- FindStat

edit this statistic or download as text // json

Identifier

St001081: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1,2]=>1
[2,1]=>1
[1,2,3]=>1
[1,3,2]=>2
[2,1,3]=>1
[2,3,1]=>1
[3,1,2]=>1
[3,2,1]=>1
[1,2,3,4]=>1
[1,2,4,3]=>2
[1,3,2,4]=>2
[1,3,4,2]=>3
[1,4,2,3]=>3
[1,4,3,2]=>2
[2,1,3,4]=>1
[2,1,4,3]=>4
[2,3,1,4]=>1
[2,3,4,1]=>1
[2,4,1,3]=>1
[2,4,3,1]=>1
[3,1,2,4]=>1
[3,1,4,2]=>1
[3,2,1,4]=>1
[3,2,4,1]=>1
[3,4,1,2]=>4
[3,4,2,1]=>1
[4,1,2,3]=>1
[4,1,3,2]=>1
[4,2,1,3]=>1
[4,2,3,1]=>1
[4,3,1,2]=>1
[4,3,2,1]=>4
[1,2,3,4,5]=>1
[1,2,3,5,4]=>2
[1,2,4,3,5]=>2
[1,2,4,5,3]=>3
[1,2,5,3,4]=>3
[1,2,5,4,3]=>2
[1,3,2,4,5]=>2
[1,3,2,5,4]=>24
[1,3,4,2,5]=>3
[1,3,4,5,2]=>4
[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]=>2
[1,4,3,5,2]=>3
[1,4,5,2,3]=>24
[1,4,5,3,2]=>4
[1,5,2,3,4]=>4
[1,5,2,4,3]=>3
[1,5,3,2,4]=>3
[1,5,3,4,2]=>2
[1,5,4,2,3]=>4
[1,5,4,3,2]=>24
[2,1,3,4,5]=>1
[2,1,3,5,4]=>4
[2,1,4,3,5]=>4
[2,1,4,5,3]=>6
[2,1,5,3,4]=>6
[2,1,5,4,3]=>4
[2,3,1,4,5]=>1
[2,3,1,5,4]=>6
[2,3,4,1,5]=>1
[2,3,4,5,1]=>1
[2,3,5,1,4]=>1
[2,3,5,4,1]=>1
[2,4,1,3,5]=>1
[2,4,1,5,3]=>1
[2,4,3,1,5]=>1
[2,4,3,5,1]=>1
[2,4,5,1,3]=>6
[2,4,5,3,1]=>1
[2,5,1,3,4]=>1
[2,5,1,4,3]=>1
[2,5,3,1,4]=>1
[2,5,3,4,1]=>1
[2,5,4,1,3]=>1
[2,5,4,3,1]=>6
[3,1,2,4,5]=>1
[3,1,2,5,4]=>6
[3,1,4,2,5]=>1
[3,1,4,5,2]=>1
[3,1,5,2,4]=>1
[3,1,5,4,2]=>1
[3,2,1,4,5]=>1
[3,2,1,5,4]=>4
[3,2,4,1,5]=>1
[3,2,4,5,1]=>1
[3,2,5,1,4]=>1
[3,2,5,4,1]=>1
[3,4,1,2,5]=>4
[3,4,1,5,2]=>6
[3,4,2,1,5]=>1
[3,4,2,5,1]=>1
[3,4,5,1,2]=>1
[3,4,5,2,1]=>6
[3,5,1,2,4]=>6
[3,5,1,4,2]=>4
[3,5,2,1,4]=>1
[3,5,2,4,1]=>1
[3,5,4,1,2]=>6
[3,5,4,2,1]=>1
[4,1,2,3,5]=>1
[4,1,2,5,3]=>1
[4,1,3,2,5]=>1
[4,1,3,5,2]=>1
[4,1,5,2,3]=>6
[4,1,5,3,2]=>1
[4,2,1,3,5]=>1
[4,2,1,5,3]=>1
[4,2,3,1,5]=>1
[4,2,3,5,1]=>1
[4,2,5,1,3]=>4
[4,2,5,3,1]=>1
[4,3,1,2,5]=>1
[4,3,1,5,2]=>1
[4,3,2,1,5]=>4
[4,3,2,5,1]=>6
[4,3,5,1,2]=>6
[4,3,5,2,1]=>1
[4,5,1,2,3]=>1
[4,5,1,3,2]=>6
[4,5,2,1,3]=>6
[4,5,2,3,1]=>1
[4,5,3,1,2]=>4
[4,5,3,2,1]=>1
[5,1,2,3,4]=>1
[5,1,2,4,3]=>1
[5,1,3,2,4]=>1
[5,1,3,4,2]=>1
[5,1,4,2,3]=>1
[5,1,4,3,2]=>6
[5,2,1,3,4]=>1
[5,2,1,4,3]=>1
[5,2,3,1,4]=>1
[5,2,3,4,1]=>1
[5,2,4,1,3]=>1
[5,2,4,3,1]=>4
[5,3,1,2,4]=>1
[5,3,1,4,2]=>1
[5,3,2,1,4]=>6
[5,3,2,4,1]=>4
[5,3,4,1,2]=>1
[5,3,4,2,1]=>6
[5,4,1,2,3]=>6
[5,4,1,3,2]=>1
[5,4,2,1,3]=>1
[5,4,2,3,1]=>6
[5,4,3,1,2]=>1
[5,4,3,2,1]=>4
[1,2,3,4,5,6]=>1
[1,2,3,4,6,5]=>2
[1,2,3,5,4,6]=>2
[1,2,3,5,6,4]=>3
[1,2,3,6,4,5]=>3
[1,2,3,6,5,4]=>2
[1,2,4,3,5,6]=>2
[1,2,4,3,6,5]=>24
[1,2,4,5,3,6]=>3
[1,2,4,5,6,3]=>4
[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]=>2
[1,2,5,4,6,3]=>3
[1,2,5,6,3,4]=>24
[1,2,5,6,4,3]=>4
[1,2,6,3,4,5]=>4
[1,2,6,3,5,4]=>3
[1,2,6,4,3,5]=>3
[1,2,6,4,5,3]=>2
[1,2,6,5,3,4]=>4
[1,2,6,5,4,3]=>24
[1,3,2,4,5,6]=>2
[1,3,2,4,6,5]=>24
[1,3,2,5,4,6]=>24
[1,3,2,5,6,4]=>42
[1,3,2,6,4,5]=>42
[1,3,2,6,5,4]=>24
[1,3,4,2,5,6]=>3
[1,3,4,2,6,5]=>42
[1,3,4,5,2,6]=>4
[1,3,4,5,6,2]=>5
[1,3,4,6,2,5]=>5
[1,3,4,6,5,2]=>4
[1,3,5,2,4,6]=>4
[1,3,5,2,6,4]=>5
[1,3,5,4,2,6]=>3
[1,3,5,4,6,2]=>4
[1,3,5,6,2,4]=>42
[1,3,5,6,4,2]=>5
[1,3,6,2,4,5]=>5
[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]=>5
[1,3,6,5,4,2]=>42
[1,4,2,3,5,6]=>3
[1,4,2,3,6,5]=>42
[1,4,2,5,3,6]=>4
[1,4,2,5,6,3]=>5
[1,4,2,6,3,5]=>5
[1,4,2,6,5,3]=>4
[1,4,3,2,5,6]=>2
[1,4,3,2,6,5]=>24
[1,4,3,5,2,6]=>3
[1,4,3,5,6,2]=>4
[1,4,3,6,2,5]=>4
[1,4,3,6,5,2]=>3
[1,4,5,2,3,6]=>24
[1,4,5,2,6,3]=>42
[1,4,5,3,2,6]=>4
[1,4,5,3,6,2]=>5
[1,4,5,6,2,3]=>5
[1,4,5,6,3,2]=>42
[1,4,6,2,3,5]=>42
[1,4,6,2,5,3]=>24
[1,4,6,3,2,5]=>5
[1,4,6,3,5,2]=>4
[1,4,6,5,2,3]=>42
[1,4,6,5,3,2]=>5
[1,5,2,3,4,6]=>4
[1,5,2,3,6,4]=>5
[1,5,2,4,3,6]=>3
[1,5,2,4,6,3]=>4
[1,5,2,6,3,4]=>42
[1,5,2,6,4,3]=>5
[1,5,3,2,4,6]=>3
[1,5,3,2,6,4]=>4
[1,5,3,4,2,6]=>2
[1,5,3,4,6,2]=>3
[1,5,3,6,2,4]=>24
[1,5,3,6,4,2]=>4
[1,5,4,2,3,6]=>4
[1,5,4,2,6,3]=>5
[1,5,4,3,2,6]=>24
[1,5,4,3,6,2]=>42
[1,5,4,6,2,3]=>42
[1,5,4,6,3,2]=>5
[1,5,6,2,3,4]=>5
[1,5,6,2,4,3]=>42
[1,5,6,3,2,4]=>42
[1,5,6,3,4,2]=>5
[1,5,6,4,2,3]=>24
[1,5,6,4,3,2]=>4
[1,6,2,3,4,5]=>5
[1,6,2,3,5,4]=>4
[1,6,2,4,3,5]=>4
[1,6,2,4,5,3]=>3
[1,6,2,5,3,4]=>5
[1,6,2,5,4,3]=>42
[1,6,3,2,4,5]=>4
[1,6,3,2,5,4]=>3
[1,6,3,4,2,5]=>3
[1,6,3,4,5,2]=>2
[1,6,3,5,2,4]=>4
[1,6,3,5,4,2]=>24
[1,6,4,2,3,5]=>5
[1,6,4,2,5,3]=>4
[1,6,4,3,2,5]=>42
[1,6,4,3,5,2]=>24
[1,6,4,5,2,3]=>5
[1,6,4,5,3,2]=>42
[1,6,5,2,3,4]=>42
[1,6,5,2,4,3]=>5
[1,6,5,3,2,4]=>5
[1,6,5,3,4,2]=>42
[1,6,5,4,2,3]=>4
[1,6,5,4,3,2]=>24
[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]=>6
[2,1,3,6,4,5]=>6
[2,1,3,6,5,4]=>4
[2,1,4,3,5,6]=>4
[2,1,4,3,6,5]=>56
[2,1,4,5,3,6]=>6
[2,1,4,5,6,3]=>8
[2,1,4,6,3,5]=>8
[2,1,4,6,5,3]=>6
[2,1,5,3,4,6]=>6
[2,1,5,3,6,4]=>8
[2,1,5,4,3,6]=>4
[2,1,5,4,6,3]=>6
[2,1,5,6,3,4]=>56
[2,1,5,6,4,3]=>8
[2,1,6,3,4,5]=>8
[2,1,6,3,5,4]=>6
[2,1,6,4,3,5]=>6
[2,1,6,4,5,3]=>4
[2,1,6,5,3,4]=>8
[2,1,6,5,4,3]=>56
[2,3,1,4,5,6]=>1
[2,3,1,4,6,5]=>6
[2,3,1,5,4,6]=>6
[2,3,1,5,6,4]=>9
[2,3,1,6,4,5]=>9
[2,3,1,6,5,4]=>6
[2,3,4,1,5,6]=>1
[2,3,4,1,6,5]=>8
[2,3,4,5,1,6]=>1
[2,3,4,5,6,1]=>1
[2,3,4,6,1,5]=>1
[2,3,4,6,5,1]=>1
[2,3,5,1,4,6]=>1
[2,3,5,1,6,4]=>1
[2,3,5,4,1,6]=>1
[2,3,5,4,6,1]=>1
[2,3,5,6,1,4]=>8
[2,3,5,6,4,1]=>1
[2,3,6,1,4,5]=>1
[2,3,6,1,5,4]=>1
[2,3,6,4,1,5]=>1
[2,3,6,4,5,1]=>1
[2,3,6,5,1,4]=>1
[2,3,6,5,4,1]=>8
[2,4,1,3,5,6]=>1
[2,4,1,3,6,5]=>8
[2,4,1,5,3,6]=>1
[2,4,1,5,6,3]=>1
[2,4,1,6,3,5]=>1
[2,4,1,6,5,3]=>1
[2,4,3,1,5,6]=>1
[2,4,3,1,6,5]=>6
[2,4,3,5,1,6]=>1
[2,4,3,5,6,1]=>1
[2,4,3,6,1,5]=>1
[2,4,3,6,5,1]=>1
[2,4,5,1,3,6]=>6
[2,4,5,1,6,3]=>9
[2,4,5,3,1,6]=>1
[2,4,5,3,6,1]=>1
[2,4,5,6,1,3]=>1
[2,4,5,6,3,1]=>8
[2,4,6,1,3,5]=>9
[2,4,6,1,5,3]=>6
[2,4,6,3,1,5]=>1
[2,4,6,3,5,1]=>1
[2,4,6,5,1,3]=>8
[2,4,6,5,3,1]=>1
[2,5,1,3,4,6]=>1
[2,5,1,3,6,4]=>1
[2,5,1,4,3,6]=>1
[2,5,1,4,6,3]=>1
[2,5,1,6,3,4]=>8
[2,5,1,6,4,3]=>1
[2,5,3,1,4,6]=>1
[2,5,3,1,6,4]=>1
[2,5,3,4,1,6]=>1
[2,5,3,4,6,1]=>1
[2,5,3,6,1,4]=>6
[2,5,3,6,4,1]=>1
[2,5,4,1,3,6]=>1
[2,5,4,1,6,3]=>1
[2,5,4,3,1,6]=>6
[2,5,4,3,6,1]=>8
[2,5,4,6,1,3]=>9
[2,5,4,6,3,1]=>1
[2,5,6,1,3,4]=>1
[2,5,6,1,4,3]=>8
[2,5,6,3,1,4]=>9
[2,5,6,3,4,1]=>1
[2,5,6,4,1,3]=>6
[2,5,6,4,3,1]=>1
[2,6,1,3,4,5]=>1
[2,6,1,3,5,4]=>1
[2,6,1,4,3,5]=>1
[2,6,1,4,5,3]=>1
[2,6,1,5,3,4]=>1
[2,6,1,5,4,3]=>8
[2,6,3,1,4,5]=>1
[2,6,3,1,5,4]=>1
[2,6,3,4,1,5]=>1
[2,6,3,4,5,1]=>1
[2,6,3,5,1,4]=>1
[2,6,3,5,4,1]=>6
[2,6,4,1,3,5]=>1
[2,6,4,1,5,3]=>1
[2,6,4,3,1,5]=>8
[2,6,4,3,5,1]=>6
[2,6,4,5,1,3]=>1
[2,6,4,5,3,1]=>9
[2,6,5,1,3,4]=>8
[2,6,5,1,4,3]=>1
[2,6,5,3,1,4]=>1
[2,6,5,3,4,1]=>9
[2,6,5,4,1,3]=>1
[2,6,5,4,3,1]=>6
[3,1,2,4,5,6]=>1
[3,1,2,4,6,5]=>6
[3,1,2,5,4,6]=>6
[3,1,2,5,6,4]=>9
[3,1,2,6,4,5]=>9
[3,1,2,6,5,4]=>6
[3,1,4,2,5,6]=>1
[3,1,4,2,6,5]=>8
[3,1,4,5,2,6]=>1
[3,1,4,5,6,2]=>1
[3,1,4,6,2,5]=>1
[3,1,4,6,5,2]=>1
[3,1,5,2,4,6]=>1
[3,1,5,2,6,4]=>1
[3,1,5,4,2,6]=>1
[3,1,5,4,6,2]=>1
[3,1,5,6,2,4]=>8
[3,1,5,6,4,2]=>1
[3,1,6,2,4,5]=>1
[3,1,6,2,5,4]=>1
[3,1,6,4,2,5]=>1
[3,1,6,4,5,2]=>1
[3,1,6,5,2,4]=>1
[3,1,6,5,4,2]=>8
[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]=>6
[3,2,1,6,4,5]=>6
[3,2,1,6,5,4]=>4
[3,2,4,1,5,6]=>1
[3,2,4,1,6,5]=>6
[3,2,4,5,1,6]=>1
[3,2,4,5,6,1]=>1
[3,2,4,6,1,5]=>1
[3,2,4,6,5,1]=>1
[3,2,5,1,4,6]=>1
[3,2,5,1,6,4]=>1
[3,2,5,4,1,6]=>1
[3,2,5,4,6,1]=>1
[3,2,5,6,1,4]=>6
[3,2,5,6,4,1]=>1
[3,2,6,1,4,5]=>1
[3,2,6,1,5,4]=>1
[3,2,6,4,1,5]=>1
[3,2,6,4,5,1]=>1
[3,2,6,5,1,4]=>1
[3,2,6,5,4,1]=>6
[3,4,1,2,5,6]=>4
[3,4,1,2,6,5]=>56
[3,4,1,5,2,6]=>6
[3,4,1,5,6,2]=>8
[3,4,1,6,2,5]=>8
[3,4,1,6,5,2]=>6
[3,4,2,1,5,6]=>1
[3,4,2,1,6,5]=>8
[3,4,2,5,1,6]=>1
[3,4,2,5,6,1]=>1
[3,4,2,6,1,5]=>1
[3,4,2,6,5,1]=>1
[3,4,5,1,2,6]=>1
[3,4,5,1,6,2]=>1
[3,4,5,2,1,6]=>6
[3,4,5,2,6,1]=>8
[3,4,5,6,1,2]=>9
[3,4,5,6,2,1]=>1
[3,4,6,1,2,5]=>1
[3,4,6,1,5,2]=>1
[3,4,6,2,1,5]=>8
[3,4,6,2,5,1]=>6
[3,4,6,5,1,2]=>1
[3,4,6,5,2,1]=>9
[3,5,1,2,4,6]=>6
[3,5,1,2,6,4]=>8
[3,5,1,4,2,6]=>4
[3,5,1,4,6,2]=>6
[3,5,1,6,2,4]=>56
[3,5,1,6,4,2]=>8
[3,5,2,1,4,6]=>1
[3,5,2,1,6,4]=>1
[3,5,2,4,1,6]=>1
[3,5,2,4,6,1]=>1
[3,5,2,6,1,4]=>8
[3,5,2,6,4,1]=>1
[3,5,4,1,2,6]=>6
[3,5,4,1,6,2]=>9
[3,5,4,2,1,6]=>1
[3,5,4,2,6,1]=>1
[3,5,4,6,1,2]=>1
[3,5,4,6,2,1]=>8
[3,5,6,1,2,4]=>8
[3,5,6,1,4,2]=>1
[3,5,6,2,1,4]=>1
[3,5,6,2,4,1]=>9
[3,5,6,4,1,2]=>1
[3,5,6,4,2,1]=>6
[3,6,1,2,4,5]=>8
[3,6,1,2,5,4]=>6
[3,6,1,4,2,5]=>6
[3,6,1,4,5,2]=>4
[3,6,1,5,2,4]=>8
[3,6,1,5,4,2]=>56
[3,6,2,1,4,5]=>1
[3,6,2,1,5,4]=>1
[3,6,2,4,1,5]=>1
[3,6,2,4,5,1]=>1
[3,6,2,5,1,4]=>1
[3,6,2,5,4,1]=>8
[3,6,4,1,2,5]=>9
[3,6,4,1,5,2]=>6
[3,6,4,2,1,5]=>1
[3,6,4,2,5,1]=>1
[3,6,4,5,1,2]=>8
[3,6,4,5,2,1]=>1
[3,6,5,1,2,4]=>1
[3,6,5,1,4,2]=>8
[3,6,5,2,1,4]=>9
[3,6,5,2,4,1]=>1
[3,6,5,4,1,2]=>6
[3,6,5,4,2,1]=>1
[4,1,2,3,5,6]=>1
[4,1,2,3,6,5]=>8
[4,1,2,5,3,6]=>1
[4,1,2,5,6,3]=>1
[4,1,2,6,3,5]=>1
[4,1,2,6,5,3]=>1
[4,1,3,2,5,6]=>1
[4,1,3,2,6,5]=>6
[4,1,3,5,2,6]=>1
[4,1,3,5,6,2]=>1
[4,1,3,6,2,5]=>1
[4,1,3,6,5,2]=>1
[4,1,5,2,3,6]=>6
[4,1,5,2,6,3]=>9
[4,1,5,3,2,6]=>1
[4,1,5,3,6,2]=>1
[4,1,5,6,2,3]=>1
[4,1,5,6,3,2]=>8
[4,1,6,2,3,5]=>9
[4,1,6,2,5,3]=>6
[4,1,6,3,2,5]=>1
[4,1,6,3,5,2]=>1
[4,1,6,5,2,3]=>8
[4,1,6,5,3,2]=>1
[4,2,1,3,5,6]=>1
[4,2,1,3,6,5]=>6
[4,2,1,5,3,6]=>1
[4,2,1,5,6,3]=>1
[4,2,1,6,3,5]=>1
[4,2,1,6,5,3]=>1
[4,2,3,1,5,6]=>1
[4,2,3,1,6,5]=>4
[4,2,3,5,1,6]=>1
[4,2,3,5,6,1]=>1
[4,2,3,6,1,5]=>1
[4,2,3,6,5,1]=>1
[4,2,5,1,3,6]=>4
[4,2,5,1,6,3]=>6
[4,2,5,3,1,6]=>1
[4,2,5,3,6,1]=>1
[4,2,5,6,1,3]=>1
[4,2,5,6,3,1]=>6
[4,2,6,1,3,5]=>6
[4,2,6,1,5,3]=>4
[4,2,6,3,1,5]=>1
[4,2,6,3,5,1]=>1
[4,2,6,5,1,3]=>6
[4,2,6,5,3,1]=>1
[4,3,1,2,5,6]=>1
[4,3,1,2,6,5]=>8
[4,3,1,5,2,6]=>1
[4,3,1,5,6,2]=>1
[4,3,1,6,2,5]=>1
[4,3,1,6,5,2]=>1
[4,3,2,1,5,6]=>4
[4,3,2,1,6,5]=>56
[4,3,2,5,1,6]=>6
[4,3,2,5,6,1]=>8
[4,3,2,6,1,5]=>8
[4,3,2,6,5,1]=>6
[4,3,5,1,2,6]=>6
[4,3,5,1,6,2]=>8
[4,3,5,2,1,6]=>1
[4,3,5,2,6,1]=>1
[4,3,5,6,1,2]=>1
[4,3,5,6,2,1]=>9
[4,3,6,1,2,5]=>8
[4,3,6,1,5,2]=>6
[4,3,6,2,1,5]=>1
[4,3,6,2,5,1]=>1
[4,3,6,5,1,2]=>9
[4,3,6,5,2,1]=>1
[4,5,1,2,3,6]=>1
[4,5,1,2,6,3]=>1
[4,5,1,3,2,6]=>6
[4,5,1,3,6,2]=>9
[4,5,1,6,2,3]=>8
[4,5,1,6,3,2]=>1
[4,5,2,1,3,6]=>6
[4,5,2,1,6,3]=>8
[4,5,2,3,1,6]=>1
[4,5,2,3,6,1]=>1
[4,5,2,6,1,3]=>1
[4,5,2,6,3,1]=>9
[4,5,3,1,2,6]=>4
[4,5,3,1,6,2]=>6
[4,5,3,2,1,6]=>1
[4,5,3,2,6,1]=>1
[4,5,3,6,1,2]=>1
[4,5,3,6,2,1]=>6
[4,5,6,1,2,3]=>56
[4,5,6,1,3,2]=>8
[4,5,6,2,1,3]=>8
[4,5,6,2,3,1]=>1
[4,5,6,3,1,2]=>1
[4,5,6,3,2,1]=>8
[4,6,1,2,3,5]=>1
[4,6,1,2,5,3]=>1
[4,6,1,3,2,5]=>9
[4,6,1,3,5,2]=>6
[4,6,1,5,2,3]=>1
[4,6,1,5,3,2]=>8
[4,6,2,1,3,5]=>8
[4,6,2,1,5,3]=>6
[4,6,2,3,1,5]=>1
[4,6,2,3,5,1]=>1
[4,6,2,5,1,3]=>9
[4,6,2,5,3,1]=>1
[4,6,3,1,2,5]=>6
[4,6,3,1,5,2]=>4
[4,6,3,2,1,5]=>1
[4,6,3,2,5,1]=>1
[4,6,3,5,1,2]=>6
[4,6,3,5,2,1]=>1
[4,6,5,1,2,3]=>8
[4,6,5,1,3,2]=>56
[4,6,5,2,1,3]=>1
[4,6,5,2,3,1]=>8
[4,6,5,3,1,2]=>8
[4,6,5,3,2,1]=>1
[5,1,2,3,4,6]=>1
[5,1,2,3,6,4]=>1
[5,1,2,4,3,6]=>1
[5,1,2,4,6,3]=>1
[5,1,2,6,3,4]=>8
[5,1,2,6,4,3]=>1
[5,1,3,2,4,6]=>1
[5,1,3,2,6,4]=>1
[5,1,3,4,2,6]=>1
[5,1,3,4,6,2]=>1
[5,1,3,6,2,4]=>6
[5,1,3,6,4,2]=>1
[5,1,4,2,3,6]=>1
[5,1,4,2,6,3]=>1
[5,1,4,3,2,6]=>6
[5,1,4,3,6,2]=>8
[5,1,4,6,2,3]=>9
[5,1,4,6,3,2]=>1
[5,1,6,2,3,4]=>1
[5,1,6,2,4,3]=>8
[5,1,6,3,2,4]=>9
[5,1,6,3,4,2]=>1
[5,1,6,4,2,3]=>6
[5,1,6,4,3,2]=>1
[5,2,1,3,4,6]=>1
[5,2,1,3,6,4]=>1
[5,2,1,4,3,6]=>1
[5,2,1,4,6,3]=>1
[5,2,1,6,3,4]=>6
[5,2,1,6,4,3]=>1
[5,2,3,1,4,6]=>1
[5,2,3,1,6,4]=>1
[5,2,3,4,1,6]=>1
[5,2,3,4,6,1]=>1
[5,2,3,6,1,4]=>4
[5,2,3,6,4,1]=>1
[5,2,4,1,3,6]=>1
[5,2,4,1,6,3]=>1
[5,2,4,3,1,6]=>4
[5,2,4,3,6,1]=>6
[5,2,4,6,1,3]=>6
[5,2,4,6,3,1]=>1
[5,2,6,1,3,4]=>1
[5,2,6,1,4,3]=>6
[5,2,6,3,1,4]=>6
[5,2,6,3,4,1]=>1
[5,2,6,4,1,3]=>4
[5,2,6,4,3,1]=>1
[5,3,1,2,4,6]=>1
[5,3,1,2,6,4]=>1
[5,3,1,4,2,6]=>1
[5,3,1,4,6,2]=>1
[5,3,1,6,2,4]=>8
[5,3,1,6,4,2]=>1
[5,3,2,1,4,6]=>6
[5,3,2,1,6,4]=>8
[5,3,2,4,1,6]=>4
[5,3,2,4,6,1]=>6
[5,3,2,6,1,4]=>56
[5,3,2,6,4,1]=>8
[5,3,4,1,2,6]=>1
[5,3,4,1,6,2]=>1
[5,3,4,2,1,6]=>6
[5,3,4,2,6,1]=>9
[5,3,4,6,1,2]=>8
[5,3,4,6,2,1]=>1
[5,3,6,1,2,4]=>1
[5,3,6,1,4,2]=>9
[5,3,6,2,1,4]=>8
[5,3,6,2,4,1]=>1
[5,3,6,4,1,2]=>6
[5,3,6,4,2,1]=>1
[5,4,1,2,3,6]=>6
[5,4,1,2,6,3]=>8
[5,4,1,3,2,6]=>1
[5,4,1,3,6,2]=>1
[5,4,1,6,2,3]=>1
[5,4,1,6,3,2]=>9
[5,4,2,1,3,6]=>1
[5,4,2,1,6,3]=>1
[5,4,2,3,1,6]=>6
[5,4,2,3,6,1]=>9
[5,4,2,6,1,3]=>8
[5,4,2,6,3,1]=>1
[5,4,3,1,2,6]=>1
[5,4,3,1,6,2]=>1
[5,4,3,2,1,6]=>4
[5,4,3,2,6,1]=>6
[5,4,3,6,1,2]=>6
[5,4,3,6,2,1]=>1
[5,4,6,1,2,3]=>8
[5,4,6,1,3,2]=>1
[5,4,6,2,1,3]=>56
[5,4,6,2,3,1]=>8
[5,4,6,3,1,2]=>8
[5,4,6,3,2,1]=>1
[5,6,1,2,3,4]=>9
[5,6,1,2,4,3]=>1
[5,6,1,3,2,4]=>1
[5,6,1,3,4,2]=>8
[5,6,1,4,2,3]=>1
[5,6,1,4,3,2]=>6
[5,6,2,1,3,4]=>1
[5,6,2,1,4,3]=>9
[5,6,2,3,1,4]=>8
[5,6,2,3,4,1]=>1
[5,6,2,4,1,3]=>6
[5,6,2,4,3,1]=>1
[5,6,3,1,2,4]=>1
[5,6,3,1,4,2]=>6
[5,6,3,2,1,4]=>6
[5,6,3,2,4,1]=>1
[5,6,3,4,1,2]=>4
[5,6,3,4,2,1]=>1
[5,6,4,1,2,3]=>1
[5,6,4,1,3,2]=>8
[5,6,4,2,1,3]=>8
[5,6,4,2,3,1]=>1
[5,6,4,3,1,2]=>56
[5,6,4,3,2,1]=>8
[6,1,2,3,4,5]=>1
[6,1,2,3,5,4]=>1
[6,1,2,4,3,5]=>1
[6,1,2,4,5,3]=>1
[6,1,2,5,3,4]=>1
[6,1,2,5,4,3]=>8
[6,1,3,2,4,5]=>1
[6,1,3,2,5,4]=>1
[6,1,3,4,2,5]=>1
[6,1,3,4,5,2]=>1
[6,1,3,5,2,4]=>1
[6,1,3,5,4,2]=>6
[6,1,4,2,3,5]=>1
[6,1,4,2,5,3]=>1
[6,1,4,3,2,5]=>8
[6,1,4,3,5,2]=>6
[6,1,4,5,2,3]=>1
[6,1,4,5,3,2]=>9
[6,1,5,2,3,4]=>8
[6,1,5,2,4,3]=>1
[6,1,5,3,2,4]=>1
[6,1,5,3,4,2]=>9
[6,1,5,4,2,3]=>1
[6,1,5,4,3,2]=>6
[6,2,1,3,4,5]=>1
[6,2,1,3,5,4]=>1
[6,2,1,4,3,5]=>1
[6,2,1,4,5,3]=>1
[6,2,1,5,3,4]=>1
[6,2,1,5,4,3]=>6
[6,2,3,1,4,5]=>1
[6,2,3,1,5,4]=>1
[6,2,3,4,1,5]=>1
[6,2,3,4,5,1]=>1
[6,2,3,5,1,4]=>1
[6,2,3,5,4,1]=>4
[6,2,4,1,3,5]=>1
[6,2,4,1,5,3]=>1
[6,2,4,3,1,5]=>6
[6,2,4,3,5,1]=>4
[6,2,4,5,1,3]=>1
[6,2,4,5,3,1]=>6
[6,2,5,1,3,4]=>6
[6,2,5,1,4,3]=>1
[6,2,5,3,1,4]=>1
[6,2,5,3,4,1]=>6
[6,2,5,4,1,3]=>1
[6,2,5,4,3,1]=>4
[6,3,1,2,4,5]=>1
[6,3,1,2,5,4]=>1
[6,3,1,4,2,5]=>1
[6,3,1,4,5,2]=>1
[6,3,1,5,2,4]=>1
[6,3,1,5,4,2]=>8
[6,3,2,1,4,5]=>8
[6,3,2,1,5,4]=>6
[6,3,2,4,1,5]=>6
[6,3,2,4,5,1]=>4
[6,3,2,5,1,4]=>8
[6,3,2,5,4,1]=>56
[6,3,4,1,2,5]=>1
[6,3,4,1,5,2]=>1
[6,3,4,2,1,5]=>9
[6,3,4,2,5,1]=>6
[6,3,4,5,1,2]=>1
[6,3,4,5,2,1]=>8
[6,3,5,1,2,4]=>9
[6,3,5,1,4,2]=>1
[6,3,5,2,1,4]=>1
[6,3,5,2,4,1]=>8
[6,3,5,4,1,2]=>1
[6,3,5,4,2,1]=>6
[6,4,1,2,3,5]=>8
[6,4,1,2,5,3]=>6
[6,4,1,3,2,5]=>1
[6,4,1,3,5,2]=>1
[6,4,1,5,2,3]=>9
[6,4,1,5,3,2]=>1
[6,4,2,1,3,5]=>1
[6,4,2,1,5,3]=>1
[6,4,2,3,1,5]=>9
[6,4,2,3,5,1]=>6
[6,4,2,5,1,3]=>1
[6,4,2,5,3,1]=>8
[6,4,3,1,2,5]=>1
[6,4,3,1,5,2]=>1
[6,4,3,2,1,5]=>6
[6,4,3,2,5,1]=>4
[6,4,3,5,1,2]=>1
[6,4,3,5,2,1]=>6
[6,4,5,1,2,3]=>1
[6,4,5,1,3,2]=>8
[6,4,5,2,1,3]=>8
[6,4,5,2,3,1]=>56
[6,4,5,3,1,2]=>1
[6,4,5,3,2,1]=>8
[6,5,1,2,3,4]=>1
[6,5,1,2,4,3]=>9
[6,5,1,3,2,4]=>8
[6,5,1,3,4,2]=>1
[6,5,1,4,2,3]=>6
[6,5,1,4,3,2]=>1
[6,5,2,1,3,4]=>9
[6,5,2,1,4,3]=>1
[6,5,2,3,1,4]=>1
[6,5,2,3,4,1]=>8
[6,5,2,4,1,3]=>1
[6,5,2,4,3,1]=>6
[6,5,3,1,2,4]=>6
[6,5,3,1,4,2]=>1
[6,5,3,2,1,4]=>1
[6,5,3,2,4,1]=>6
[6,5,3,4,1,2]=>1
[6,5,3,4,2,1]=>4
[6,5,4,1,2,3]=>8
[6,5,4,1,3,2]=>1
[6,5,4,2,1,3]=>1
[6,5,4,2,3,1]=>8
[6,5,4,3,1,2]=>8
[6,5,4,3,2,1]=>56
[1,2,3,4,5,6,7]=>1
[1,2,3,4,5,7,6]=>2
[1,2,3,4,6,5,7]=>2
[1,2,3,4,6,7,5]=>3
[1,2,3,4,7,5,6]=>3
[1,2,3,4,7,6,5]=>2
[1,2,3,5,4,6,7]=>2
[1,2,3,5,4,7,6]=>24
[1,2,3,5,6,4,7]=>3
[1,2,3,5,6,7,4]=>4
[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]=>2
[1,2,3,6,5,7,4]=>3
[1,2,3,6,7,4,5]=>24
[1,2,3,6,7,5,4]=>4
[1,2,3,7,4,5,6]=>4
[1,2,3,7,4,6,5]=>3
[1,2,3,7,5,4,6]=>3
[1,2,3,7,5,6,4]=>2
[1,2,3,7,6,4,5]=>4
[1,2,3,7,6,5,4]=>24
[1,2,4,3,5,6,7]=>2
[1,2,4,3,5,7,6]=>24
[1,2,4,3,6,5,7]=>24
[1,2,4,3,6,7,5]=>42
[1,2,4,3,7,5,6]=>42
[1,2,4,3,7,6,5]=>24
[1,2,4,5,3,6,7]=>3
[1,2,4,5,3,7,6]=>42
[1,2,4,5,6,3,7]=>4
[1,2,4,5,6,7,3]=>5
[1,2,4,5,7,3,6]=>5
[1,2,4,5,7,6,3]=>4
[1,2,4,6,3,5,7]=>4
[1,2,4,6,3,7,5]=>5
[1,2,4,6,5,3,7]=>3
[1,2,4,6,5,7,3]=>4
[1,2,4,6,7,3,5]=>42
[1,2,4,6,7,5,3]=>5
[1,2,4,7,3,5,6]=>5
[1,2,4,7,3,6,5]=>4
[1,2,4,7,5,3,6]=>4
[1,2,4,7,5,6,3]=>3
[1,2,4,7,6,3,5]=>5
[1,2,4,7,6,5,3]=>42
[1,2,5,3,4,6,7]=>3
[1,2,5,3,4,7,6]=>42
[1,2,5,3,6,4,7]=>4
[1,2,5,3,6,7,4]=>5
[1,2,5,3,7,4,6]=>5
[1,2,5,3,7,6,4]=>4
[1,2,5,4,3,6,7]=>2
[1,2,5,4,3,7,6]=>24
[1,2,5,4,6,3,7]=>3
[1,2,5,4,6,7,3]=>4
[1,2,5,4,7,3,6]=>4
[1,2,5,4,7,6,3]=>3
[1,2,5,6,3,4,7]=>24
[1,2,5,6,3,7,4]=>42
[1,2,5,6,4,3,7]=>4
[1,2,5,6,4,7,3]=>5
[1,2,5,6,7,3,4]=>5
[1,2,5,6,7,4,3]=>42
[1,2,5,7,3,4,6]=>42
[1,2,5,7,3,6,4]=>24
[1,2,5,7,4,3,6]=>5
[1,2,5,7,4,6,3]=>4
[1,2,5,7,6,3,4]=>42
[1,2,5,7,6,4,3]=>5
[1,2,6,3,4,5,7]=>4
[1,2,6,3,4,7,5]=>5
[1,2,6,3,5,4,7]=>3
[1,2,6,3,5,7,4]=>4
[1,2,6,3,7,4,5]=>42
[1,2,6,3,7,5,4]=>5
[1,2,6,4,3,5,7]=>3
[1,2,6,4,3,7,5]=>4
[1,2,6,4,5,3,7]=>2
[1,2,6,4,5,7,3]=>3
[1,2,6,4,7,3,5]=>24
[1,2,6,4,7,5,3]=>4
[1,2,6,5,3,4,7]=>4
[1,2,6,5,3,7,4]=>5
[1,2,6,5,4,3,7]=>24
[1,2,6,5,4,7,3]=>42
[1,2,6,5,7,3,4]=>42
[1,2,6,5,7,4,3]=>5
[1,2,6,7,3,4,5]=>5
[1,2,6,7,3,5,4]=>42
[1,2,6,7,4,3,5]=>42
[1,2,6,7,4,5,3]=>5
[1,2,6,7,5,3,4]=>24
[1,2,6,7,5,4,3]=>4
[1,2,7,3,4,5,6]=>5
[1,2,7,3,4,6,5]=>4
[1,2,7,3,5,4,6]=>4
[1,2,7,3,5,6,4]=>3
[1,2,7,3,6,4,5]=>5
[1,2,7,3,6,5,4]=>42
[1,2,7,4,3,5,6]=>4
[1,2,7,4,3,6,5]=>3
[1,2,7,4,5,3,6]=>3
[1,2,7,4,5,6,3]=>2
[1,2,7,4,6,3,5]=>4
[1,2,7,4,6,5,3]=>24
[1,2,7,5,3,4,6]=>5
[1,2,7,5,3,6,4]=>4
[1,2,7,5,4,3,6]=>42
[1,2,7,5,4,6,3]=>24
[1,2,7,5,6,3,4]=>5
[1,2,7,5,6,4,3]=>42
[1,2,7,6,3,4,5]=>42
[1,2,7,6,3,5,4]=>5
[1,2,7,6,4,3,5]=>5
[1,2,7,6,4,5,3]=>42
[1,2,7,6,5,3,4]=>4
[1,2,7,6,5,4,3]=>24
[1,3,2,4,5,6,7]=>2
[1,3,2,4,5,7,6]=>24
[1,3,2,4,6,5,7]=>24
[1,3,2,4,6,7,5]=>42
[1,3,2,4,7,5,6]=>42
[1,3,2,4,7,6,5]=>24
[1,3,2,5,4,6,7]=>24
[1,3,2,5,4,7,6]=>576
[1,3,2,5,6,4,7]=>42
[1,3,2,5,6,7,4]=>64
[1,3,2,5,7,4,6]=>64
[1,3,2,5,7,6,4]=>42
[1,3,2,6,4,5,7]=>42
[1,3,2,6,4,7,5]=>64
[1,3,2,6,5,4,7]=>24
[1,3,2,6,5,7,4]=>42
[1,3,2,6,7,4,5]=>576
[1,3,2,6,7,5,4]=>64
[1,3,2,7,4,5,6]=>64
[1,3,2,7,4,6,5]=>42
[1,3,2,7,5,4,6]=>42
[1,3,2,7,5,6,4]=>24
[1,3,2,7,6,4,5]=>64
[1,3,2,7,6,5,4]=>576
[1,3,4,2,5,6,7]=>3
[1,3,4,2,5,7,6]=>42
[1,3,4,2,6,5,7]=>42
[1,3,4,2,6,7,5]=>72
[1,3,4,2,7,5,6]=>72
[1,3,4,2,7,6,5]=>42
[1,3,4,5,2,6,7]=>4
[1,3,4,5,2,7,6]=>64
[1,3,4,5,6,2,7]=>5
[1,3,4,5,6,7,2]=>6
[1,3,4,5,7,2,6]=>6
[1,3,4,5,7,6,2]=>5
[1,3,4,6,2,5,7]=>5
[1,3,4,6,2,7,5]=>6
[1,3,4,6,5,2,7]=>4
[1,3,4,6,5,7,2]=>5
[1,3,4,6,7,2,5]=>64
[1,3,4,6,7,5,2]=>6
[1,3,4,7,2,5,6]=>6
[1,3,4,7,2,6,5]=>5
[1,3,4,7,5,2,6]=>5
[1,3,4,7,5,6,2]=>4
[1,3,4,7,6,2,5]=>6
[1,3,4,7,6,5,2]=>64
[1,3,5,2,4,6,7]=>4
[1,3,5,2,4,7,6]=>64
[1,3,5,2,6,4,7]=>5
[1,3,5,2,6,7,4]=>6
[1,3,5,2,7,4,6]=>6
[1,3,5,2,7,6,4]=>5
[1,3,5,4,2,6,7]=>3
[1,3,5,4,2,7,6]=>42
[1,3,5,4,6,2,7]=>4
[1,3,5,4,6,7,2]=>5
[1,3,5,4,7,2,6]=>5
[1,3,5,4,7,6,2]=>4
[1,3,5,6,2,4,7]=>42
[1,3,5,6,2,7,4]=>72
[1,3,5,6,4,2,7]=>5
[1,3,5,6,4,7,2]=>6
[1,3,5,6,7,2,4]=>6
[1,3,5,6,7,4,2]=>64
[1,3,5,7,2,4,6]=>72
[1,3,5,7,2,6,4]=>42
[1,3,5,7,4,2,6]=>6
[1,3,5,7,4,6,2]=>5
[1,3,5,7,6,2,4]=>64
[1,3,5,7,6,4,2]=>6
[1,3,6,2,4,5,7]=>5
[1,3,6,2,4,7,5]=>6
[1,3,6,2,5,4,7]=>4
[1,3,6,2,5,7,4]=>5
[1,3,6,2,7,4,5]=>64
[1,3,6,2,7,5,4]=>6
[1,3,6,4,2,5,7]=>4
[1,3,6,4,2,7,5]=>5
[1,3,6,4,5,2,7]=>3
[1,3,6,4,5,7,2]=>4
[1,3,6,4,7,2,5]=>42
[1,3,6,4,7,5,2]=>5
[1,3,6,5,2,4,7]=>5
[1,3,6,5,2,7,4]=>6
[1,3,6,5,4,2,7]=>42
[1,3,6,5,4,7,2]=>64
[1,3,6,5,7,2,4]=>72
[1,3,6,5,7,4,2]=>6
[1,3,6,7,2,4,5]=>6
[1,3,6,7,2,5,4]=>64
[1,3,6,7,4,2,5]=>72
[1,3,6,7,4,5,2]=>6
[1,3,6,7,5,2,4]=>42
[1,3,6,7,5,4,2]=>5
[1,3,7,2,4,5,6]=>6
[1,3,7,2,4,6,5]=>5
[1,3,7,2,5,4,6]=>5
[1,3,7,2,5,6,4]=>4
[1,3,7,2,6,4,5]=>6
[1,3,7,2,6,5,4]=>64
[1,3,7,4,2,5,6]=>5
[1,3,7,4,2,6,5]=>4
[1,3,7,4,5,2,6]=>4
[1,3,7,4,5,6,2]=>3
[1,3,7,4,6,2,5]=>5
[1,3,7,4,6,5,2]=>42
[1,3,7,5,2,4,6]=>6
[1,3,7,5,2,6,4]=>5
[1,3,7,5,4,2,6]=>64
[1,3,7,5,4,6,2]=>42
[1,3,7,5,6,2,4]=>6
[1,3,7,5,6,4,2]=>72
[1,3,7,6,2,4,5]=>64
[1,3,7,6,2,5,4]=>6
[1,3,7,6,4,2,5]=>6
[1,3,7,6,4,5,2]=>72
[1,3,7,6,5,2,4]=>5
[1,3,7,6,5,4,2]=>42
[1,4,2,3,5,6,7]=>3
[1,4,2,3,5,7,6]=>42
[1,4,2,3,6,5,7]=>42
[1,4,2,3,6,7,5]=>72
[1,4,2,3,7,5,6]=>72
[1,4,2,3,7,6,5]=>42
[1,4,2,5,3,6,7]=>4
[1,4,2,5,3,7,6]=>64
[1,4,2,5,6,3,7]=>5
[1,4,2,5,6,7,3]=>6
[1,4,2,5,7,3,6]=>6
[1,4,2,5,7,6,3]=>5
[1,4,2,6,3,5,7]=>5
[1,4,2,6,3,7,5]=>6
[1,4,2,6,5,3,7]=>4
[1,4,2,6,5,7,3]=>5
[1,4,2,6,7,3,5]=>64
[1,4,2,6,7,5,3]=>6
[1,4,2,7,3,5,6]=>6
[1,4,2,7,3,6,5]=>5
[1,4,2,7,5,3,6]=>5
[1,4,2,7,5,6,3]=>4
[1,4,2,7,6,3,5]=>6
[1,4,2,7,6,5,3]=>64
[1,4,3,2,5,6,7]=>2
[1,4,3,2,5,7,6]=>24
[1,4,3,2,6,5,7]=>24
[1,4,3,2,6,7,5]=>42
[1,4,3,2,7,5,6]=>42
[1,4,3,2,7,6,5]=>24
[1,4,3,5,2,6,7]=>3
[1,4,3,5,2,7,6]=>42
[1,4,3,5,6,2,7]=>4
[1,4,3,5,6,7,2]=>5
[1,4,3,5,7,2,6]=>5
[1,4,3,5,7,6,2]=>4
[1,4,3,6,2,5,7]=>4
[1,4,3,6,2,7,5]=>5
[1,4,3,6,5,2,7]=>3
[1,4,3,6,5,7,2]=>4
[1,4,3,6,7,2,5]=>42
[1,4,3,6,7,5,2]=>5
[1,4,3,7,2,5,6]=>5
[1,4,3,7,2,6,5]=>4
[1,4,3,7,5,2,6]=>4
[1,4,3,7,5,6,2]=>3
[1,4,3,7,6,2,5]=>5
[1,4,3,7,6,5,2]=>42
[1,4,5,2,3,6,7]=>24
[1,4,5,2,3,7,6]=>576
[1,4,5,2,6,3,7]=>42
[1,4,5,2,6,7,3]=>64
[1,4,5,2,7,3,6]=>64
[1,4,5,2,7,6,3]=>42
[1,4,5,3,2,6,7]=>4
[1,4,5,3,2,7,6]=>64
[1,4,5,3,6,2,7]=>5
[1,4,5,3,6,7,2]=>6
[1,4,5,3,7,2,6]=>6
[1,4,5,3,7,6,2]=>5
[1,4,5,6,2,3,7]=>5
[1,4,5,6,2,7,3]=>6
[1,4,5,6,3,2,7]=>42
[1,4,5,6,3,7,2]=>64
[1,4,5,6,7,2,3]=>72
[1,4,5,6,7,3,2]=>6
[1,4,5,7,2,3,6]=>6
[1,4,5,7,2,6,3]=>5
[1,4,5,7,3,2,6]=>64
[1,4,5,7,3,6,2]=>42
[1,4,5,7,6,2,3]=>6
[1,4,5,7,6,3,2]=>72
[1,4,6,2,3,5,7]=>42
[1,4,6,2,3,7,5]=>64
[1,4,6,2,5,3,7]=>24
[1,4,6,2,5,7,3]=>42
[1,4,6,2,7,3,5]=>576
[1,4,6,2,7,5,3]=>64
[1,4,6,3,2,5,7]=>5
[1,4,6,3,2,7,5]=>6
[1,4,6,3,5,2,7]=>4
[1,4,6,3,5,7,2]=>5
[1,4,6,3,7,2,5]=>64
[1,4,6,3,7,5,2]=>6
[1,4,6,5,2,3,7]=>42
[1,4,6,5,2,7,3]=>72
[1,4,6,5,3,2,7]=>5
[1,4,6,5,3,7,2]=>6
                    

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Description

The number of minimal length factorizations of a permutation into star transpositions.
For a permutation $\pi\in\mathfrak S_n$ a minimal length factorization into star transpositions is a factorization of the form
$$\pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n,$$
where $\tau_a = (1,a)$ for $2 \leq a \leq n$ and $k$ is minimal.
[1, lem.2.1] shows that the minimal length of such a factorization 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 $1$, see St001077The prefix exchange distance of a permutation..
[2, cor.2] shows that the number of such minimal factorizations is
$$ \frac{(n+m-2(k+1))!}{(n-k)!}\ell_1\cdots\ell_m, $$
where $\ell_1,\dots,\ell_m$ is the cycle type of $\pi$ and $k$ is the number of fixed point different from $1$.

References

[1] Irving, J., Rattan, A. Minimal factorizations of permutations into star transpositions MathSciNet:2721480

Code

def statistic(pi):
    L = pi.cycle_type()
    k = pi.number_of_fixed_points()
    if pi(1) == 1:
        k -= 1
    m = len(L)    
    n = len(pi)
    return factorial(n+m-2*(k+1))/factorial(n-k)*prod(L)

# alternative code for checking
@cached_function
def number_of_minimal_star_factorizations_dict(n):
    S = Permutations(n)
    g = [S(Permutation((1, i))) for i in range(2, n+1)]

    result = dict()
    result[S(Permutation([]))] = 1
    l = 1 # the length of the minimal factorization
    while len(result) < factorial(n):
        result_l = dict()
        for w in cartesian_product([g]*l):
            pi = prod(w)
            if pi not in result:
                result_l[pi] = result_l.get(pi, 0) + 1
        for pi, v in result_l.items():
            result[pi] = v
        l += 1
    return result

def statistic(pi):
    d = number_of_minimal_star_factorizations_dict(len(pi))
    return d[pi]

Created

Jan 09, 2018 at 21:52 by Martin Rubey

Updated

Jan 09, 2018 at 21:52 by Martin Rubey