edit this statistic or download as text // json
Identifier
Values
=>
[1]=>0 [1,2]=>0 [2,1]=>0 [1,2,3]=>0 [1,3,2]=>0 [2,1,3]=>1 [2,3,1]=>0 [3,1,2]=>0 [3,2,1]=>0 [1,2,3,4]=>0 [1,2,4,3]=>0 [1,3,2,4]=>1 [1,3,4,2]=>0 [1,4,2,3]=>0 [1,4,3,2]=>0 [2,1,3,4]=>2 [2,1,4,3]=>2 [2,3,1,4]=>1 [2,3,4,1]=>0 [2,4,1,3]=>0 [2,4,3,1]=>0 [3,1,2,4]=>2 [3,1,4,2]=>1 [3,2,1,4]=>1 [3,2,4,1]=>1 [3,4,1,2]=>0 [3,4,2,1]=>0 [4,1,2,3]=>1 [4,1,3,2]=>0 [4,2,1,3]=>0 [4,2,3,1]=>0 [4,3,1,2]=>0 [4,3,2,1]=>0 [1,2,3,4,5]=>0 [1,2,3,5,4]=>0 [1,2,4,3,5]=>1 [1,2,4,5,3]=>0 [1,2,5,3,4]=>0 [1,2,5,4,3]=>0 [1,3,2,4,5]=>2 [1,3,2,5,4]=>2 [1,3,4,2,5]=>1 [1,3,4,5,2]=>0 [1,3,5,2,4]=>0 [1,3,5,4,2]=>0 [1,4,2,3,5]=>2 [1,4,2,5,3]=>1 [1,4,3,2,5]=>1 [1,4,3,5,2]=>1 [1,4,5,2,3]=>0 [1,4,5,3,2]=>0 [1,5,2,3,4]=>1 [1,5,2,4,3]=>0 [1,5,3,2,4]=>0 [1,5,3,4,2]=>0 [1,5,4,2,3]=>0 [1,5,4,3,2]=>0 [2,1,3,4,5]=>3 [2,1,3,5,4]=>3 [2,1,4,3,5]=>4 [2,1,4,5,3]=>3 [2,1,5,3,4]=>3 [2,1,5,4,3]=>3 [2,3,1,4,5]=>2 [2,3,1,5,4]=>2 [2,3,4,1,5]=>1 [2,3,4,5,1]=>0 [2,3,5,1,4]=>0 [2,3,5,4,1]=>0 [2,4,1,3,5]=>2 [2,4,1,5,3]=>1 [2,4,3,1,5]=>1 [2,4,3,5,1]=>1 [2,4,5,1,3]=>0 [2,4,5,3,1]=>0 [2,5,1,3,4]=>1 [2,5,1,4,3]=>0 [2,5,3,1,4]=>0 [2,5,3,4,1]=>0 [2,5,4,1,3]=>0 [2,5,4,3,1]=>0 [3,1,2,4,5]=>4 [3,1,2,5,4]=>4 [3,1,4,2,5]=>3 [3,1,4,5,2]=>2 [3,1,5,2,4]=>2 [3,1,5,4,2]=>2 [3,2,1,4,5]=>2 [3,2,1,5,4]=>2 [3,2,4,1,5]=>3 [3,2,4,5,1]=>2 [3,2,5,1,4]=>2 [3,2,5,4,1]=>2 [3,4,1,2,5]=>2 [3,4,1,5,2]=>1 [3,4,2,1,5]=>1 [3,4,2,5,1]=>1 [3,4,5,1,2]=>0 [3,4,5,2,1]=>0 [3,5,1,2,4]=>1 [3,5,1,4,2]=>0 [3,5,2,1,4]=>0 [3,5,2,4,1]=>0 [3,5,4,1,2]=>0 [3,5,4,2,1]=>0 [4,1,2,3,5]=>4 [4,1,2,5,3]=>3 [4,1,3,2,5]=>2 [4,1,3,5,2]=>2 [4,1,5,2,3]=>1 [4,1,5,3,2]=>1 [4,2,1,3,5]=>2 [4,2,1,5,3]=>1 [4,2,3,1,5]=>2 [4,2,3,5,1]=>2 [4,2,5,1,3]=>1 [4,2,5,3,1]=>1 [4,3,1,2,5]=>2 [4,3,1,5,2]=>1 [4,3,2,1,5]=>1 [4,3,2,5,1]=>1 [4,3,5,1,2]=>1 [4,3,5,2,1]=>1 [4,5,1,2,3]=>1 [4,5,1,3,2]=>0 [4,5,2,1,3]=>0 [4,5,2,3,1]=>0 [4,5,3,1,2]=>0 [4,5,3,2,1]=>0 [5,1,2,3,4]=>3 [5,1,2,4,3]=>2 [5,1,3,2,4]=>1 [5,1,3,4,2]=>1 [5,1,4,2,3]=>0 [5,1,4,3,2]=>0 [5,2,1,3,4]=>1 [5,2,1,4,3]=>0 [5,2,3,1,4]=>1 [5,2,3,4,1]=>1 [5,2,4,1,3]=>0 [5,2,4,3,1]=>0 [5,3,1,2,4]=>1 [5,3,1,4,2]=>0 [5,3,2,1,4]=>0 [5,3,2,4,1]=>0 [5,3,4,1,2]=>0 [5,3,4,2,1]=>0 [5,4,1,2,3]=>1 [5,4,1,3,2]=>0 [5,4,2,1,3]=>0 [5,4,2,3,1]=>0 [5,4,3,1,2]=>0 [5,4,3,2,1]=>0 [1,2,3,4,5,6]=>0 [1,2,3,4,6,5]=>0 [1,2,3,5,4,6]=>1 [1,2,3,5,6,4]=>0 [1,2,3,6,4,5]=>0 [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]=>1 [1,2,4,5,6,3]=>0 [1,2,4,6,3,5]=>0 [1,2,4,6,5,3]=>0 [1,2,5,3,4,6]=>2 [1,2,5,3,6,4]=>1 [1,2,5,4,3,6]=>1 [1,2,5,4,6,3]=>1 [1,2,5,6,3,4]=>0 [1,2,5,6,4,3]=>0 [1,2,6,3,4,5]=>1 [1,2,6,3,5,4]=>0 [1,2,6,4,3,5]=>0 [1,2,6,4,5,3]=>0 [1,2,6,5,3,4]=>0 [1,2,6,5,4,3]=>0 [1,3,2,4,5,6]=>3 [1,3,2,4,6,5]=>3 [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]=>3 [1,3,4,2,5,6]=>2 [1,3,4,2,6,5]=>2 [1,3,4,5,2,6]=>1 [1,3,4,5,6,2]=>0 [1,3,4,6,2,5]=>0 [1,3,4,6,5,2]=>0 [1,3,5,2,4,6]=>2 [1,3,5,2,6,4]=>1 [1,3,5,4,2,6]=>1 [1,3,5,4,6,2]=>1 [1,3,5,6,2,4]=>0 [1,3,5,6,4,2]=>0 [1,3,6,2,4,5]=>1 [1,3,6,2,5,4]=>0 [1,3,6,4,2,5]=>0 [1,3,6,4,5,2]=>0 [1,3,6,5,2,4]=>0 [1,3,6,5,4,2]=>0 [1,4,2,3,5,6]=>4 [1,4,2,3,6,5]=>4 [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]=>2 [1,4,3,2,5,6]=>2 [1,4,3,2,6,5]=>2 [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]=>2 [1,4,5,2,3,6]=>2 [1,4,5,2,6,3]=>1 [1,4,5,3,2,6]=>1 [1,4,5,3,6,2]=>1 [1,4,5,6,2,3]=>0 [1,4,5,6,3,2]=>0 [1,4,6,2,3,5]=>1 [1,4,6,2,5,3]=>0 [1,4,6,3,2,5]=>0 [1,4,6,3,5,2]=>0 [1,4,6,5,2,3]=>0 [1,4,6,5,3,2]=>0 [1,5,2,3,4,6]=>4 [1,5,2,3,6,4]=>3 [1,5,2,4,3,6]=>2 [1,5,2,4,6,3]=>2 [1,5,2,6,3,4]=>1 [1,5,2,6,4,3]=>1 [1,5,3,2,4,6]=>2 [1,5,3,2,6,4]=>1 [1,5,3,4,2,6]=>2 [1,5,3,4,6,2]=>2 [1,5,3,6,2,4]=>1 [1,5,3,6,4,2]=>1 [1,5,4,2,3,6]=>2 [1,5,4,2,6,3]=>1 [1,5,4,3,2,6]=>1 [1,5,4,3,6,2]=>1 [1,5,4,6,2,3]=>1 [1,5,4,6,3,2]=>1 [1,5,6,2,3,4]=>1 [1,5,6,2,4,3]=>0 [1,5,6,3,2,4]=>0 [1,5,6,3,4,2]=>0 [1,5,6,4,2,3]=>0 [1,5,6,4,3,2]=>0 [1,6,2,3,4,5]=>3 [1,6,2,3,5,4]=>2 [1,6,2,4,3,5]=>1 [1,6,2,4,5,3]=>1 [1,6,2,5,3,4]=>0 [1,6,2,5,4,3]=>0 [1,6,3,2,4,5]=>1 [1,6,3,2,5,4]=>0 [1,6,3,4,2,5]=>1 [1,6,3,4,5,2]=>1 [1,6,3,5,2,4]=>0 [1,6,3,5,4,2]=>0 [1,6,4,2,3,5]=>1 [1,6,4,2,5,3]=>0 [1,6,4,3,2,5]=>0 [1,6,4,3,5,2]=>0 [1,6,4,5,2,3]=>0 [1,6,4,5,3,2]=>0 [1,6,5,2,3,4]=>1 [1,6,5,2,4,3]=>0 [1,6,5,3,2,4]=>0 [1,6,5,3,4,2]=>0 [1,6,5,4,2,3]=>0 [1,6,5,4,3,2]=>0 [2,1,3,4,5,6]=>4 [2,1,3,4,6,5]=>4 [2,1,3,5,4,6]=>5 [2,1,3,5,6,4]=>4 [2,1,3,6,4,5]=>4 [2,1,3,6,5,4]=>4 [2,1,4,3,5,6]=>6 [2,1,4,3,6,5]=>6 [2,1,4,5,3,6]=>5 [2,1,4,5,6,3]=>4 [2,1,4,6,3,5]=>4 [2,1,4,6,5,3]=>4 [2,1,5,3,4,6]=>6 [2,1,5,3,6,4]=>5 [2,1,5,4,3,6]=>5 [2,1,5,4,6,3]=>5 [2,1,5,6,3,4]=>4 [2,1,5,6,4,3]=>4 [2,1,6,3,4,5]=>5 [2,1,6,3,5,4]=>4 [2,1,6,4,3,5]=>4 [2,1,6,4,5,3]=>4 [2,1,6,5,3,4]=>4 [2,1,6,5,4,3]=>4 [2,3,1,4,5,6]=>3 [2,3,1,4,6,5]=>3 [2,3,1,5,4,6]=>4 [2,3,1,5,6,4]=>3 [2,3,1,6,4,5]=>3 [2,3,1,6,5,4]=>3 [2,3,4,1,5,6]=>2 [2,3,4,1,6,5]=>2 [2,3,4,5,1,6]=>1 [2,3,4,5,6,1]=>0 [2,3,4,6,1,5]=>0 [2,3,4,6,5,1]=>0 [2,3,5,1,4,6]=>2 [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]=>0 [2,3,5,6,4,1]=>0 [2,3,6,1,4,5]=>1 [2,3,6,1,5,4]=>0 [2,3,6,4,1,5]=>0 [2,3,6,4,5,1]=>0 [2,3,6,5,1,4]=>0 [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]=>2 [2,4,1,6,3,5]=>2 [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]=>2 [2,4,3,6,1,5]=>2 [2,4,3,6,5,1]=>2 [2,4,5,1,3,6]=>2 [2,4,5,1,6,3]=>1 [2,4,5,3,1,6]=>1 [2,4,5,3,6,1]=>1 [2,4,5,6,1,3]=>0 [2,4,5,6,3,1]=>0 [2,4,6,1,3,5]=>1 [2,4,6,1,5,3]=>0 [2,4,6,3,1,5]=>0 [2,4,6,3,5,1]=>0 [2,4,6,5,1,3]=>0 [2,4,6,5,3,1]=>0 [2,5,1,3,4,6]=>4 [2,5,1,3,6,4]=>3 [2,5,1,4,3,6]=>2 [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]=>2 [2,5,3,1,6,4]=>1 [2,5,3,4,1,6]=>2 [2,5,3,4,6,1]=>2 [2,5,3,6,1,4]=>1 [2,5,3,6,4,1]=>1 [2,5,4,1,3,6]=>2 [2,5,4,1,6,3]=>1 [2,5,4,3,1,6]=>1 [2,5,4,3,6,1]=>1 [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]=>0 [2,5,6,3,1,4]=>0 [2,5,6,3,4,1]=>0 [2,5,6,4,1,3]=>0 [2,5,6,4,3,1]=>0 [2,6,1,3,4,5]=>3 [2,6,1,3,5,4]=>2 [2,6,1,4,3,5]=>1 [2,6,1,4,5,3]=>1 [2,6,1,5,3,4]=>0 [2,6,1,5,4,3]=>0 [2,6,3,1,4,5]=>1 [2,6,3,1,5,4]=>0 [2,6,3,4,1,5]=>1 [2,6,3,4,5,1]=>1 [2,6,3,5,1,4]=>0 [2,6,3,5,4,1]=>0 [2,6,4,1,3,5]=>1 [2,6,4,1,5,3]=>0 [2,6,4,3,1,5]=>0 [2,6,4,3,5,1]=>0 [2,6,4,5,1,3]=>0 [2,6,4,5,3,1]=>0 [2,6,5,1,3,4]=>1 [2,6,5,1,4,3]=>0 [2,6,5,3,1,4]=>0 [2,6,5,3,4,1]=>0 [2,6,5,4,1,3]=>0 [2,6,5,4,3,1]=>0 [3,1,2,4,5,6]=>6 [3,1,2,4,6,5]=>6 [3,1,2,5,4,6]=>7 [3,1,2,5,6,4]=>6 [3,1,2,6,4,5]=>6 [3,1,2,6,5,4]=>6 [3,1,4,2,5,6]=>5 [3,1,4,2,6,5]=>5 [3,1,4,5,2,6]=>4 [3,1,4,5,6,2]=>3 [3,1,4,6,2,5]=>3 [3,1,4,6,5,2]=>3 [3,1,5,2,4,6]=>5 [3,1,5,2,6,4]=>4 [3,1,5,4,2,6]=>4 [3,1,5,4,6,2]=>4 [3,1,5,6,2,4]=>3 [3,1,5,6,4,2]=>3 [3,1,6,2,4,5]=>4 [3,1,6,2,5,4]=>3 [3,1,6,4,2,5]=>3 [3,1,6,4,5,2]=>3 [3,1,6,5,2,4]=>3 [3,1,6,5,4,2]=>3 [3,2,1,4,5,6]=>3 [3,2,1,4,6,5]=>3 [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]=>3 [3,2,4,1,5,6]=>5 [3,2,4,1,6,5]=>5 [3,2,4,5,1,6]=>4 [3,2,4,5,6,1]=>3 [3,2,4,6,1,5]=>3 [3,2,4,6,5,1]=>3 [3,2,5,1,4,6]=>5 [3,2,5,1,6,4]=>4 [3,2,5,4,1,6]=>4 [3,2,5,4,6,1]=>4 [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]=>3 [3,2,6,4,1,5]=>3 [3,2,6,4,5,1]=>3 [3,2,6,5,1,4]=>3 [3,2,6,5,4,1]=>3 [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]=>2 [3,4,1,6,5,2]=>2 [3,4,2,1,5,6]=>2 [3,4,2,1,6,5]=>2 [3,4,2,5,1,6]=>3 [3,4,2,5,6,1]=>2 [3,4,2,6,1,5]=>2 [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]=>1 [3,4,5,2,6,1]=>1 [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]=>0 [3,4,6,2,1,5]=>0 [3,4,6,2,5,1]=>0 [3,4,6,5,1,2]=>0 [3,4,6,5,2,1]=>0 [3,5,1,2,4,6]=>4 [3,5,1,2,6,4]=>3 [3,5,1,4,2,6]=>2 [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]=>2 [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]=>1 [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]=>0 [3,5,6,2,1,4]=>0 [3,5,6,2,4,1]=>0 [3,5,6,4,1,2]=>0 [3,5,6,4,2,1]=>0 [3,6,1,2,4,5]=>3 [3,6,1,2,5,4]=>2 [3,6,1,4,2,5]=>1 [3,6,1,4,5,2]=>1 [3,6,1,5,2,4]=>0 [3,6,1,5,4,2]=>0 [3,6,2,1,4,5]=>1 [3,6,2,1,5,4]=>0 [3,6,2,4,1,5]=>1 [3,6,2,4,5,1]=>1 [3,6,2,5,1,4]=>0 [3,6,2,5,4,1]=>0 [3,6,4,1,2,5]=>1 [3,6,4,1,5,2]=>0 [3,6,4,2,1,5]=>0 [3,6,4,2,5,1]=>0 [3,6,4,5,1,2]=>0 [3,6,4,5,2,1]=>0 [3,6,5,1,2,4]=>1 [3,6,5,1,4,2]=>0 [3,6,5,2,1,4]=>0 [3,6,5,2,4,1]=>0 [3,6,5,4,1,2]=>0 [3,6,5,4,2,1]=>0 [4,1,2,3,5,6]=>7 [4,1,2,3,6,5]=>7 [4,1,2,5,3,6]=>6 [4,1,2,5,6,3]=>5 [4,1,2,6,3,5]=>5 [4,1,2,6,5,3]=>5 [4,1,3,2,5,6]=>4 [4,1,3,2,6,5]=>4 [4,1,3,5,2,6]=>5 [4,1,3,5,6,2]=>4 [4,1,3,6,2,5]=>4 [4,1,3,6,5,2]=>4 [4,1,5,2,3,6]=>4 [4,1,5,2,6,3]=>3 [4,1,5,3,2,6]=>3 [4,1,5,3,6,2]=>3 [4,1,5,6,2,3]=>2 [4,1,5,6,3,2]=>2 [4,1,6,2,3,5]=>3 [4,1,6,2,5,3]=>2 [4,1,6,3,2,5]=>2 [4,1,6,3,5,2]=>2 [4,1,6,5,2,3]=>2 [4,1,6,5,3,2]=>2 [4,2,1,3,5,6]=>4 [4,2,1,3,6,5]=>4 [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]=>2 [4,2,3,1,5,6]=>4 [4,2,3,1,6,5]=>4 [4,2,3,5,1,6]=>5 [4,2,3,5,6,1]=>4 [4,2,3,6,1,5]=>4 [4,2,3,6,5,1]=>4 [4,2,5,1,3,6]=>4 [4,2,5,1,6,3]=>3 [4,2,5,3,1,6]=>3 [4,2,5,3,6,1]=>3 [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]=>2 [4,2,6,3,1,5]=>2 [4,2,6,3,5,1]=>2 [4,2,6,5,1,3]=>2 [4,2,6,5,3,1]=>2 [4,3,1,2,5,6]=>4 [4,3,1,2,6,5]=>4 [4,3,1,5,2,6]=>3 [4,3,1,5,6,2]=>2 [4,3,1,6,2,5]=>2 [4,3,1,6,5,2]=>2 [4,3,2,1,5,6]=>2 [4,3,2,1,6,5]=>2 [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]=>2 [4,3,5,1,2,6]=>4 [4,3,5,1,6,2]=>3 [4,3,5,2,1,6]=>3 [4,3,5,2,6,1]=>3 [4,3,5,6,1,2]=>2 [4,3,5,6,2,1]=>2 [4,3,6,1,2,5]=>3 [4,3,6,1,5,2]=>2 [4,3,6,2,1,5]=>2 [4,3,6,2,5,1]=>2 [4,3,6,5,1,2]=>2 [4,3,6,5,2,1]=>2 [4,5,1,2,3,6]=>4 [4,5,1,2,6,3]=>3 [4,5,1,3,2,6]=>2 [4,5,1,3,6,2]=>2 [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]=>2 [4,5,2,6,1,3]=>1 [4,5,2,6,3,1]=>1 [4,5,3,1,2,6]=>2 [4,5,3,1,6,2]=>1 [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]=>1 [4,5,6,1,2,3]=>1 [4,5,6,1,3,2]=>0 [4,5,6,2,1,3]=>0 [4,5,6,2,3,1]=>0 [4,5,6,3,1,2]=>0 [4,5,6,3,2,1]=>0 [4,6,1,2,3,5]=>3 [4,6,1,2,5,3]=>2 [4,6,1,3,2,5]=>1 [4,6,1,3,5,2]=>1 [4,6,1,5,2,3]=>0 [4,6,1,5,3,2]=>0 [4,6,2,1,3,5]=>1 [4,6,2,1,5,3]=>0 [4,6,2,3,1,5]=>1 [4,6,2,3,5,1]=>1 [4,6,2,5,1,3]=>0 [4,6,2,5,3,1]=>0 [4,6,3,1,2,5]=>1 [4,6,3,1,5,2]=>0 [4,6,3,2,1,5]=>0 [4,6,3,2,5,1]=>0 [4,6,3,5,1,2]=>0 [4,6,3,5,2,1]=>0 [4,6,5,1,2,3]=>1 [4,6,5,1,3,2]=>0 [4,6,5,2,1,3]=>0 [4,6,5,2,3,1]=>0 [4,6,5,3,1,2]=>0 [4,6,5,3,2,1]=>0 [5,1,2,3,4,6]=>7 [5,1,2,3,6,4]=>6 [5,1,2,4,3,6]=>5 [5,1,2,4,6,3]=>5 [5,1,2,6,3,4]=>4 [5,1,2,6,4,3]=>4 [5,1,3,2,4,6]=>4 [5,1,3,2,6,4]=>3 [5,1,3,4,2,6]=>4 [5,1,3,4,6,2]=>4 [5,1,3,6,2,4]=>3 [5,1,3,6,4,2]=>3 [5,1,4,2,3,6]=>3 [5,1,4,2,6,3]=>2 [5,1,4,3,2,6]=>2 [5,1,4,3,6,2]=>2 [5,1,4,6,2,3]=>2 [5,1,4,6,3,2]=>2 [5,1,6,2,3,4]=>2 [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]=>1 [5,1,6,4,3,2]=>1 [5,2,1,3,4,6]=>4 [5,2,1,3,6,4]=>3 [5,2,1,4,3,6]=>2 [5,2,1,4,6,3]=>2 [5,2,1,6,3,4]=>1 [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]=>4 [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]=>2 [5,2,4,3,6,1]=>2 [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]=>1 [5,2,6,3,1,4]=>1 [5,2,6,3,4,1]=>1 [5,2,6,4,1,3]=>1 [5,2,6,4,3,1]=>1 [5,3,1,2,4,6]=>4 [5,3,1,2,6,4]=>3 [5,3,1,4,2,6]=>2 [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]=>2 [5,3,2,1,6,4]=>1 [5,3,2,4,1,6]=>2 [5,3,2,4,6,1]=>2 [5,3,2,6,1,4]=>1 [5,3,2,6,4,1]=>1 [5,3,4,1,2,6]=>3 [5,3,4,1,6,2]=>2 [5,3,4,2,1,6]=>2 [5,3,4,2,6,1]=>2 [5,3,4,6,1,2]=>2 [5,3,4,6,2,1]=>2 [5,3,6,1,2,4]=>2 [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]=>1 [5,3,6,4,2,1]=>1 [5,4,1,2,3,6]=>4 [5,4,1,2,6,3]=>3 [5,4,1,3,2,6]=>2 [5,4,1,3,6,2]=>2 [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]=>2 [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]=>2 [5,4,3,1,6,2]=>1 [5,4,3,2,1,6]=>1 [5,4,3,2,6,1]=>1 [5,4,3,6,1,2]=>1 [5,4,3,6,2,1]=>1 [5,4,6,1,2,3]=>2 [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]=>3 [5,6,1,2,4,3]=>2 [5,6,1,3,2,4]=>1 [5,6,1,3,4,2]=>1 [5,6,1,4,2,3]=>0 [5,6,1,4,3,2]=>0 [5,6,2,1,3,4]=>1 [5,6,2,1,4,3]=>0 [5,6,2,3,1,4]=>1 [5,6,2,3,4,1]=>1 [5,6,2,4,1,3]=>0 [5,6,2,4,3,1]=>0 [5,6,3,1,2,4]=>1 [5,6,3,1,4,2]=>0 [5,6,3,2,1,4]=>0 [5,6,3,2,4,1]=>0 [5,6,3,4,1,2]=>0 [5,6,3,4,2,1]=>0 [5,6,4,1,2,3]=>1 [5,6,4,1,3,2]=>0 [5,6,4,2,1,3]=>0 [5,6,4,2,3,1]=>0 [5,6,4,3,1,2]=>0 [5,6,4,3,2,1]=>0 [6,1,2,3,4,5]=>6 [6,1,2,3,5,4]=>5 [6,1,2,4,3,5]=>4 [6,1,2,4,5,3]=>4 [6,1,2,5,3,4]=>3 [6,1,2,5,4,3]=>3 [6,1,3,2,4,5]=>3 [6,1,3,2,5,4]=>2 [6,1,3,4,2,5]=>3 [6,1,3,4,5,2]=>3 [6,1,3,5,2,4]=>2 [6,1,3,5,4,2]=>2 [6,1,4,2,3,5]=>2 [6,1,4,2,5,3]=>1 [6,1,4,3,2,5]=>1 [6,1,4,3,5,2]=>1 [6,1,4,5,2,3]=>1 [6,1,4,5,3,2]=>1 [6,1,5,2,3,4]=>1 [6,1,5,2,4,3]=>0 [6,1,5,3,2,4]=>0 [6,1,5,3,4,2]=>0 [6,1,5,4,2,3]=>0 [6,1,5,4,3,2]=>0 [6,2,1,3,4,5]=>3 [6,2,1,3,5,4]=>2 [6,2,1,4,3,5]=>1 [6,2,1,4,5,3]=>1 [6,2,1,5,3,4]=>0 [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]=>3 [6,2,3,4,5,1]=>3 [6,2,3,5,1,4]=>2 [6,2,3,5,4,1]=>2 [6,2,4,1,3,5]=>2 [6,2,4,1,5,3]=>1 [6,2,4,3,1,5]=>1 [6,2,4,3,5,1]=>1 [6,2,4,5,1,3]=>1 [6,2,4,5,3,1]=>1 [6,2,5,1,3,4]=>1 [6,2,5,1,4,3]=>0 [6,2,5,3,1,4]=>0 [6,2,5,3,4,1]=>0 [6,2,5,4,1,3]=>0 [6,2,5,4,3,1]=>0 [6,3,1,2,4,5]=>3 [6,3,1,2,5,4]=>2 [6,3,1,4,2,5]=>1 [6,3,1,4,5,2]=>1 [6,3,1,5,2,4]=>0 [6,3,1,5,4,2]=>0 [6,3,2,1,4,5]=>1 [6,3,2,1,5,4]=>0 [6,3,2,4,1,5]=>1 [6,3,2,4,5,1]=>1 [6,3,2,5,1,4]=>0 [6,3,2,5,4,1]=>0 [6,3,4,1,2,5]=>2 [6,3,4,1,5,2]=>1 [6,3,4,2,1,5]=>1 [6,3,4,2,5,1]=>1 [6,3,4,5,1,2]=>1 [6,3,4,5,2,1]=>1 [6,3,5,1,2,4]=>1 [6,3,5,1,4,2]=>0 [6,3,5,2,1,4]=>0 [6,3,5,2,4,1]=>0 [6,3,5,4,1,2]=>0 [6,3,5,4,2,1]=>0 [6,4,1,2,3,5]=>3 [6,4,1,2,5,3]=>2 [6,4,1,3,2,5]=>1 [6,4,1,3,5,2]=>1 [6,4,1,5,2,3]=>0 [6,4,1,5,3,2]=>0 [6,4,2,1,3,5]=>1 [6,4,2,1,5,3]=>0 [6,4,2,3,1,5]=>1 [6,4,2,3,5,1]=>1 [6,4,2,5,1,3]=>0 [6,4,2,5,3,1]=>0 [6,4,3,1,2,5]=>1 [6,4,3,1,5,2]=>0 [6,4,3,2,1,5]=>0 [6,4,3,2,5,1]=>0 [6,4,3,5,1,2]=>0 [6,4,3,5,2,1]=>0 [6,4,5,1,2,3]=>1 [6,4,5,1,3,2]=>0 [6,4,5,2,1,3]=>0 [6,4,5,2,3,1]=>0 [6,4,5,3,1,2]=>0 [6,4,5,3,2,1]=>0 [6,5,1,2,3,4]=>3 [6,5,1,2,4,3]=>2 [6,5,1,3,2,4]=>1 [6,5,1,3,4,2]=>1 [6,5,1,4,2,3]=>0 [6,5,1,4,3,2]=>0 [6,5,2,1,3,4]=>1 [6,5,2,1,4,3]=>0 [6,5,2,3,1,4]=>1 [6,5,2,3,4,1]=>1 [6,5,2,4,1,3]=>0 [6,5,2,4,3,1]=>0 [6,5,3,1,2,4]=>1 [6,5,3,1,4,2]=>0 [6,5,3,2,1,4]=>0 [6,5,3,2,4,1]=>0 [6,5,3,4,1,2]=>0 [6,5,3,4,2,1]=>0 [6,5,4,1,2,3]=>1 [6,5,4,1,3,2]=>0 [6,5,4,2,1,3]=>0 [6,5,4,2,3,1]=>0 [6,5,4,3,1,2]=>0 [6,5,4,3,2,1]=>0 [1,2,3,4,5,6,7]=>0 [1,2,3,4,5,7,6]=>0 [1,2,3,4,6,5,7]=>1 [1,2,3,4,6,7,5]=>0 [1,2,3,4,7,5,6]=>0 [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]=>1 [1,2,3,5,6,7,4]=>0 [1,2,3,5,7,4,6]=>0 [1,2,3,5,7,6,4]=>0 [1,2,3,6,4,5,7]=>2 [1,2,3,6,4,7,5]=>1 [1,2,3,6,5,4,7]=>1 [1,2,3,6,5,7,4]=>1 [1,2,3,6,7,4,5]=>0 [1,2,3,6,7,5,4]=>0 [1,2,3,7,4,5,6]=>1 [1,2,3,7,4,6,5]=>0 [1,2,3,7,5,4,6]=>0 [1,2,3,7,5,6,4]=>0 [1,2,3,7,6,4,5]=>0 [1,2,3,7,6,5,4]=>0 [1,2,4,3,5,6,7]=>3 [1,2,4,3,5,7,6]=>3 [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]=>3 [1,2,4,5,3,6,7]=>2 [1,2,4,5,3,7,6]=>2 [1,2,4,5,6,3,7]=>1 [1,2,4,5,6,7,3]=>0 [1,2,4,5,7,3,6]=>0 [1,2,4,5,7,6,3]=>0 [1,2,4,6,3,5,7]=>2 [1,2,4,6,3,7,5]=>1 [1,2,4,6,5,3,7]=>1 [1,2,4,6,5,7,3]=>1 [1,2,4,6,7,3,5]=>0 [1,2,4,6,7,5,3]=>0 [1,2,4,7,3,5,6]=>1 [1,2,4,7,3,6,5]=>0 [1,2,4,7,5,3,6]=>0 [1,2,4,7,5,6,3]=>0 [1,2,4,7,6,3,5]=>0 [1,2,4,7,6,5,3]=>0 [1,2,5,3,4,6,7]=>4 [1,2,5,3,4,7,6]=>4 [1,2,5,3,6,4,7]=>3 [1,2,5,3,6,7,4]=>2 [1,2,5,3,7,4,6]=>2 [1,2,5,3,7,6,4]=>2 [1,2,5,4,3,6,7]=>2 [1,2,5,4,3,7,6]=>2 [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]=>2 [1,2,5,6,3,4,7]=>2 [1,2,5,6,3,7,4]=>1 [1,2,5,6,4,3,7]=>1 [1,2,5,6,4,7,3]=>1 [1,2,5,6,7,3,4]=>0 [1,2,5,6,7,4,3]=>0 [1,2,5,7,3,4,6]=>1 [1,2,5,7,3,6,4]=>0 [1,2,5,7,4,3,6]=>0 [1,2,5,7,4,6,3]=>0 [1,2,5,7,6,3,4]=>0 [1,2,5,7,6,4,3]=>0 [1,2,6,3,4,5,7]=>4 [1,2,6,3,4,7,5]=>3 [1,2,6,3,5,4,7]=>2 [1,2,6,3,5,7,4]=>2 [1,2,6,3,7,4,5]=>1 [1,2,6,3,7,5,4]=>1 [1,2,6,4,3,5,7]=>2 [1,2,6,4,3,7,5]=>1 [1,2,6,4,5,3,7]=>2 [1,2,6,4,5,7,3]=>2 [1,2,6,4,7,3,5]=>1 [1,2,6,4,7,5,3]=>1 [1,2,6,5,3,4,7]=>2 [1,2,6,5,3,7,4]=>1 [1,2,6,5,4,3,7]=>1 [1,2,6,5,4,7,3]=>1 [1,2,6,5,7,3,4]=>1 [1,2,6,5,7,4,3]=>1 [1,2,6,7,3,4,5]=>1 [1,2,6,7,3,5,4]=>0 [1,2,6,7,4,3,5]=>0 [1,2,6,7,4,5,3]=>0 [1,2,6,7,5,3,4]=>0 [1,2,6,7,5,4,3]=>0 [1,2,7,3,4,5,6]=>3 [1,2,7,3,4,6,5]=>2 [1,2,7,3,5,4,6]=>1 [1,2,7,3,5,6,4]=>1 [1,2,7,3,6,4,5]=>0 [1,2,7,3,6,5,4]=>0 [1,2,7,4,3,5,6]=>1 [1,2,7,4,3,6,5]=>0 [1,2,7,4,5,3,6]=>1 [1,2,7,4,5,6,3]=>1 [1,2,7,4,6,3,5]=>0 [1,2,7,4,6,5,3]=>0 [1,2,7,5,3,4,6]=>1 [1,2,7,5,3,6,4]=>0 [1,2,7,5,4,3,6]=>0 [1,2,7,5,4,6,3]=>0 [1,2,7,5,6,3,4]=>0 [1,2,7,5,6,4,3]=>0 [1,2,7,6,3,4,5]=>1 [1,2,7,6,3,5,4]=>0 [1,2,7,6,4,3,5]=>0 [1,2,7,6,4,5,3]=>0 [1,2,7,6,5,3,4]=>0 [1,2,7,6,5,4,3]=>0 [1,3,2,4,5,6,7]=>4 [1,3,2,4,5,7,6]=>4 [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]=>4 [1,3,2,5,4,6,7]=>6 [1,3,2,5,4,7,6]=>6 [1,3,2,5,6,4,7]=>5 [1,3,2,5,6,7,4]=>4 [1,3,2,5,7,4,6]=>4 [1,3,2,5,7,6,4]=>4 [1,3,2,6,4,5,7]=>6 [1,3,2,6,4,7,5]=>5 [1,3,2,6,5,4,7]=>5 [1,3,2,6,5,7,4]=>5 [1,3,2,6,7,4,5]=>4 [1,3,2,6,7,5,4]=>4 [1,3,2,7,4,5,6]=>5 [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]=>4 [1,3,2,7,6,5,4]=>4 [1,3,4,2,5,6,7]=>3 [1,3,4,2,5,7,6]=>3 [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]=>3 [1,3,4,5,2,6,7]=>2 [1,3,4,5,2,7,6]=>2 [1,3,4,5,6,2,7]=>1 [1,3,4,5,6,7,2]=>0 [1,3,4,5,7,2,6]=>0 [1,3,4,5,7,6,2]=>0 [1,3,4,6,2,5,7]=>2 [1,3,4,6,2,7,5]=>1 [1,3,4,6,5,2,7]=>1 [1,3,4,6,5,7,2]=>1 [1,3,4,6,7,2,5]=>0 [1,3,4,6,7,5,2]=>0 [1,3,4,7,2,5,6]=>1 [1,3,4,7,2,6,5]=>0 [1,3,4,7,5,2,6]=>0 [1,3,4,7,5,6,2]=>0 [1,3,4,7,6,2,5]=>0 [1,3,4,7,6,5,2]=>0 [1,3,5,2,4,6,7]=>4 [1,3,5,2,4,7,6]=>4 [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]=>2 [1,3,5,4,2,6,7]=>2 [1,3,5,4,2,7,6]=>2 [1,3,5,4,6,2,7]=>3 [1,3,5,4,6,7,2]=>2 [1,3,5,4,7,2,6]=>2 [1,3,5,4,7,6,2]=>2 [1,3,5,6,2,4,7]=>2 [1,3,5,6,2,7,4]=>1 [1,3,5,6,4,2,7]=>1 [1,3,5,6,4,7,2]=>1 [1,3,5,6,7,2,4]=>0 [1,3,5,6,7,4,2]=>0 [1,3,5,7,2,4,6]=>1 [1,3,5,7,2,6,4]=>0 [1,3,5,7,4,2,6]=>0 [1,3,5,7,4,6,2]=>0 [1,3,5,7,6,2,4]=>0 [1,3,5,7,6,4,2]=>0 [1,3,6,2,4,5,7]=>4 [1,3,6,2,4,7,5]=>3 [1,3,6,2,5,4,7]=>2 [1,3,6,2,5,7,4]=>2 [1,3,6,2,7,4,5]=>1 [1,3,6,2,7,5,4]=>1 [1,3,6,4,2,5,7]=>2 [1,3,6,4,2,7,5]=>1 [1,3,6,4,5,2,7]=>2 [1,3,6,4,5,7,2]=>2 [1,3,6,4,7,2,5]=>1 [1,3,6,4,7,5,2]=>1 [1,3,6,5,2,4,7]=>2 [1,3,6,5,2,7,4]=>1 [1,3,6,5,4,2,7]=>1 [1,3,6,5,4,7,2]=>1 [1,3,6,5,7,2,4]=>1 [1,3,6,5,7,4,2]=>1 [1,3,6,7,2,4,5]=>1 [1,3,6,7,2,5,4]=>0 [1,3,6,7,4,2,5]=>0 [1,3,6,7,4,5,2]=>0 [1,3,6,7,5,2,4]=>0 [1,3,6,7,5,4,2]=>0 [1,3,7,2,4,5,6]=>3 [1,3,7,2,4,6,5]=>2 [1,3,7,2,5,4,6]=>1 [1,3,7,2,5,6,4]=>1 [1,3,7,2,6,4,5]=>0 [1,3,7,2,6,5,4]=>0 [1,3,7,4,2,5,6]=>1 [1,3,7,4,2,6,5]=>0 [1,3,7,4,5,2,6]=>1 [1,3,7,4,5,6,2]=>1 [1,3,7,4,6,2,5]=>0 [1,3,7,4,6,5,2]=>0 [1,3,7,5,2,4,6]=>1 [1,3,7,5,2,6,4]=>0 [1,3,7,5,4,2,6]=>0 [1,3,7,5,4,6,2]=>0 [1,3,7,5,6,2,4]=>0 [1,3,7,5,6,4,2]=>0 [1,3,7,6,2,4,5]=>1 [1,3,7,6,2,5,4]=>0 [1,3,7,6,4,2,5]=>0 [1,3,7,6,4,5,2]=>0 [1,3,7,6,5,2,4]=>0 [1,3,7,6,5,4,2]=>0 [1,4,2,3,5,6,7]=>6 [1,4,2,3,5,7,6]=>6 [1,4,2,3,6,5,7]=>7 [1,4,2,3,6,7,5]=>6 [1,4,2,3,7,5,6]=>6 [1,4,2,3,7,6,5]=>6 [1,4,2,5,3,6,7]=>5 [1,4,2,5,3,7,6]=>5 [1,4,2,5,6,3,7]=>4 [1,4,2,5,6,7,3]=>3 [1,4,2,5,7,3,6]=>3 [1,4,2,5,7,6,3]=>3 [1,4,2,6,3,5,7]=>5 [1,4,2,6,3,7,5]=>4 [1,4,2,6,5,3,7]=>4 [1,4,2,6,5,7,3]=>4 [1,4,2,6,7,3,5]=>3 [1,4,2,6,7,5,3]=>3 [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]=>3 [1,4,3,2,5,6,7]=>3 [1,4,3,2,5,7,6]=>3 [1,4,3,2,6,5,7]=>4 [1,4,3,2,6,7,5]=>3 [1,4,3,2,7,5,6]=>3 [1,4,3,2,7,6,5]=>3 [1,4,3,5,2,6,7]=>5 [1,4,3,5,2,7,6]=>5 [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]=>3 [1,4,3,6,2,5,7]=>5 [1,4,3,6,2,7,5]=>4 [1,4,3,6,5,2,7]=>4 [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]=>4 [1,4,3,7,2,6,5]=>3 [1,4,3,7,5,2,6]=>3 [1,4,3,7,5,6,2]=>3 [1,4,3,7,6,2,5]=>3 [1,4,3,7,6,5,2]=>3 [1,4,5,2,3,6,7]=>4 [1,4,5,2,3,7,6]=>4 [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]=>2 [1,4,5,3,2,6,7]=>2 [1,4,5,3,2,7,6]=>2 [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]=>2 [1,4,5,6,2,3,7]=>2 [1,4,5,6,2,7,3]=>1 [1,4,5,6,3,2,7]=>1 [1,4,5,6,3,7,2]=>1 [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]=>0 [1,4,5,7,3,2,6]=>0 [1,4,5,7,3,6,2]=>0 [1,4,5,7,6,2,3]=>0 [1,4,5,7,6,3,2]=>0 [1,4,6,2,3,5,7]=>4 [1,4,6,2,3,7,5]=>3 [1,4,6,2,5,3,7]=>2 [1,4,6,2,5,7,3]=>2 [1,4,6,2,7,3,5]=>1 [1,4,6,2,7,5,3]=>1 [1,4,6,3,2,5,7]=>2 [1,4,6,3,2,7,5]=>1 [1,4,6,3,5,2,7]=>2 [1,4,6,3,5,7,2]=>2 [1,4,6,3,7,2,5]=>1 [1,4,6,3,7,5,2]=>1 [1,4,6,5,2,3,7]=>2 [1,4,6,5,2,7,3]=>1 [1,4,6,5,3,2,7]=>1
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Description
The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
References
[1] Berman, Y., Tenner, B. E. Pattern-functions, statistics, and shallow permutations arXiv:2110.11146
Code
def vincular_occurrences_iterator(perm, pat, columns):
    perm = Permutation(perm)    
    for pos in perm.pattern_positions(pat):
        if all(pos[i-1]+1 == pos[i] for i in columns):
            yield tuple(pos)

from sage.combinat.permutation import to_standard
def arrow_occurrences_iterator(perm, pat, columns, arrows):
    perm = Permutation(perm)
    s = set(pat + [p for p, _ in arrows] + [q for _, q in arrows])
    k = max(s)
    assert min(s) == 1 and len(s) == k
    nu = to_standard(pat)
    sigma = perm.fundamental_transformation_inverse()
    for pos in vincular_occurrences_iterator(perm, nu, columns):
        x = [None]*k
        for i, a in enumerate(pos):
            x[pat[i]-1] = perm[a]
        for p, q in arrows:
            if x[p-1] is None and x[q-1] is None:
                raise ValueError("doesn't work for %s %s %s %s" % (perm, pat, columns, arrows))
            if x[p-1] is None:
                x[p-1] = sigma.inverse()(x[q-1])
            elif x[q-1] is None:
                x[q-1] = sigma(x[p-1])
        if (all(x[i] < x[i+1] for i in range(len(x)-1))
            and all(sigma(x[p-1]) == x[q-1] for p, q in arrows)):
            yield tuple(pos)

def arrow_occurrences(perm, pat, columns, arrows):
    return list(arrow_occurrences_iterator(perm, pat, columns, arrows))

def statistic(pi):
    return len(arrow_occurrences(pi, [1,3], [], [[1,2]]))
Created
Oct 23, 2021 at 00:39 by Martin Rubey
Updated
Oct 23, 2021 at 00:39 by Martin Rubey