edit this statistic or download as text // json
Identifier
Values
=>
[]=>0 [1]=>0 [1,2]=>0 [2,1]=>0 [1,2,3]=>0 [1,3,2]=>0 [2,1,3]=>0 [2,3,1]=>0 [3,1,2]=>0 [3,2,1]=>1 [1,2,3,4]=>0 [1,2,4,3]=>0 [1,3,2,4]=>0 [1,3,4,2]=>0 [1,4,2,3]=>0 [1,4,3,2]=>1 [2,1,3,4]=>0 [2,1,4,3]=>0 [2,3,1,4]=>0 [2,3,4,1]=>0 [2,4,1,3]=>0 [2,4,3,1]=>1 [3,1,2,4]=>0 [3,1,4,2]=>0 [3,2,1,4]=>1 [3,2,4,1]=>1 [3,4,1,2]=>1 [3,4,2,1]=>2 [4,1,2,3]=>0 [4,1,3,2]=>1 [4,2,1,3]=>1 [4,2,3,1]=>2 [4,3,1,2]=>2 [4,3,2,1]=>2 [1,2,3,4,5]=>0 [1,2,3,5,4]=>0 [1,2,4,3,5]=>0 [1,2,4,5,3]=>0 [1,2,5,3,4]=>0 [1,2,5,4,3]=>1 [1,3,2,4,5]=>0 [1,3,2,5,4]=>0 [1,3,4,2,5]=>0 [1,3,4,5,2]=>0 [1,3,5,2,4]=>0 [1,3,5,4,2]=>1 [1,4,2,3,5]=>0 [1,4,2,5,3]=>0 [1,4,3,2,5]=>1 [1,4,3,5,2]=>1 [1,4,5,2,3]=>1 [1,4,5,3,2]=>2 [1,5,2,3,4]=>0 [1,5,2,4,3]=>1 [1,5,3,2,4]=>1 [1,5,3,4,2]=>2 [1,5,4,2,3]=>2 [1,5,4,3,2]=>2 [2,1,3,4,5]=>0 [2,1,3,5,4]=>0 [2,1,4,3,5]=>0 [2,1,4,5,3]=>0 [2,1,5,3,4]=>0 [2,1,5,4,3]=>1 [2,3,1,4,5]=>0 [2,3,1,5,4]=>0 [2,3,4,1,5]=>0 [2,3,4,5,1]=>0 [2,3,5,1,4]=>0 [2,3,5,4,1]=>1 [2,4,1,3,5]=>0 [2,4,1,5,3]=>0 [2,4,3,1,5]=>1 [2,4,3,5,1]=>1 [2,4,5,1,3]=>1 [2,4,5,3,1]=>2 [2,5,1,3,4]=>0 [2,5,1,4,3]=>1 [2,5,3,1,4]=>1 [2,5,3,4,1]=>2 [2,5,4,1,3]=>2 [2,5,4,3,1]=>2 [3,1,2,4,5]=>0 [3,1,2,5,4]=>0 [3,1,4,2,5]=>0 [3,1,4,5,2]=>0 [3,1,5,2,4]=>0 [3,1,5,4,2]=>1 [3,2,1,4,5]=>1 [3,2,1,5,4]=>1 [3,2,4,1,5]=>1 [3,2,4,5,1]=>1 [3,2,5,1,4]=>1 [3,2,5,4,1]=>2 [3,4,1,2,5]=>1 [3,4,1,5,2]=>1 [3,4,2,1,5]=>2 [3,4,2,5,1]=>2 [3,4,5,1,2]=>1 [3,4,5,2,1]=>2 [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]=>2 [3,5,4,2,1]=>3 [4,1,2,3,5]=>0 [4,1,2,5,3]=>0 [4,1,3,2,5]=>1 [4,1,3,5,2]=>1 [4,1,5,2,3]=>1 [4,1,5,3,2]=>2 [4,2,1,3,5]=>1 [4,2,1,5,3]=>1 [4,2,3,1,5]=>2 [4,2,3,5,1]=>2 [4,2,5,1,3]=>2 [4,2,5,3,1]=>3 [4,3,1,2,5]=>2 [4,3,1,5,2]=>2 [4,3,2,1,5]=>2 [4,3,2,5,1]=>2 [4,3,5,1,2]=>2 [4,3,5,2,1]=>3 [4,5,1,2,3]=>1 [4,5,1,3,2]=>2 [4,5,2,1,3]=>2 [4,5,2,3,1]=>3 [4,5,3,1,2]=>3 [4,5,3,2,1]=>4 [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]=>2 [5,2,1,3,4]=>1 [5,2,1,4,3]=>2 [5,2,3,1,4]=>2 [5,2,3,4,1]=>3 [5,2,4,1,3]=>3 [5,2,4,3,1]=>3 [5,3,1,2,4]=>2 [5,3,1,4,2]=>3 [5,3,2,1,4]=>2 [5,3,2,4,1]=>3 [5,3,4,1,2]=>3 [5,3,4,2,1]=>4 [5,4,1,2,3]=>2 [5,4,1,3,2]=>3 [5,4,2,1,3]=>3 [5,4,2,3,1]=>4 [5,4,3,1,2]=>4 [5,4,3,2,1]=>5 [1,2,3,4,5,6]=>0 [1,2,3,4,6,5]=>0 [1,2,3,5,4,6]=>0 [1,2,3,5,6,4]=>0 [1,2,3,6,4,5]=>0 [1,2,3,6,5,4]=>1 [1,2,4,3,5,6]=>0 [1,2,4,3,6,5]=>0 [1,2,4,5,3,6]=>0 [1,2,4,5,6,3]=>0 [1,2,4,6,3,5]=>0 [1,2,4,6,5,3]=>1 [1,2,5,3,4,6]=>0 [1,2,5,3,6,4]=>0 [1,2,5,4,3,6]=>1 [1,2,5,4,6,3]=>1 [1,2,5,6,3,4]=>1 [1,2,5,6,4,3]=>2 [1,2,6,3,4,5]=>0 [1,2,6,3,5,4]=>1 [1,2,6,4,3,5]=>1 [1,2,6,4,5,3]=>2 [1,2,6,5,3,4]=>2 [1,2,6,5,4,3]=>2 [1,3,2,4,5,6]=>0 [1,3,2,4,6,5]=>0 [1,3,2,5,4,6]=>0 [1,3,2,5,6,4]=>0 [1,3,2,6,4,5]=>0 [1,3,2,6,5,4]=>1 [1,3,4,2,5,6]=>0 [1,3,4,2,6,5]=>0 [1,3,4,5,2,6]=>0 [1,3,4,5,6,2]=>0 [1,3,4,6,2,5]=>0 [1,3,4,6,5,2]=>1 [1,3,5,2,4,6]=>0 [1,3,5,2,6,4]=>0 [1,3,5,4,2,6]=>1 [1,3,5,4,6,2]=>1 [1,3,5,6,2,4]=>1 [1,3,5,6,4,2]=>2 [1,3,6,2,4,5]=>0 [1,3,6,2,5,4]=>1 [1,3,6,4,2,5]=>1 [1,3,6,4,5,2]=>2 [1,3,6,5,2,4]=>2 [1,3,6,5,4,2]=>2 [1,4,2,3,5,6]=>0 [1,4,2,3,6,5]=>0 [1,4,2,5,3,6]=>0 [1,4,2,5,6,3]=>0 [1,4,2,6,3,5]=>0 [1,4,2,6,5,3]=>1 [1,4,3,2,5,6]=>1 [1,4,3,2,6,5]=>1 [1,4,3,5,2,6]=>1 [1,4,3,5,6,2]=>1 [1,4,3,6,2,5]=>1 [1,4,3,6,5,2]=>2 [1,4,5,2,3,6]=>1 [1,4,5,2,6,3]=>1 [1,4,5,3,2,6]=>2 [1,4,5,3,6,2]=>2 [1,4,5,6,2,3]=>1 [1,4,5,6,3,2]=>2 [1,4,6,2,3,5]=>1 [1,4,6,2,5,3]=>2 [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]=>3 [1,5,2,3,4,6]=>0 [1,5,2,3,6,4]=>0 [1,5,2,4,3,6]=>1 [1,5,2,4,6,3]=>1 [1,5,2,6,3,4]=>1 [1,5,2,6,4,3]=>2 [1,5,3,2,4,6]=>1 [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]=>2 [1,5,3,6,4,2]=>3 [1,5,4,2,3,6]=>2 [1,5,4,2,6,3]=>2 [1,5,4,3,2,6]=>2 [1,5,4,3,6,2]=>2 [1,5,4,6,2,3]=>2 [1,5,4,6,3,2]=>3 [1,5,6,2,3,4]=>1 [1,5,6,2,4,3]=>2 [1,5,6,3,2,4]=>2 [1,5,6,3,4,2]=>3 [1,5,6,4,2,3]=>3 [1,5,6,4,3,2]=>4 [1,6,2,3,4,5]=>0 [1,6,2,3,5,4]=>1 [1,6,2,4,3,5]=>1 [1,6,2,4,5,3]=>2 [1,6,2,5,3,4]=>2 [1,6,2,5,4,3]=>2 [1,6,3,2,4,5]=>1 [1,6,3,2,5,4]=>2 [1,6,3,4,2,5]=>2 [1,6,3,4,5,2]=>3 [1,6,3,5,2,4]=>3 [1,6,3,5,4,2]=>3 [1,6,4,2,3,5]=>2 [1,6,4,2,5,3]=>3 [1,6,4,3,2,5]=>2 [1,6,4,3,5,2]=>3 [1,6,4,5,2,3]=>3 [1,6,4,5,3,2]=>4 [1,6,5,2,3,4]=>2 [1,6,5,2,4,3]=>3 [1,6,5,3,2,4]=>3 [1,6,5,3,4,2]=>4 [1,6,5,4,2,3]=>4 [1,6,5,4,3,2]=>5 [2,1,3,4,5,6]=>0 [2,1,3,4,6,5]=>0 [2,1,3,5,4,6]=>0 [2,1,3,5,6,4]=>0 [2,1,3,6,4,5]=>0 [2,1,3,6,5,4]=>1 [2,1,4,3,5,6]=>0 [2,1,4,3,6,5]=>0 [2,1,4,5,3,6]=>0 [2,1,4,5,6,3]=>0 [2,1,4,6,3,5]=>0 [2,1,4,6,5,3]=>1 [2,1,5,3,4,6]=>0 [2,1,5,3,6,4]=>0 [2,1,5,4,3,6]=>1 [2,1,5,4,6,3]=>1 [2,1,5,6,3,4]=>1 [2,1,5,6,4,3]=>2 [2,1,6,3,4,5]=>0 [2,1,6,3,5,4]=>1 [2,1,6,4,3,5]=>1 [2,1,6,4,5,3]=>2 [2,1,6,5,3,4]=>2 [2,1,6,5,4,3]=>2 [2,3,1,4,5,6]=>0 [2,3,1,4,6,5]=>0 [2,3,1,5,4,6]=>0 [2,3,1,5,6,4]=>0 [2,3,1,6,4,5]=>0 [2,3,1,6,5,4]=>1 [2,3,4,1,5,6]=>0 [2,3,4,1,6,5]=>0 [2,3,4,5,1,6]=>0 [2,3,4,5,6,1]=>0 [2,3,4,6,1,5]=>0 [2,3,4,6,5,1]=>1 [2,3,5,1,4,6]=>0 [2,3,5,1,6,4]=>0 [2,3,5,4,1,6]=>1 [2,3,5,4,6,1]=>1 [2,3,5,6,1,4]=>1 [2,3,5,6,4,1]=>2 [2,3,6,1,4,5]=>0 [2,3,6,1,5,4]=>1 [2,3,6,4,1,5]=>1 [2,3,6,4,5,1]=>2 [2,3,6,5,1,4]=>2 [2,3,6,5,4,1]=>2 [2,4,1,3,5,6]=>0 [2,4,1,3,6,5]=>0 [2,4,1,5,3,6]=>0 [2,4,1,5,6,3]=>0 [2,4,1,6,3,5]=>0 [2,4,1,6,5,3]=>1 [2,4,3,1,5,6]=>1 [2,4,3,1,6,5]=>1 [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]=>2 [2,4,5,1,3,6]=>1 [2,4,5,1,6,3]=>1 [2,4,5,3,1,6]=>2 [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]=>1 [2,4,6,1,5,3]=>2 [2,4,6,3,1,5]=>2 [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]=>0 [2,5,1,3,6,4]=>0 [2,5,1,4,3,6]=>1 [2,5,1,6,3,4]=>1 [2,5,1,6,4,3]=>2 [2,5,3,1,4,6]=>1 [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]=>2 [2,5,3,6,4,1]=>3 [2,5,4,1,3,6]=>2 [2,5,4,1,6,3]=>2 [2,5,4,3,1,6]=>2 [2,5,4,3,6,1]=>2 [2,5,4,6,1,3]=>2 [2,5,4,6,3,1]=>3 [2,5,6,1,3,4]=>1 [2,5,6,1,4,3]=>2 [2,5,6,3,1,4]=>2 [2,5,6,3,4,1]=>3 [2,5,6,4,1,3]=>3 [2,5,6,4,3,1]=>4 [2,6,1,3,4,5]=>0 [2,6,1,3,5,4]=>1 [2,6,1,4,3,5]=>1 [2,6,1,4,5,3]=>2 [2,6,1,5,3,4]=>2 [2,6,1,5,4,3]=>2 [2,6,3,1,4,5]=>1 [2,6,3,4,1,5]=>2 [2,6,3,4,5,1]=>3 [2,6,3,5,1,4]=>3 [2,6,3,5,4,1]=>3 [2,6,4,1,3,5]=>2 [2,6,4,1,5,3]=>3 [2,6,4,3,5,1]=>3 [2,6,4,5,1,3]=>3 [2,6,4,5,3,1]=>4 [2,6,5,1,4,3]=>3 [2,6,5,3,1,4]=>3 [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]=>0 [3,1,2,4,6,5]=>0 [3,1,2,5,4,6]=>0 [3,1,2,5,6,4]=>0 [3,1,2,6,4,5]=>0 [3,1,2,6,5,4]=>1 [3,1,4,2,5,6]=>0 [3,1,4,2,6,5]=>0 [3,1,4,5,2,6]=>0 [3,1,4,5,6,2]=>0 [3,1,4,6,2,5]=>0 [3,1,4,6,5,2]=>1 [3,1,5,2,4,6]=>0 [3,1,5,2,6,4]=>0 [3,1,5,4,2,6]=>1 [3,1,5,4,6,2]=>1 [3,1,5,6,2,4]=>1 [3,1,5,6,4,2]=>2 [3,1,6,2,4,5]=>0 [3,1,6,2,5,4]=>1 [3,1,6,4,2,5]=>1 [3,1,6,4,5,2]=>2 [3,1,6,5,2,4]=>2 [3,1,6,5,4,2]=>2 [3,2,1,4,5,6]=>1 [3,2,1,4,6,5]=>1 [3,2,1,5,4,6]=>1 [3,2,1,5,6,4]=>1 [3,2,1,6,4,5]=>1 [3,2,1,6,5,4]=>2 [3,2,4,1,5,6]=>1 [3,2,4,1,6,5]=>1 [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]=>2 [3,2,5,1,4,6]=>1 [3,2,5,1,6,4]=>1 [3,2,5,4,1,6]=>2 [3,2,5,4,6,1]=>2 [3,2,5,6,1,4]=>2 [3,2,5,6,4,1]=>3 [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]=>3 [3,2,6,5,1,4]=>3 [3,2,6,5,4,1]=>3 [3,4,1,2,5,6]=>1 [3,4,1,2,6,5]=>1 [3,4,1,5,2,6]=>1 [3,4,1,5,6,2]=>1 [3,4,1,6,2,5]=>1 [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]=>2 [3,4,2,5,6,1]=>2 [3,4,2,6,1,5]=>2 [3,4,2,6,5,1]=>3 [3,4,5,1,2,6]=>1 [3,4,5,1,6,2]=>1 [3,4,5,2,1,6]=>2 [3,4,5,2,6,1]=>2 [3,4,5,6,1,2]=>2 [3,4,5,6,2,1]=>3 [3,4,6,1,2,5]=>1 [3,4,6,1,5,2]=>2 [3,4,6,2,1,5]=>2 [3,4,6,2,5,1]=>3 [3,4,6,5,1,2]=>3 [3,4,6,5,2,1]=>4 [3,5,1,2,4,6]=>1 [3,5,1,2,6,4]=>1 [3,5,1,4,2,6]=>2 [3,5,1,4,6,2]=>2 [3,5,1,6,2,4]=>2 [3,5,1,6,4,2]=>3 [3,5,2,1,4,6]=>2 [3,5,2,1,6,4]=>2 [3,5,2,4,1,6]=>3 [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]=>2 [3,5,4,1,6,2]=>2 [3,5,4,2,1,6]=>3 [3,5,4,2,6,1]=>3 [3,5,4,6,1,2]=>3 [3,5,4,6,2,1]=>4 [3,5,6,1,2,4]=>2 [3,5,6,1,4,2]=>3 [3,5,6,2,1,4]=>3 [3,5,6,2,4,1]=>4 [3,5,6,4,1,2]=>4 [3,5,6,4,2,1]=>5 [3,6,1,2,4,5]=>1 [3,6,1,2,5,4]=>2 [3,6,1,4,2,5]=>2 [3,6,1,4,5,2]=>3 [3,6,1,5,2,4]=>3 [3,6,1,5,4,2]=>3 [3,6,2,1,4,5]=>2 [3,6,2,1,5,4]=>3 [3,6,2,4,1,5]=>3 [3,6,2,4,5,1]=>4 [3,6,2,5,1,4]=>4 [3,6,2,5,4,1]=>4 [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]=>4 [3,6,4,5,1,2]=>4 [3,6,4,5,2,1]=>5 [3,6,5,1,2,4]=>3 [3,6,5,1,4,2]=>4 [3,6,5,2,1,4]=>4 [3,6,5,2,4,1]=>5 [3,6,5,4,1,2]=>5 [3,6,5,4,2,1]=>6 [4,1,2,3,5,6]=>0 [4,1,2,3,6,5]=>0 [4,1,2,5,3,6]=>0 [4,1,2,5,6,3]=>0 [4,1,2,6,3,5]=>0 [4,1,2,6,5,3]=>1 [4,1,3,2,5,6]=>1 [4,1,3,2,6,5]=>1 [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]=>2 [4,1,5,2,3,6]=>1 [4,1,5,2,6,3]=>1 [4,1,5,3,2,6]=>2 [4,1,5,3,6,2]=>2 [4,1,5,6,2,3]=>1 [4,1,5,6,3,2]=>2 [4,1,6,2,3,5]=>1 [4,1,6,2,5,3]=>2 [4,1,6,3,2,5]=>2 [4,1,6,3,5,2]=>3 [4,1,6,5,2,3]=>2 [4,1,6,5,3,2]=>3 [4,2,1,3,5,6]=>1 [4,2,1,3,6,5]=>1 [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]=>2 [4,2,3,1,5,6]=>2 [4,2,3,1,6,5]=>2 [4,2,3,5,1,6]=>2 [4,2,3,5,6,1]=>2 [4,2,3,6,1,5]=>2 [4,2,3,6,5,1]=>3 [4,2,5,1,3,6]=>2 [4,2,5,1,6,3]=>2 [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]=>3 [4,2,6,1,3,5]=>2 [4,2,6,1,5,3]=>3 [4,2,6,3,1,5]=>3 [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]=>2 [4,3,1,2,6,5]=>2 [4,3,1,5,2,6]=>2 [4,3,1,5,6,2]=>2 [4,3,1,6,2,5]=>2 [4,3,1,6,5,2]=>3 [4,3,2,1,5,6]=>2 [4,3,2,1,6,5]=>2 [4,3,2,5,1,6]=>2 [4,3,2,5,6,1]=>2 [4,3,2,6,1,5]=>2 [4,3,2,6,5,1]=>3 [4,3,5,1,2,6]=>2 [4,3,5,1,6,2]=>2 [4,3,5,2,1,6]=>3 [4,3,5,2,6,1]=>3 [4,3,5,6,1,2]=>3 [4,3,5,6,2,1]=>4 [4,3,6,1,2,5]=>2 [4,3,6,1,5,2]=>3 [4,3,6,2,1,5]=>3 [4,3,6,2,5,1]=>4 [4,3,6,5,1,2]=>4 [4,3,6,5,2,1]=>4 [4,5,1,2,3,6]=>1 [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]=>3 [4,5,2,1,3,6]=>2 [4,5,2,1,6,3]=>2 [4,5,2,3,1,6]=>3 [4,5,2,3,6,1]=>3 [4,5,2,6,1,3]=>3 [4,5,2,6,3,1]=>4 [4,5,3,1,2,6]=>3 [4,5,3,1,6,2]=>3 [4,5,3,2,1,6]=>4 [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]=>3 [4,5,6,1,3,2]=>4 [4,5,6,2,1,3]=>4 [4,5,6,2,3,1]=>5 [4,5,6,3,1,2]=>4 [4,5,6,3,2,1]=>5 [4,6,1,2,3,5]=>1 [4,6,1,2,5,3]=>2 [4,6,1,3,5,2]=>3 [4,6,1,5,2,3]=>3 [4,6,1,5,3,2]=>4 [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]=>4 [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]=>4 [4,6,3,2,1,5]=>4 [4,6,3,2,5,1]=>5 [4,6,3,5,1,2]=>5 [4,6,3,5,2,1]=>5 [4,6,5,1,2,3]=>4 [4,6,5,1,3,2]=>4 [4,6,5,2,1,3]=>5 [4,6,5,2,3,1]=>5 [4,6,5,3,1,2]=>5 [4,6,5,3,2,1]=>6 [5,1,2,3,4,6]=>0 [5,1,2,3,6,4]=>0 [5,1,2,4,3,6]=>1 [5,1,2,4,6,3]=>1 [5,1,2,6,3,4]=>1 [5,1,2,6,4,3]=>2 [5,1,3,2,4,6]=>1 [5,1,3,2,6,4]=>1 [5,1,3,4,2,6]=>2 [5,1,3,4,6,2]=>2 [5,1,3,6,2,4]=>2 [5,1,3,6,4,2]=>3 [5,1,4,2,3,6]=>2 [5,1,4,2,6,3]=>2 [5,1,4,3,2,6]=>2 [5,1,4,6,3,2]=>3 [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]=>1 [5,2,1,3,6,4]=>1 [5,2,1,4,3,6]=>2 [5,2,1,4,6,3]=>2 [5,2,1,6,3,4]=>2 [5,2,1,6,4,3]=>3 [5,2,3,1,4,6]=>2 [5,2,3,1,6,4]=>2 [5,2,3,4,1,6]=>3 [5,2,3,4,6,1]=>3 [5,2,3,6,1,4]=>3 [5,2,3,6,4,1]=>4 [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]=>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]=>4 [5,2,6,4,1,3]=>4 [5,2,6,4,3,1]=>5 [5,3,1,2,4,6]=>2 [5,3,1,2,6,4]=>2 [5,3,1,4,2,6]=>3 [5,3,1,4,6,2]=>3 [5,3,1,6,2,4]=>3 [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]=>3 [5,3,2,6,1,4]=>3 [5,3,2,6,4,1]=>4 [5,3,4,1,2,6]=>3 [5,3,4,1,6,2]=>3 [5,3,4,2,1,6]=>4 [5,3,4,2,6,1]=>4 [5,3,4,6,1,2]=>4 [5,3,4,6,2,1]=>5 [5,3,6,1,2,4]=>3 [5,3,6,1,4,2]=>4 [5,3,6,2,1,4]=>4 [5,3,6,2,4,1]=>4 [5,3,6,4,1,2]=>5 [5,3,6,4,2,1]=>5 [5,4,1,2,3,6]=>2 [5,4,1,2,6,3]=>2 [5,4,1,3,2,6]=>3 [5,4,1,6,2,3]=>3 [5,4,1,6,3,2]=>4 [5,4,2,1,3,6]=>3 [5,4,2,1,6,3]=>3 [5,4,2,3,1,6]=>4 [5,4,2,3,6,1]=>4 [5,4,2,6,1,3]=>4 [5,4,2,6,3,1]=>5 [5,4,3,1,2,6]=>4 [5,4,3,1,6,2]=>4 [5,4,3,2,1,6]=>5 [5,4,3,2,6,1]=>5 [5,4,3,6,1,2]=>5 [5,4,3,6,2,1]=>6 [5,4,6,1,2,3]=>4 [5,4,6,1,3,2]=>5 [5,4,6,2,1,3]=>4 [5,4,6,2,3,1]=>5 [5,4,6,3,1,2]=>5 [5,4,6,3,2,1]=>6 [5,6,1,2,3,4]=>2 [5,6,1,2,4,3]=>3 [5,6,1,3,2,4]=>3 [5,6,1,3,4,2]=>4 [5,6,1,4,2,3]=>4 [5,6,1,4,3,2]=>5 [5,6,2,1,3,4]=>3 [5,6,2,1,4,3]=>4 [5,6,2,3,1,4]=>4 [5,6,2,3,4,1]=>4 [5,6,2,4,1,3]=>5 [5,6,2,4,3,1]=>5 [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]=>5 [5,6,3,4,1,2]=>6 [5,6,3,4,2,1]=>6 [5,6,4,1,2,3]=>4 [5,6,4,1,3,2]=>5 [5,6,4,2,1,3]=>5 [5,6,4,2,3,1]=>6 [5,6,4,3,1,2]=>6 [5,6,4,3,2,1]=>7 [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]=>2 [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]=>3 [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]=>2 [6,1,4,3,5,2]=>3 [6,1,4,5,2,3]=>3 [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]=>3 [6,2,3,1,4,5]=>2 [6,2,3,1,5,4]=>3 [6,2,3,4,1,5]=>3 [6,2,3,4,5,1]=>4 [6,2,3,5,1,4]=>4 [6,2,3,5,4,1]=>4 [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]=>4 [6,2,4,5,1,3]=>4 [6,2,4,5,3,1]=>5 [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]=>5 [6,2,5,4,1,3]=>5 [6,2,5,4,3,1]=>6 [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]=>4 [6,3,1,5,2,4]=>4 [6,3,1,5,4,2]=>4 [6,3,2,1,4,5]=>2 [6,3,2,1,5,4]=>3 [6,3,2,4,1,5]=>3 [6,3,2,4,5,1]=>4 [6,3,2,5,1,4]=>4 [6,3,2,5,4,1]=>4 [6,3,4,1,2,5]=>3 [6,3,4,1,5,2]=>4 [6,3,4,2,1,5]=>4 [6,3,4,2,5,1]=>5 [6,3,4,5,1,2]=>4 [6,3,4,5,2,1]=>5 [6,3,5,1,2,4]=>4 [6,3,5,1,4,2]=>4 [6,3,5,2,1,4]=>5 [6,3,5,2,4,1]=>5 [6,3,5,4,1,2]=>5 [6,3,5,4,2,1]=>6 [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]=>4 [6,4,1,5,3,2]=>5 [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]=>5 [6,4,2,5,1,3]=>4 [6,4,2,5,3,1]=>5 [6,4,3,1,2,5]=>4 [6,4,3,1,5,2]=>5 [6,4,3,2,1,5]=>5 [6,4,3,2,5,1]=>6 [6,4,3,5,1,2]=>5 [6,4,3,5,2,1]=>6 [6,4,5,1,2,3]=>5 [6,4,5,1,3,2]=>5 [6,4,5,2,1,3]=>5 [6,4,5,2,3,1]=>6 [6,4,5,3,1,2]=>6 [6,4,5,3,2,1]=>7 [6,5,1,2,3,4]=>3 [6,5,1,2,4,3]=>4 [6,5,1,3,2,4]=>4 [6,5,1,3,4,2]=>5 [6,5,1,4,2,3]=>5 [6,5,1,4,3,2]=>6 [6,5,2,1,3,4]=>4 [6,5,2,1,4,3]=>4 [6,5,2,3,1,4]=>5 [6,5,2,3,4,1]=>5 [6,5,2,4,1,3]=>5 [6,5,2,4,3,1]=>6 [6,5,3,1,2,4]=>5 [6,5,3,1,4,2]=>5 [6,5,3,2,1,4]=>6
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Description
The maximum defect over any reduced expression for a permutation and any subexpression.
References
[1] Deodhar, V. V. A combinatorial setting for questions in Kazhdan-Lusztig theory MathSciNet:1065215
[2] Billey, S. C., Warrington, G. S. Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations MathSciNet:1826948
[3] Jones, B., Woo, A. Mask formulas for cograssmannian Kazhdan-Lusztig polynomials arXiv:1011.1110
Code
# WARNING: takes a long time even for permutations of 5.

def defect_set(sigma, w):
    """
    INPUT:

    - w is a reduced word for a permutation
    - sigma is the mask, given as a binary word of length l(w)

    OUTPUT:
    
    - the defect set, that is the set of positions j such that
    l(w^{\sigma[j-1]} w_j) < l(w^{\sigma[j-1]}) 

    """
    res = []
    S = Permutations(max(w)+1)
    w_S = [S.simple_reflection(e) for e in w]
    w_up_to_j = S([])
    for j in range(len(w)-1):
        if sigma[j] == 1:
            w_up_to_j = w_up_to_j.left_action_product(w_S[j])

        if (w_up_to_j.left_action_product(w_S[j+1])).length() < w_up_to_j.length():
            res += [j+2]
    return res

def statistic(pi):
    """
    sage: statistic(Permutation([4,3,5,2,1]))
    3
    sage: statistic(Permutation([5,4,2,1,3]))
    3
    """
    k = pi.length()
    if not k:
        return 0
    masks = cartesian_product([[0,1]]*k)
    return max([len(defect_set(m, w)) 
                for m in masks
                for w in pi.reduced_words()])
Created
Feb 13, 2013 at 23:53 by Sara Billey
Updated
Mar 31, 2019 at 21:08 by Martin Rubey