,6,3,1,4]=>6
[2,5,6,3,4,1]=>5
[2,5,6,4,1,3]=>5
[2,5,6,4,3,1]=>4
[2,6,1,3,4,5]=>5
[2,6,1,3,5,4]=>4
[2,6,1,4,3,5]=>4
[2,6,1,4,5,3]=>3
[2,6,1,5,3,4]=>5
[2,6,1,5,4,3]=>6
[2,6,3,1,4,5]=>4
[2,6,3,1,5,4]=>3
[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]=>5
[2,6,4,1,3,5]=>5
[2,6,4,1,5,3]=>4
[2,6,4,3,1,5]=>6
[2,6,4,3,5,1]=>5
[2,6,4,5,1,3]=>5
[2,6,4,5,3,1]=>6
[2,6,5,1,3,4]=>6
[2,6,5,1,4,3]=>5
[2,6,5,3,1,4]=>5
[2,6,5,3,4,1]=>6
[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]=>5
[3,1,2,5,4,6]=>5
[3,1,2,5,6,4]=>6
[3,1,2,6,4,5]=>6
[3,1,2,6,5,4]=>5
[3,1,4,2,5,6]=>3
[3,1,4,2,6,5]=>6
[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]=>4
[3,1,5,2,4,6]=>4
[3,1,5,2,6,4]=>5
[3,1,5,4,2,6]=>3
[3,1,5,4,6,2]=>4
[3,1,5,6,2,4]=>6
[3,1,5,6,4,2]=>5
[3,1,6,2,4,5]=>5
[3,1,6,2,5,4]=>4
[3,1,6,4,2,5]=>4
[3,1,6,4,5,2]=>3
[3,1,6,5,2,4]=>5
[3,1,6,5,4,2]=>6
[3,2,1,4,5,6]=>1
[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]=>4
[3,2,4,1,5,6]=>2
[3,2,4,1,6,5]=>5
[3,2,4,5,1,6]=>3
[3,2,4,5,6,1]=>4
[3,2,4,6,1,5]=>4
[3,2,4,6,5,1]=>3
[3,2,5,1,4,6]=>3
[3,2,5,1,6,4]=>4
[3,2,5,4,1,6]=>2
[3,2,5,4,6,1]=>3
[3,2,5,6,1,4]=>5
[3,2,5,6,4,1]=>4
[3,2,6,1,4,5]=>4
[3,2,6,1,5,4]=>3
[3,2,6,4,1,5]=>3
[3,2,6,4,5,1]=>2
[3,2,6,5,1,4]=>4
[3,2,6,5,4,1]=>5
[3,4,1,2,5,6]=>4
[3,4,1,2,6,5]=>7
[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]=>5
[3,4,2,1,5,6]=>3
[3,4,2,1,6,5]=>6
[3,4,2,5,1,6]=>4
[3,4,2,5,6,1]=>5
[3,4,2,6,1,5]=>5
[3,4,2,6,5,1]=>4
[3,4,5,1,2,6]=>4
[3,4,5,1,6,2]=>5
[3,4,5,2,1,6]=>5
[3,4,5,2,6,1]=>6
[3,4,5,6,1,2]=>6
[3,4,5,6,2,1]=>5
[3,4,6,1,2,5]=>5
[3,4,6,1,5,2]=>4
[3,4,6,2,1,5]=>6
[3,4,6,2,5,1]=>5
[3,4,6,5,1,2]=>5
[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]=>4
[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]=>4
[3,5,2,1,6,4]=>5
[3,5,2,4,1,6]=>3
[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]=>5
[3,5,4,1,6,2]=>6
[3,5,4,2,1,6]=>4
[3,5,4,2,6,1]=>5
[3,5,4,6,1,2]=>5
[3,5,4,6,2,1]=>6
[3,5,6,1,2,4]=>6
[3,5,6,1,4,2]=>5
[3,5,6,2,1,4]=>5
[3,5,6,2,4,1]=>6
[3,5,6,4,1,2]=>4
[3,5,6,4,2,1]=>5
[3,6,1,2,4,5]=>6
[3,6,1,2,5,4]=>5
[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]=>7
[3,6,2,1,4,5]=>5
[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]=>5
[3,6,2,5,4,1]=>6
[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]=>5
[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]=>5
[3,6,5,4,2,1]=>4
[4,1,2,3,5,6]=>3
[4,1,2,3,6,5]=>6
[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]=>4
[4,1,3,2,5,6]=>2
[4,1,3,2,6,5]=>5
[4,1,3,5,2,6]=>3
[4,1,3,5,6,2]=>4
[4,1,3,6,2,5]=>4
[4,1,3,6,5,2]=>3
[4,1,5,2,3,6]=>5
[4,1,5,2,6,3]=>6
[4,1,5,3,2,6]=>4
[4,1,5,3,6,2]=>5
[4,1,5,6,2,3]=>5
[4,1,5,6,3,2]=>6
[4,1,6,2,3,5]=>6
[4,1,6,2,5,3]=>5
[4,1,6,3,2,5]=>5
[4,1,6,3,5,2]=>4
[4,1,6,5,2,3]=>6
[4,1,6,5,3,2]=>5
[4,2,1,3,5,6]=>2
[4,2,1,3,6,5]=>5
[4,2,1,5,3,6]=>3
[4,2,1,5,6,3]=>4
[4,2,1,6,3,5]=>4
[4,2,1,6,5,3]=>3
[4,2,3,1,5,6]=>1
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>2
[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]=>4
[4,2,5,1,6,3]=>5
[4,2,5,3,1,6]=>3
[4,2,5,3,6,1]=>4
[4,2,5,6,1,3]=>4
[4,2,5,6,3,1]=>5
[4,2,6,1,3,5]=>5
[4,2,6,1,5,3]=>4
[4,2,6,3,1,5]=>4
[4,2,6,3,5,1]=>3
[4,2,6,5,1,3]=>5
[4,2,6,5,3,1]=>4
[4,3,1,2,5,6]=>3
[4,3,1,2,6,5]=>6
[4,3,1,5,2,6]=>4
[4,3,1,5,6,2]=>5
[4,3,1,6,2,5]=>5
[4,3,1,6,5,2]=>4
[4,3,2,1,5,6]=>4
[4,3,2,1,6,5]=>7
[4,3,2,5,1,6]=>5
[4,3,2,5,6,1]=>6
[4,3,2,6,1,5]=>6
[4,3,2,6,5,1]=>5
[4,3,5,1,2,6]=>5
[4,3,5,1,6,2]=>6
[4,3,5,2,1,6]=>4
[4,3,5,2,6,1]=>5
[4,3,5,6,1,2]=>5
[4,3,5,6,2,1]=>6
[4,3,6,1,2,5]=>6
[4,3,6,1,5,2]=>5
[4,3,6,2,1,5]=>5
[4,3,6,2,5,1]=>4
[4,3,6,5,1,2]=>6
[4,3,6,5,2,1]=>5
[4,5,1,2,3,6]=>4
[4,5,1,2,6,3]=>5
[4,5,1,3,2,6]=>5
[4,5,1,3,6,2]=>6
[4,5,1,6,2,3]=>6
[4,5,1,6,3,2]=>5
[4,5,2,1,3,6]=>5
[4,5,2,1,6,3]=>6
[4,5,2,3,1,6]=>4
[4,5,2,3,6,1]=>5
[4,5,2,6,1,3]=>5
[4,5,2,6,3,1]=>6
[4,5,3,1,2,6]=>4
[4,5,3,1,6,2]=>5
[4,5,3,2,1,6]=>3
[4,5,3,2,6,1]=>4
[4,5,3,6,1,2]=>4
[4,5,3,6,2,1]=>5
[4,5,6,1,2,3]=>7
[4,5,6,1,3,2]=>6
[4,5,6,2,1,3]=>6
[4,5,6,2,3,1]=>5
[4,5,6,3,1,2]=>5
[4,5,6,3,2,1]=>6
[4,6,1,2,3,5]=>5
[4,6,1,2,5,3]=>4
[4,6,1,3,2,5]=>6
[4,6,1,3,5,2]=>5
[4,6,1,5,2,3]=>5
[4,6,1,5,3,2]=>6
[4,6,2,1,3,5]=>6
[4,6,2,1,5,3]=>5
[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]=>5
[4,6,3,1,5,2]=>4
[4,6,3,2,1,5]=>4
[4,6,3,2,5,1]=>3
[4,6,3,5,1,2]=>5
[4,6,3,5,2,1]=>4
[4,6,5,1,2,3]=>6
[4,6,5,1,3,2]=>7
[4,6,5,2,1,3]=>5
[4,6,5,2,3,1]=>6
[4,6,5,3,1,2]=>6
[4,6,5,3,2,1]=>5
[5,1,2,3,4,6]=>4
[5,1,2,3,6,4]=>5
[5,1,2,4,3,6]=>3
[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]=>3
[5,1,3,2,6,4]=>4
[5,1,3,4,2,6]=>2
[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]=>4
[5,1,4,2,6,3]=>5
[5,1,4,3,2,6]=>5
[5,1,4,3,6,2]=>6
[5,1,4,6,2,3]=>6
[5,1,4,6,3,2]=>5
[5,1,6,2,3,4]=>5
[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]=>5
[5,1,6,4,3,2]=>4
[5,2,1,3,4,6]=>3
[5,2,1,3,6,4]=>4
[5,2,1,4,3,6]=>2
[5,2,1,4,6,3]=>3
[5,2,1,6,3,4]=>5
[5,2,1,6,4,3]=>4
[5,2,3,1,4,6]=>2
[5,2,3,1,6,4]=>3
[5,2,3,4,1,6]=>1
[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]=>3
[5,2,4,1,6,3]=>4
[5,2,4,3,1,6]=>4
[5,2,4,3,6,1]=>5
[5,2,4,6,1,3]=>5
[5,2,4,6,3,1]=>4
[5,2,6,1,3,4]=>4
[5,2,6,1,4,3]=>5
[5,2,6,3,1,4]=>5
[5,2,6,3,4,1]=>4
[5,2,6,4,1,3]=>4
[5,2,6,4,3,1]=>3
[5,3,1,2,4,6]=>4
[5,3,1,2,6,4]=>5
[5,3,1,4,2,6]=>3
[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]=>5
[5,3,2,6,1,4]=>7
[5,3,2,6,4,1]=>6
[5,3,4,1,2,6]=>4
[5,3,4,1,6,2]=>5
[5,3,4,2,1,6]=>5
[5,3,4,2,6,1]=>6
[5,3,4,6,1,2]=>6
[5,3,4,6,2,1]=>5
[5,3,6,1,2,4]=>5
[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]=>4
[5,4,1,2,3,6]=>5
[5,4,1,2,6,3]=>6
[5,4,1,3,2,6]=>4
[5,4,1,3,6,2]=>5
[5,4,1,6,2,3]=>5
[5,4,1,6,3,2]=>6
[5,4,2,1,3,6]=>4
[5,4,2,1,6,3]=>5
[5,4,2,3,1,6]=>5
[5,4,2,3,6,1]=>6
[5,4,2,6,1,3]=>6
[5,4,2,6,3,1]=>5
[5,4,3,1,2,6]=>3
[5,4,3,1,6,2]=>4
[5,4,3,2,1,6]=>4
[5,4,3,2,6,1]=>5
[5,4,3,6,1,2]=>5
[5,4,3,6,2,1]=>4
[5,4,6,1,2,3]=>6
[5,4,6,1,3,2]=>5
[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]=>5
[5,6,1,2,3,4]=>6
[5,6,1,2,4,3]=>5
[5,6,1,3,2,4]=>5
[5,6,1,3,4,2]=>6
[5,6,1,4,2,3]=>4
[5,6,1,4,3,2]=>5
[5,6,2,1,3,4]=>5
[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]=>4
[5,6,3,1,2,4]=>4
[5,6,3,1,4,2]=>5
[5,6,3,2,1,4]=>5
[5,6,3,2,4,1]=>4
[5,6,3,4,1,2]=>4
[5,6,3,4,2,1]=>3
[5,6,4,1,2,3]=>5
[5,6,4,1,3,2]=>6
[5,6,4,2,1,3]=>6
[5,6,4,2,3,1]=>5
[5,6,4,3,1,2]=>7
[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]=>6
[6,1,3,2,4,5]=>4
[6,1,3,2,5,4]=>3
[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]=>5
[6,1,4,2,3,5]=>5
[6,1,4,2,5,3]=>4
[6,1,4,3,2,5]=>6
[6,1,4,3,5,2]=>5
[6,1,4,5,2,3]=>5
[6,1,4,5,3,2]=>6
[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]=>6
[6,1,5,4,2,3]=>4
[6,1,5,4,3,2]=>5
[6,2,1,3,4,5]=>4
[6,2,1,3,5,4]=>3
[6,2,1,4,3,5]=>3
[6,2,1,4,5,3]=>2
[6,2,1,5,3,4]=>4
[6,2,1,5,4,3]=>5
[6,2,3,1,4,5]=>3
[6,2,3,1,5,4]=>2
[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]=>4
[6,2,4,1,3,5]=>4
[6,2,4,1,5,3]=>3
[6,2,4,3,1,5]=>5
[6,2,4,3,5,1]=>4
[6,2,4,5,1,3]=>4
[6,2,4,5,3,1]=>5
[6,2,5,1,3,4]=>5
[6,2,5,1,4,3]=>4
[6,2,5,3,1,4]=>4
[6,2,5,3,4,1]=>5
[6,2,5,4,1,3]=>3
[6,2,5,4,3,1]=>4
[6,3,1,2,4,5]=>5
[6,3,1,2,5,4]=>4
[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]=>6
[6,3,2,1,4,5]=>6
[6,3,2,1,5,4]=>5
[6,3,2,4,1,5]=>5
[6,3,2,4,5,1]=>4
[6,3,2,5,1,4]=>6
[6,3,2,5,4,1]=>7
[6,3,4,1,2,5]=>5
[6,3,4,1,5,2]=>4
[6,3,4,2,1,5]=>6
[6,3,4,2,5,1]=>5
[6,3,4,5,1,2]=>5
[6,3,4,5,2,1]=>6
[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]=>6
[6,3,5,4,1,2]=>4
[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]=>4
[6,4,2,3,1,5]=>6
[6,4,2,3,5,1]=>5
[6,4,2,5,1,3]=>5
[6,4,2,5,3,1]=>6
[6,4,3,1,2,5]=>4
[6,4,3,1,5,2]=>3
[6,4,3,2,1,5]=>5
[6,4,3,2,5,1]=>4
[6,4,3,5,1,2]=>4
[6,4,3,5,2,1]=>5
[6,4,5,1,2,3]=>5
[6,4,5,1,3,2]=>6
[6,4,5,2,1,3]=>6
[6,4,5,2,3,1]=>7
[6,4,5,3,1,2]=>5
[6,4,5,3,2,1]=>6
[6,5,1,2,3,4]=>5
[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]=>5
[6,5,1,4,3,2]=>4
[6,5,2,1,3,4]=>6
[6,5,2,1,4,3]=>5
[6,5,2,3,1,4]=>5
[6,5,2,3,4,1]=>6
[6,5,2,4,1,3]=>4
[6,5,2,4,3,1]=>5
[6,5,3,1,2,4]=>5
[6,5,3,1,4,2]=>4
[6,5,3,2,1,4]=>4
[6,5,3,2,4,1]=>5
[6,5,3,4,1,2]=>3
[6,5,3,4,2,1]=>4
[6,5,4,1,2,3]=>6
[6,5,4,1,3,2]=>5
[6,5,4,2,1,3]=>5
[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]=>3
[1,2,3,4,6,5,7]=>3
[1,2,3,4,6,7,5]=>4
[1,2,3,4,7,5,6]=>4
[1,2,3,4,7,6,5]=>3
[1,2,3,5,4,6,7]=>3
[1,2,3,5,4,7,6]=>6
[1,2,3,5,6,4,7]=>4
[1,2,3,5,6,7,4]=>5
[1,2,3,5,7,4,6]=>5
[1,2,3,5,7,6,4]=>4
[1,2,3,6,4,5,7]=>4
[1,2,3,6,4,7,5]=>5
[1,2,3,6,5,4,7]=>3
[1,2,3,6,5,7,4]=>4
[1,2,3,6,7,4,5]=>6
[1,2,3,6,7,5,4]=>5
[1,2,3,7,4,5,6]=>5
[1,2,3,7,4,6,5]=>4
[1,2,3,7,5,4,6]=>4
[1,2,3,7,5,6,4]=>3
[1,2,3,7,6,4,5]=>5
[1,2,3,7,6,5,4]=>6
[1,2,4,3,5,6,7]=>3
[1,2,4,3,5,7,6]=>6
[1,2,4,3,6,5,7]=>6
[1,2,4,3,6,7,5]=>7
[1,2,4,3,7,5,6]=>7
[1,2,4,3,7,6,5]=>6
[1,2,4,5,3,6,7]=>4
[1,2,4,5,3,7,6]=>7
[1,2,4,5,6,3,7]=>5
[1,2,4,5,6,7,3]=>6
[1,2,4,5,7,3,6]=>6
[1,2,4,5,7,6,3]=>5
[1,2,4,6,3,5,7]=>5
[1,2,4,6,3,7,5]=>6
[1,2,4,6,5,3,7]=>4
[1,2,4,6,5,7,3]=>5
[1,2,4,6,7,3,5]=>7
[1,2,4,6,7,5,3]=>6
[1,2,4,7,3,5,6]=>6
[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]=>7
[1,2,5,3,4,6,7]=>4
[1,2,5,3,4,7,6]=>7
[1,2,5,3,6,4,7]=>5
[1,2,5,3,6,7,4]=>6
[1,2,5,3,7,4,6]=>6
[1,2,5,3,7,6,4]=>5
[1,2,5,4,3,6,7]=>3
[1,2,5,4,3,7,6]=>6
[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]=>4
[1,2,5,6,3,4,7]=>6
[1,2,5,6,3,7,4]=>7
[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]=>7
[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]=>7
[1,2,5,7,6,4,3]=>6
[1,2,6,3,4,5,7]=>5
[1,2,6,3,4,7,5]=>6
[1,2,6,3,5,4,7]=>4
[1,2,6,3,5,7,4]=>5
[1,2,6,3,7,4,5]=>7
[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]=>3
[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]=>7
[1,2,6,5,7,3,4]=>7
[1,2,6,5,7,4,3]=>6
[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]=>6
[1,2,6,7,5,3,4]=>6
[1,2,6,7,5,4,3]=>5
[1,2,7,3,4,5,6]=>6
[1,2,7,3,4,6,5]=>5
[1,2,7,3,5,4,6]=>5
[1,2,7,3,5,6,4]=>4
[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]=>4
[1,2,7,4,5,3,6]=>4
[1,2,7,4,5,6,3]=>3
[1,2,7,4,6,3,5]=>5
[1,2,7,4,6,5,3]=>6
[1,2,7,5,3,4,6]=>6
[1,2,7,5,3,6,4]=>5
[1,2,7,5,4,3,6]=>7
[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]=>7
[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]=>5
[1,2,7,6,5,4,3]=>6
[1,3,2,4,5,6,7]=>3
[1,3,2,4,5,7,6]=>6
[1,3,2,4,6,5,7]=>6
[1,3,2,4,6,7,5]=>7
[1,3,2,4,7,5,6]=>7
[1,3,2,4,7,6,5]=>6
[1,3,2,5,4,6,7]=>6
[1,3,2,5,4,7,6]=>9
[1,3,2,5,6,4,7]=>7
[1,3,2,5,6,7,4]=>8
[1,3,2,5,7,4,6]=>8
[1,3,2,5,7,6,4]=>7
[1,3,2,6,4,5,7]=>7
[1,3,2,6,4,7,5]=>8
[1,3,2,6,5,4,7]=>6
[1,3,2,6,5,7,4]=>7
[1,3,2,6,7,4,5]=>9
[1,3,2,6,7,5,4]=>8
[1,3,2,7,4,5,6]=>8
[1,3,2,7,4,6,5]=>7
[1,3,2,7,5,4,6]=>7
[1,3,2,7,5,6,4]=>6
[1,3,2,7,6,4,5]=>8
[1,3,2,7,6,5,4]=>9
[1,3,4,2,5,6,7]=>4
[1,3,4,2,5,7,6]=>7
[1,3,4,2,6,5,7]=>7
[1,3,4,2,6,7,5]=>8
[1,3,4,2,7,5,6]=>8
[1,3,4,2,7,6,5]=>7
[1,3,4,5,2,6,7]=>5
[1,3,4,5,2,7,6]=>8
[1,3,4,5,6,2,7]=>6
[1,3,4,5,6,7,2]=>7
[1,3,4,5,7,2,6]=>7
[1,3,4,5,7,6,2]=>6
[1,3,4,6,2,5,7]=>6
[1,3,4,6,2,7,5]=>7
[1,3,4,6,5,2,7]=>5
[1,3,4,6,5,7,2]=>6
[1,3,4,6,7,2,5]=>8
[1,3,4,6,7,5,2]=>7
[1,3,4,7,2,5,6]=>7
[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]=>8
[1,3,5,2,4,6,7]=>5
[1,3,5,2,4,7,6]=>8
[1,3,5,2,6,4,7]=>6
[1,3,5,2,6,7,4]=>7
[1,3,5,2,7,4,6]=>7
[1,3,5,2,7,6,4]=>6
[1,3,5,4,2,6,7]=>4
[1,3,5,4,2,7,6]=>7
[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]=>5
[1,3,5,6,2,4,7]=>7
[1,3,5,6,2,7,4]=>8
[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]=>8
[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]=>6
[1,3,6,2,4,7,5]=>7
[1,3,6,2,5,4,7]=>5
[1,3,6,2,5,7,4]=>6
[1,3,6,2,7,4,5]=>8
[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]=>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]=>8
[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]=>7
[1,3,6,7,5,4,2]=>6
[1,3,7,2,4,5,6]=>7
[1,3,7,2,4,6,5]=>6
[1,3,7,2,5,4,6]=>6
[1,3,7,2,5,6,4]=>5
[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]=>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]=>7
[1,3,7,5,2,4,6]=>7
[1,3,7,5,2,6,4]=>6
[1,3,7,5,4,2,6]=>8
[1,3,7,5,4,6,2]=>7
[1,3,7,5,6,2,4]=>7
[1,3,7,5,6,4,2]=>8
[1,3,7,6,2,4,5]=>8
[1,3,7,6,2,5,4]=>7
[1,3,7,6,4,2,5]=>7
[1,3,7,6,4,5,2]=>8
[1,3,7,6,5,2,4]=>6
[1,3,7,6,5,4,2]=>7
[1,4,2,3,5,6,7]=>4
[1,4,2,3,5,7,6]=>7
[1,4,2,3,6,5,7]=>7
[1,4,2,3,6,7,5]=>8
[1,4,2,3,7,5,6]=>8
[1,4,2,3,7,6,5]=>7
[1,4,2,5,3,6,7]=>5
[1,4,2,5,3,7,6]=>8
[1,4,2,5,6,3,7]=>6
[1,4,2,5,6,7,3]=>7
[1,4,2,5,7,3,6]=>7
[1,4,2,5,7,6,3]=>6
[1,4,2,6,3,5,7]=>6
[1,4,2,6,3,7,5]=>7
[1,4,2,6,5,3,7]=>5
[1,4,2,6,5,7,3]=>6
[1,4,2,6,7,3,5]=>8
[1,4,2,6,7,5,3]=>7
[1,4,2,7,3,5,6]=>7
[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]=>8
[1,4,3,2,5,6,7]=>3
[1,4,3,2,5,7,6]=>6
[1,4,3,2,6,5,7]=>6
[1,4,3,2,6,7,5]=>7
[1,4,3,2,7,5,6]=>7
[1,4,3,2,7,6,5]=>6
[1,4,3,5,2,6,7]=>4
[1,4,3,5,2,7,6]=>7
[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]=>5
[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]=>5
[1,4,3,6,7,2,5]=>7
[1,4,3,6,7,5,2]=>6
[1,4,3,7,2,5,6]=>6
[1,4,3,7,2,6,5]=>5
[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]=>7
[1,4,5,2,3,6,7]=>6
[1,4,5,2,3,7,6]=>9
[1,4,5,2,6,3,7]=>7
[1,4,5,2,6,7,3]=>8
[1,4,5,2,7,3,6]=>8
[1,4,5,2,7,6,3]=>7
[1,4,5,3,2,6,7]=>5
[1,4,5,3,2,7,6]=>8
[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]=>6
[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]=>7
[1,4,5,7,2,3,6]=>7
[1,4,5,7,2,6,3]=>6
[1,4,5,7,3,2,6]=>8
[1,4,5,7,3,6,2]=>7
[1,4,5,7,6,2,3]=>7
[1,4,5,7,6,3,2]=>8
[1,4,6,2,3,5,7]=>7
[1,4,6,2,3,7,5]=>8
[1,4,6,2,5,3,7]=>6
[1,4,6,2,5,7,3]=>7
[1,4,6,2,7,3,5]=>9
[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]=>5
[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]=>6
[1,4,6,5,3,7,2]=>7
                    

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Description

The prefix exchange distance of a permutation.
This is the number of star transpositions needed to write a permutation.
In symbols, for a permutation $\pi\in\mathfrak S_n$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (1,a)$ for $2 \leq a \leq n$.
[1, Lem. 2.1] shows that the this length is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$.
One may find in [2] explicit formulas for its generating function and a combinatorial proof that it is asymptotically normal.

References

[1] Pak, I. Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and $k$-ary trees MathSciNet:1691876
[2] Grusea, S., Labarre, A. Asymptotic normality and combinatorial aspects of the prefix exchange distance distribution MathSciNet:3497998

Code

def statistic(pi):
    ct = pi.cycle_tuples()
    one = None # cycle containing 1
    two = 0    # other cycles of length at least 2
    fix = 0    # other fixed points
    for c in ct:
        if one is None and 1 in c:
            one = True
        elif len(c) > 1:
            two += 1
        else:
            fix += 1

    return len(pi) + two - fix - 1

# alternative code for checking the lemma
def statistic_alt1(pi):
    def gens(n):
        S = Permutations(n)
        return [S(Permutation((1, i))) for i in range(2, n+1)]

    G = PermutationGroup(gens(len(pi)))
    pi = G(pi)
    w = gap.Factorization(G._gap_(), pi._gap_())
    return w.Length()

Created

Jan 08, 2018 at 23:01 by Martin Rubey

Updated

Aug 15, 2019 at 12:55 by Christian Stump