edit this statistic or download as text // json
Identifier
Values
=>
[1]=>1 [1,2]=>1 [2,1]=>2 [1,2,3]=>1 [1,3,2]=>2 [2,1,3]=>3 [2,3,1]=>2 [3,1,2]=>3 [3,2,1]=>2 [1,2,3,4]=>1 [1,2,4,3]=>2 [1,3,2,4]=>3 [1,3,4,2]=>2 [1,4,2,3]=>3 [1,4,3,2]=>2 [2,1,3,4]=>3 [2,1,4,3]=>4 [2,3,1,4]=>3 [2,3,4,1]=>2 [2,4,1,3]=>3 [2,4,3,1]=>2 [3,1,2,4]=>3 [3,1,4,2]=>4 [3,2,1,4]=>3 [3,2,4,1]=>4 [3,4,1,2]=>3 [3,4,2,1]=>2 [4,1,2,3]=>3 [4,1,3,2]=>4 [4,2,1,3]=>3 [4,2,3,1]=>4 [4,3,1,2]=>3 [4,3,2,1]=>2 [1,2,3,4,5]=>1 [1,2,3,5,4]=>2 [1,2,4,3,5]=>3 [1,2,4,5,3]=>2 [1,2,5,3,4]=>3 [1,2,5,4,3]=>2 [1,3,2,4,5]=>3 [1,3,2,5,4]=>4 [1,3,4,2,5]=>3 [1,3,4,5,2]=>2 [1,3,5,2,4]=>3 [1,3,5,4,2]=>2 [1,4,2,3,5]=>3 [1,4,2,5,3]=>4 [1,4,3,2,5]=>3 [1,4,3,5,2]=>4 [1,4,5,2,3]=>3 [1,4,5,3,2]=>2 [1,5,2,3,4]=>3 [1,5,2,4,3]=>4 [1,5,3,2,4]=>3 [1,5,3,4,2]=>4 [1,5,4,2,3]=>3 [1,5,4,3,2]=>2 [2,1,3,4,5]=>3 [2,1,3,5,4]=>4 [2,1,4,3,5]=>5 [2,1,4,5,3]=>4 [2,1,5,3,4]=>5 [2,1,5,4,3]=>4 [2,3,1,4,5]=>3 [2,3,1,5,4]=>4 [2,3,4,1,5]=>3 [2,3,4,5,1]=>2 [2,3,5,1,4]=>3 [2,3,5,4,1]=>2 [2,4,1,3,5]=>3 [2,4,1,5,3]=>4 [2,4,3,1,5]=>3 [2,4,3,5,1]=>4 [2,4,5,1,3]=>3 [2,4,5,3,1]=>2 [2,5,1,3,4]=>3 [2,5,1,4,3]=>4 [2,5,3,1,4]=>3 [2,5,3,4,1]=>4 [2,5,4,1,3]=>3 [2,5,4,3,1]=>2 [3,1,2,4,5]=>3 [3,1,2,5,4]=>4 [3,1,4,2,5]=>5 [3,1,4,5,2]=>4 [3,1,5,2,4]=>5 [3,1,5,4,2]=>4 [3,2,1,4,5]=>3 [3,2,1,5,4]=>4 [3,2,4,1,5]=>5 [3,2,4,5,1]=>4 [3,2,5,1,4]=>5 [3,2,5,4,1]=>4 [3,4,1,2,5]=>3 [3,4,1,5,2]=>4 [3,4,2,1,5]=>3 [3,4,2,5,1]=>4 [3,4,5,1,2]=>3 [3,4,5,2,1]=>2 [3,5,1,2,4]=>3 [3,5,1,4,2]=>4 [3,5,2,1,4]=>3 [3,5,2,4,1]=>4 [3,5,4,1,2]=>3 [3,5,4,2,1]=>2 [4,1,2,3,5]=>3 [4,1,2,5,3]=>4 [4,1,3,2,5]=>5 [4,1,3,5,2]=>4 [4,1,5,2,3]=>5 [4,1,5,3,2]=>4 [4,2,1,3,5]=>3 [4,2,1,5,3]=>4 [4,2,3,1,5]=>5 [4,2,3,5,1]=>4 [4,2,5,1,3]=>5 [4,2,5,3,1]=>4 [4,3,1,2,5]=>3 [4,3,1,5,2]=>4 [4,3,2,1,5]=>3 [4,3,2,5,1]=>4 [4,3,5,1,2]=>5 [4,3,5,2,1]=>4 [4,5,1,2,3]=>3 [4,5,1,3,2]=>4 [4,5,2,1,3]=>3 [4,5,2,3,1]=>4 [4,5,3,1,2]=>3 [4,5,3,2,1]=>2 [5,1,2,3,4]=>3 [5,1,2,4,3]=>4 [5,1,3,2,4]=>5 [5,1,3,4,2]=>4 [5,1,4,2,3]=>5 [5,1,4,3,2]=>4 [5,2,1,3,4]=>3 [5,2,1,4,3]=>4 [5,2,3,1,4]=>5 [5,2,3,4,1]=>4 [5,2,4,1,3]=>5 [5,2,4,3,1]=>4 [5,3,1,2,4]=>3 [5,3,1,4,2]=>4 [5,3,2,1,4]=>3 [5,3,2,4,1]=>4 [5,3,4,1,2]=>5 [5,3,4,2,1]=>4 [5,4,1,2,3]=>3 [5,4,1,3,2]=>4 [5,4,2,1,3]=>3 [5,4,2,3,1]=>4 [5,4,3,1,2]=>3 [5,4,3,2,1]=>2 [1,2,3,4,5,6]=>1 [1,2,3,4,6,5]=>2 [1,2,3,5,4,6]=>3 [1,2,3,5,6,4]=>2 [1,2,3,6,4,5]=>3 [1,2,3,6,5,4]=>2 [1,2,4,3,5,6]=>3 [1,2,4,3,6,5]=>4 [1,2,4,5,3,6]=>3 [1,2,4,5,6,3]=>2 [1,2,4,6,3,5]=>3 [1,2,4,6,5,3]=>2 [1,2,5,3,4,6]=>3 [1,2,5,3,6,4]=>4 [1,2,5,4,3,6]=>3 [1,2,5,4,6,3]=>4 [1,2,5,6,3,4]=>3 [1,2,5,6,4,3]=>2 [1,2,6,3,4,5]=>3 [1,2,6,3,5,4]=>4 [1,2,6,4,3,5]=>3 [1,2,6,4,5,3]=>4 [1,2,6,5,3,4]=>3 [1,2,6,5,4,3]=>2 [1,3,2,4,5,6]=>3 [1,3,2,4,6,5]=>4 [1,3,2,5,4,6]=>5 [1,3,2,5,6,4]=>4 [1,3,2,6,4,5]=>5 [1,3,2,6,5,4]=>4 [1,3,4,2,5,6]=>3 [1,3,4,2,6,5]=>4 [1,3,4,5,2,6]=>3 [1,3,4,5,6,2]=>2 [1,3,4,6,2,5]=>3 [1,3,4,6,5,2]=>2 [1,3,5,2,4,6]=>3 [1,3,5,2,6,4]=>4 [1,3,5,4,2,6]=>3 [1,3,5,4,6,2]=>4 [1,3,5,6,2,4]=>3 [1,3,5,6,4,2]=>2 [1,3,6,2,4,5]=>3 [1,3,6,2,5,4]=>4 [1,3,6,4,2,5]=>3 [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]=>4 [1,4,2,5,3,6]=>5 [1,4,2,5,6,3]=>4 [1,4,2,6,3,5]=>5 [1,4,2,6,5,3]=>4 [1,4,3,2,5,6]=>3 [1,4,3,2,6,5]=>4 [1,4,3,5,2,6]=>5 [1,4,3,5,6,2]=>4 [1,4,3,6,2,5]=>5 [1,4,3,6,5,2]=>4 [1,4,5,2,3,6]=>3 [1,4,5,2,6,3]=>4 [1,4,5,3,2,6]=>3 [1,4,5,3,6,2]=>4 [1,4,5,6,2,3]=>3 [1,4,5,6,3,2]=>2 [1,4,6,2,3,5]=>3 [1,4,6,2,5,3]=>4 [1,4,6,3,2,5]=>3 [1,4,6,3,5,2]=>4 [1,4,6,5,2,3]=>3 [1,4,6,5,3,2]=>2 [1,5,2,3,4,6]=>3 [1,5,2,3,6,4]=>4 [1,5,2,4,3,6]=>5 [1,5,2,4,6,3]=>4 [1,5,2,6,3,4]=>5 [1,5,2,6,4,3]=>4 [1,5,3,2,4,6]=>3 [1,5,3,2,6,4]=>4 [1,5,3,4,2,6]=>5 [1,5,3,4,6,2]=>4 [1,5,3,6,2,4]=>5 [1,5,3,6,4,2]=>4 [1,5,4,2,3,6]=>3 [1,5,4,2,6,3]=>4 [1,5,4,3,2,6]=>3 [1,5,4,3,6,2]=>4 [1,5,4,6,2,3]=>5 [1,5,4,6,3,2]=>4 [1,5,6,2,3,4]=>3 [1,5,6,2,4,3]=>4 [1,5,6,3,2,4]=>3 [1,5,6,3,4,2]=>4 [1,5,6,4,2,3]=>3 [1,5,6,4,3,2]=>2 [1,6,2,3,4,5]=>3 [1,6,2,3,5,4]=>4 [1,6,2,4,3,5]=>5 [1,6,2,4,5,3]=>4 [1,6,2,5,3,4]=>5 [1,6,2,5,4,3]=>4 [1,6,3,2,4,5]=>3 [1,6,3,2,5,4]=>4 [1,6,3,4,2,5]=>5 [1,6,3,4,5,2]=>4 [1,6,3,5,2,4]=>5 [1,6,3,5,4,2]=>4 [1,6,4,2,3,5]=>3 [1,6,4,2,5,3]=>4 [1,6,4,3,2,5]=>3 [1,6,4,3,5,2]=>4 [1,6,4,5,2,3]=>5 [1,6,4,5,3,2]=>4 [1,6,5,2,3,4]=>3 [1,6,5,2,4,3]=>4 [1,6,5,3,2,4]=>3 [1,6,5,3,4,2]=>4 [1,6,5,4,2,3]=>3 [1,6,5,4,3,2]=>2 [2,1,3,4,5,6]=>3 [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]=>5 [2,1,3,6,5,4]=>4 [2,1,4,3,5,6]=>5 [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]=>5 [2,1,4,6,5,3]=>4 [2,1,5,3,4,6]=>5 [2,1,5,3,6,4]=>6 [2,1,5,4,3,6]=>5 [2,1,5,4,6,3]=>6 [2,1,5,6,3,4]=>5 [2,1,5,6,4,3]=>4 [2,1,6,3,4,5]=>5 [2,1,6,3,5,4]=>6 [2,1,6,4,3,5]=>5 [2,1,6,4,5,3]=>6 [2,1,6,5,3,4]=>5 [2,1,6,5,4,3]=>4 [2,3,1,4,5,6]=>3 [2,3,1,4,6,5]=>4 [2,3,1,5,4,6]=>5 [2,3,1,5,6,4]=>4 [2,3,1,6,4,5]=>5 [2,3,1,6,5,4]=>4 [2,3,4,1,5,6]=>3 [2,3,4,1,6,5]=>4 [2,3,4,5,1,6]=>3 [2,3,4,5,6,1]=>2 [2,3,4,6,1,5]=>3 [2,3,4,6,5,1]=>2 [2,3,5,1,4,6]=>3 [2,3,5,1,6,4]=>4 [2,3,5,4,1,6]=>3 [2,3,5,4,6,1]=>4 [2,3,5,6,1,4]=>3 [2,3,5,6,4,1]=>2 [2,3,6,1,4,5]=>3 [2,3,6,1,5,4]=>4 [2,3,6,4,1,5]=>3 [2,3,6,4,5,1]=>4 [2,3,6,5,1,4]=>3 [2,3,6,5,4,1]=>2 [2,4,1,3,5,6]=>3 [2,4,1,3,6,5]=>4 [2,4,1,5,3,6]=>5 [2,4,1,5,6,3]=>4 [2,4,1,6,3,5]=>5 [2,4,1,6,5,3]=>4 [2,4,3,1,5,6]=>3 [2,4,3,1,6,5]=>4 [2,4,3,5,1,6]=>5 [2,4,3,5,6,1]=>4 [2,4,3,6,1,5]=>5 [2,4,3,6,5,1]=>4 [2,4,5,1,3,6]=>3 [2,4,5,1,6,3]=>4 [2,4,5,3,1,6]=>3 [2,4,5,3,6,1]=>4 [2,4,5,6,1,3]=>3 [2,4,5,6,3,1]=>2 [2,4,6,1,3,5]=>3 [2,4,6,1,5,3]=>4 [2,4,6,3,1,5]=>3 [2,4,6,3,5,1]=>4 [2,4,6,5,1,3]=>3 [2,4,6,5,3,1]=>2 [2,5,1,3,4,6]=>3 [2,5,1,3,6,4]=>4 [2,5,1,4,3,6]=>5 [2,5,1,4,6,3]=>4 [2,5,1,6,3,4]=>5 [2,5,1,6,4,3]=>4 [2,5,3,1,4,6]=>3 [2,5,3,1,6,4]=>4 [2,5,3,4,1,6]=>5 [2,5,3,4,6,1]=>4 [2,5,3,6,1,4]=>5 [2,5,3,6,4,1]=>4 [2,5,4,1,3,6]=>3 [2,5,4,1,6,3]=>4 [2,5,4,3,1,6]=>3 [2,5,4,3,6,1]=>4 [2,5,4,6,1,3]=>5 [2,5,4,6,3,1]=>4 [2,5,6,1,3,4]=>3 [2,5,6,1,4,3]=>4 [2,5,6,3,1,4]=>3 [2,5,6,3,4,1]=>4 [2,5,6,4,1,3]=>3 [2,5,6,4,3,1]=>2 [2,6,1,3,4,5]=>3 [2,6,1,3,5,4]=>4 [2,6,1,4,3,5]=>5 [2,6,1,4,5,3]=>4 [2,6,1,5,3,4]=>5 [2,6,1,5,4,3]=>4 [2,6,3,1,4,5]=>3 [2,6,3,1,5,4]=>4 [2,6,3,4,1,5]=>5 [2,6,3,4,5,1]=>4 [2,6,3,5,1,4]=>5 [2,6,3,5,4,1]=>4 [2,6,4,1,3,5]=>3 [2,6,4,1,5,3]=>4 [2,6,4,3,1,5]=>3 [2,6,4,3,5,1]=>4 [2,6,4,5,1,3]=>5 [2,6,4,5,3,1]=>4 [2,6,5,1,3,4]=>3 [2,6,5,1,4,3]=>4 [2,6,5,3,1,4]=>3 [2,6,5,3,4,1]=>4 [2,6,5,4,1,3]=>3 [2,6,5,4,3,1]=>2 [3,1,2,4,5,6]=>3 [3,1,2,4,6,5]=>4 [3,1,2,5,4,6]=>5 [3,1,2,5,6,4]=>4 [3,1,2,6,4,5]=>5 [3,1,2,6,5,4]=>4 [3,1,4,2,5,6]=>5 [3,1,4,2,6,5]=>6 [3,1,4,5,2,6]=>5 [3,1,4,5,6,2]=>4 [3,1,4,6,2,5]=>5 [3,1,4,6,5,2]=>4 [3,1,5,2,4,6]=>5 [3,1,5,2,6,4]=>6 [3,1,5,4,2,6]=>5 [3,1,5,4,6,2]=>6 [3,1,5,6,2,4]=>5 [3,1,5,6,4,2]=>4 [3,1,6,2,4,5]=>5 [3,1,6,2,5,4]=>6 [3,1,6,4,2,5]=>5 [3,1,6,4,5,2]=>6 [3,1,6,5,2,4]=>5 [3,1,6,5,4,2]=>4 [3,2,1,4,5,6]=>3 [3,2,1,4,6,5]=>4 [3,2,1,5,4,6]=>5 [3,2,1,5,6,4]=>4 [3,2,1,6,4,5]=>5 [3,2,1,6,5,4]=>4 [3,2,4,1,5,6]=>5 [3,2,4,1,6,5]=>6 [3,2,4,5,1,6]=>5 [3,2,4,5,6,1]=>4 [3,2,4,6,1,5]=>5 [3,2,4,6,5,1]=>4 [3,2,5,1,4,6]=>5 [3,2,5,1,6,4]=>6 [3,2,5,4,1,6]=>5 [3,2,5,4,6,1]=>6 [3,2,5,6,1,4]=>5 [3,2,5,6,4,1]=>4 [3,2,6,1,4,5]=>5 [3,2,6,1,5,4]=>6 [3,2,6,4,1,5]=>5 [3,2,6,4,5,1]=>6 [3,2,6,5,1,4]=>5 [3,2,6,5,4,1]=>4 [3,4,1,2,5,6]=>3 [3,4,1,2,6,5]=>4 [3,4,1,5,2,6]=>5 [3,4,1,5,6,2]=>4 [3,4,1,6,2,5]=>5 [3,4,1,6,5,2]=>4 [3,4,2,1,5,6]=>3 [3,4,2,1,6,5]=>4 [3,4,2,5,1,6]=>5 [3,4,2,5,6,1]=>4 [3,4,2,6,1,5]=>5 [3,4,2,6,5,1]=>4 [3,4,5,1,2,6]=>3 [3,4,5,1,6,2]=>4 [3,4,5,2,1,6]=>3 [3,4,5,2,6,1]=>4 [3,4,5,6,1,2]=>3 [3,4,5,6,2,1]=>2 [3,4,6,1,2,5]=>3 [3,4,6,1,5,2]=>4 [3,4,6,2,1,5]=>3 [3,4,6,2,5,1]=>4 [3,4,6,5,1,2]=>3 [3,4,6,5,2,1]=>2 [3,5,1,2,4,6]=>3 [3,5,1,2,6,4]=>4 [3,5,1,4,2,6]=>5 [3,5,1,4,6,2]=>4 [3,5,1,6,2,4]=>5 [3,5,1,6,4,2]=>4 [3,5,2,1,4,6]=>3 [3,5,2,1,6,4]=>4 [3,5,2,4,1,6]=>5 [3,5,2,4,6,1]=>4 [3,5,2,6,1,4]=>5 [3,5,2,6,4,1]=>4 [3,5,4,1,2,6]=>3 [3,5,4,1,6,2]=>4 [3,5,4,2,1,6]=>3 [3,5,4,2,6,1]=>4 [3,5,4,6,1,2]=>5 [3,5,4,6,2,1]=>4 [3,5,6,1,2,4]=>3 [3,5,6,1,4,2]=>4 [3,5,6,2,1,4]=>3 [3,5,6,2,4,1]=>4 [3,5,6,4,1,2]=>3 [3,5,6,4,2,1]=>2 [3,6,1,2,4,5]=>3 [3,6,1,2,5,4]=>4 [3,6,1,4,2,5]=>5 [3,6,1,4,5,2]=>4 [3,6,1,5,2,4]=>5 [3,6,1,5,4,2]=>4 [3,6,2,1,4,5]=>3 [3,6,2,1,5,4]=>4 [3,6,2,4,1,5]=>5 [3,6,2,4,5,1]=>4 [3,6,2,5,1,4]=>5 [3,6,2,5,4,1]=>4 [3,6,4,1,2,5]=>3 [3,6,4,1,5,2]=>4 [3,6,4,2,1,5]=>3 [3,6,4,2,5,1]=>4 [3,6,4,5,1,2]=>5 [3,6,4,5,2,1]=>4 [3,6,5,1,2,4]=>3 [3,6,5,1,4,2]=>4 [3,6,5,2,1,4]=>3 [3,6,5,2,4,1]=>4 [3,6,5,4,1,2]=>3 [3,6,5,4,2,1]=>2 [4,1,2,3,5,6]=>3 [4,1,2,3,6,5]=>4 [4,1,2,5,3,6]=>5 [4,1,2,5,6,3]=>4 [4,1,2,6,3,5]=>5 [4,1,2,6,5,3]=>4 [4,1,3,2,5,6]=>5 [4,1,3,2,6,5]=>6 [4,1,3,5,2,6]=>5 [4,1,3,5,6,2]=>4 [4,1,3,6,2,5]=>5 [4,1,3,6,5,2]=>4 [4,1,5,2,3,6]=>5 [4,1,5,2,6,3]=>6 [4,1,5,3,2,6]=>5 [4,1,5,3,6,2]=>6 [4,1,5,6,2,3]=>5 [4,1,5,6,3,2]=>4 [4,1,6,2,3,5]=>5 [4,1,6,2,5,3]=>6 [4,1,6,3,2,5]=>5 [4,1,6,3,5,2]=>6 [4,1,6,5,2,3]=>5 [4,1,6,5,3,2]=>4 [4,2,1,3,5,6]=>3 [4,2,1,3,6,5]=>4 [4,2,1,5,3,6]=>5 [4,2,1,5,6,3]=>4 [4,2,1,6,3,5]=>5 [4,2,1,6,5,3]=>4 [4,2,3,1,5,6]=>5 [4,2,3,1,6,5]=>6 [4,2,3,5,1,6]=>5 [4,2,3,5,6,1]=>4 [4,2,3,6,1,5]=>5 [4,2,3,6,5,1]=>4 [4,2,5,1,3,6]=>5 [4,2,5,1,6,3]=>6 [4,2,5,3,1,6]=>5 [4,2,5,3,6,1]=>6 [4,2,5,6,1,3]=>5 [4,2,5,6,3,1]=>4 [4,2,6,1,3,5]=>5 [4,2,6,1,5,3]=>6 [4,2,6,3,1,5]=>5 [4,2,6,3,5,1]=>6 [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]=>4 [4,3,1,5,2,6]=>5 [4,3,1,5,6,2]=>4 [4,3,1,6,2,5]=>5 [4,3,1,6,5,2]=>4 [4,3,2,1,5,6]=>3 [4,3,2,1,6,5]=>4 [4,3,2,5,1,6]=>5 [4,3,2,5,6,1]=>4 [4,3,2,6,1,5]=>5 [4,3,2,6,5,1]=>4 [4,3,5,1,2,6]=>5 [4,3,5,1,6,2]=>6 [4,3,5,2,1,6]=>5 [4,3,5,2,6,1]=>6 [4,3,5,6,1,2]=>5 [4,3,5,6,2,1]=>4 [4,3,6,1,2,5]=>5 [4,3,6,1,5,2]=>6 [4,3,6,2,1,5]=>5 [4,3,6,2,5,1]=>6 [4,3,6,5,1,2]=>5 [4,3,6,5,2,1]=>4 [4,5,1,2,3,6]=>3 [4,5,1,2,6,3]=>4 [4,5,1,3,2,6]=>5 [4,5,1,3,6,2]=>4 [4,5,1,6,2,3]=>5 [4,5,1,6,3,2]=>4 [4,5,2,1,3,6]=>3 [4,5,2,1,6,3]=>4 [4,5,2,3,1,6]=>5 [4,5,2,3,6,1]=>4 [4,5,2,6,1,3]=>5 [4,5,2,6,3,1]=>4 [4,5,3,1,2,6]=>3 [4,5,3,1,6,2]=>4 [4,5,3,2,1,6]=>3 [4,5,3,2,6,1]=>4 [4,5,3,6,1,2]=>5 [4,5,3,6,2,1]=>4 [4,5,6,1,2,3]=>3 [4,5,6,1,3,2]=>4 [4,5,6,2,1,3]=>3 [4,5,6,2,3,1]=>4 [4,5,6,3,1,2]=>3 [4,5,6,3,2,1]=>2 [4,6,1,2,3,5]=>3 [4,6,1,2,5,3]=>4 [4,6,1,3,2,5]=>5 [4,6,1,3,5,2]=>4 [4,6,1,5,2,3]=>5 [4,6,1,5,3,2]=>4 [4,6,2,1,3,5]=>3 [4,6,2,1,5,3]=>4 [4,6,2,3,1,5]=>5 [4,6,2,3,5,1]=>4 [4,6,2,5,1,3]=>5 [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]=>3 [4,6,3,2,5,1]=>4 [4,6,3,5,1,2]=>5 [4,6,3,5,2,1]=>4 [4,6,5,1,2,3]=>3 [4,6,5,1,3,2]=>4 [4,6,5,2,1,3]=>3 [4,6,5,2,3,1]=>4 [4,6,5,3,1,2]=>3 [4,6,5,3,2,1]=>2 [5,1,2,3,4,6]=>3 [5,1,2,3,6,4]=>4 [5,1,2,4,3,6]=>5 [5,1,2,4,6,3]=>4 [5,1,2,6,3,4]=>5 [5,1,2,6,4,3]=>4 [5,1,3,2,4,6]=>5 [5,1,3,2,6,4]=>6 [5,1,3,4,2,6]=>5 [5,1,3,4,6,2]=>4 [5,1,3,6,2,4]=>5 [5,1,3,6,4,2]=>4 [5,1,4,2,3,6]=>5 [5,1,4,2,6,3]=>6 [5,1,4,3,2,6]=>5 [5,1,4,3,6,2]=>6 [5,1,4,6,2,3]=>5 [5,1,4,6,3,2]=>4 [5,1,6,2,3,4]=>5 [5,1,6,2,4,3]=>6 [5,1,6,3,2,4]=>5 [5,1,6,3,4,2]=>6 [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]=>5 [5,2,1,4,6,3]=>4 [5,2,1,6,3,4]=>5 [5,2,1,6,4,3]=>4 [5,2,3,1,4,6]=>5 [5,2,3,1,6,4]=>6 [5,2,3,4,1,6]=>5 [5,2,3,4,6,1]=>4 [5,2,3,6,1,4]=>5 [5,2,3,6,4,1]=>4 [5,2,4,1,3,6]=>5 [5,2,4,1,6,3]=>6 [5,2,4,3,1,6]=>5 [5,2,4,3,6,1]=>6 [5,2,4,6,1,3]=>5 [5,2,4,6,3,1]=>4 [5,2,6,1,3,4]=>5 [5,2,6,1,4,3]=>6 [5,2,6,3,1,4]=>5 [5,2,6,3,4,1]=>6 [5,2,6,4,1,3]=>5 [5,2,6,4,3,1]=>4 [5,3,1,2,4,6]=>3 [5,3,1,2,6,4]=>4 [5,3,1,4,2,6]=>5 [5,3,1,4,6,2]=>4 [5,3,1,6,2,4]=>5 [5,3,1,6,4,2]=>4 [5,3,2,1,4,6]=>3 [5,3,2,1,6,4]=>4 [5,3,2,4,1,6]=>5 [5,3,2,4,6,1]=>4 [5,3,2,6,1,4]=>5 [5,3,2,6,4,1]=>4 [5,3,4,1,2,6]=>5 [5,3,4,1,6,2]=>6 [5,3,4,2,1,6]=>5 [5,3,4,2,6,1]=>6 [5,3,4,6,1,2]=>5 [5,3,4,6,2,1]=>4 [5,3,6,1,2,4]=>5 [5,3,6,1,4,2]=>6 [5,3,6,2,1,4]=>5 [5,3,6,2,4,1]=>6 [5,3,6,4,1,2]=>5 [5,3,6,4,2,1]=>4 [5,4,1,2,3,6]=>3 [5,4,1,2,6,3]=>4 [5,4,1,3,2,6]=>5 [5,4,1,3,6,2]=>4 [5,4,1,6,2,3]=>5 [5,4,1,6,3,2]=>4 [5,4,2,1,3,6]=>3 [5,4,2,1,6,3]=>4 [5,4,2,3,1,6]=>5 [5,4,2,3,6,1]=>4 [5,4,2,6,1,3]=>5 [5,4,2,6,3,1]=>4 [5,4,3,1,2,6]=>3 [5,4,3,1,6,2]=>4 [5,4,3,2,1,6]=>3 [5,4,3,2,6,1]=>4 [5,4,3,6,1,2]=>5 [5,4,3,6,2,1]=>4 [5,4,6,1,2,3]=>5 [5,4,6,1,3,2]=>6 [5,4,6,2,1,3]=>5 [5,4,6,2,3,1]=>6 [5,4,6,3,1,2]=>5 [5,4,6,3,2,1]=>4 [5,6,1,2,3,4]=>3 [5,6,1,2,4,3]=>4 [5,6,1,3,2,4]=>5 [5,6,1,3,4,2]=>4 [5,6,1,4,2,3]=>5 [5,6,1,4,3,2]=>4 [5,6,2,1,3,4]=>3 [5,6,2,1,4,3]=>4 [5,6,2,3,1,4]=>5 [5,6,2,3,4,1]=>4 [5,6,2,4,1,3]=>5 [5,6,2,4,3,1]=>4 [5,6,3,1,2,4]=>3 [5,6,3,1,4,2]=>4 [5,6,3,2,1,4]=>3 [5,6,3,2,4,1]=>4 [5,6,3,4,1,2]=>5 [5,6,3,4,2,1]=>4 [5,6,4,1,2,3]=>3 [5,6,4,1,3,2]=>4 [5,6,4,2,1,3]=>3 [5,6,4,2,3,1]=>4 [5,6,4,3,1,2]=>3 [5,6,4,3,2,1]=>2 [6,1,2,3,4,5]=>3 [6,1,2,3,5,4]=>4 [6,1,2,4,3,5]=>5 [6,1,2,4,5,3]=>4 [6,1,2,5,3,4]=>5 [6,1,2,5,4,3]=>4 [6,1,3,2,4,5]=>5 [6,1,3,2,5,4]=>6 [6,1,3,4,2,5]=>5 [6,1,3,4,5,2]=>4 [6,1,3,5,2,4]=>5 [6,1,3,5,4,2]=>4 [6,1,4,2,3,5]=>5 [6,1,4,2,5,3]=>6 [6,1,4,3,2,5]=>5 [6,1,4,3,5,2]=>6 [6,1,4,5,2,3]=>5 [6,1,4,5,3,2]=>4 [6,1,5,2,3,4]=>5 [6,1,5,2,4,3]=>6 [6,1,5,3,2,4]=>5 [6,1,5,3,4,2]=>6 [6,1,5,4,2,3]=>5 [6,1,5,4,3,2]=>4 [6,2,1,3,4,5]=>3 [6,2,1,3,5,4]=>4 [6,2,1,4,3,5]=>5 [6,2,1,4,5,3]=>4 [6,2,1,5,3,4]=>5 [6,2,1,5,4,3]=>4 [6,2,3,1,4,5]=>5 [6,2,3,1,5,4]=>6 [6,2,3,4,1,5]=>5 [6,2,3,4,5,1]=>4 [6,2,3,5,1,4]=>5 [6,2,3,5,4,1]=>4 [6,2,4,1,3,5]=>5 [6,2,4,1,5,3]=>6 [6,2,4,3,1,5]=>5 [6,2,4,3,5,1]=>6 [6,2,4,5,1,3]=>5 [6,2,4,5,3,1]=>4 [6,2,5,1,3,4]=>5 [6,2,5,1,4,3]=>6 [6,2,5,3,1,4]=>5 [6,2,5,3,4,1]=>6 [6,2,5,4,1,3]=>5 [6,2,5,4,3,1]=>4 [6,3,1,2,4,5]=>3 [6,3,1,2,5,4]=>4 [6,3,1,4,2,5]=>5 [6,3,1,4,5,2]=>4 [6,3,1,5,2,4]=>5 [6,3,1,5,4,2]=>4 [6,3,2,1,4,5]=>3 [6,3,2,1,5,4]=>4 [6,3,2,4,1,5]=>5 [6,3,2,4,5,1]=>4 [6,3,2,5,1,4]=>5 [6,3,2,5,4,1]=>4 [6,3,4,1,2,5]=>5 [6,3,4,1,5,2]=>6 [6,3,4,2,1,5]=>5 [6,3,4,2,5,1]=>6 [6,3,4,5,1,2]=>5 [6,3,4,5,2,1]=>4 [6,3,5,1,2,4]=>5 [6,3,5,1,4,2]=>6 [6,3,5,2,1,4]=>5 [6,3,5,2,4,1]=>6 [6,3,5,4,1,2]=>5 [6,3,5,4,2,1]=>4 [6,4,1,2,3,5]=>3 [6,4,1,2,5,3]=>4 [6,4,1,3,2,5]=>5 [6,4,1,3,5,2]=>4 [6,4,1,5,2,3]=>5 [6,4,1,5,3,2]=>4 [6,4,2,1,3,5]=>3 [6,4,2,1,5,3]=>4 [6,4,2,3,1,5]=>5 [6,4,2,3,5,1]=>4 [6,4,2,5,1,3]=>5 [6,4,2,5,3,1]=>4 [6,4,3,1,2,5]=>3 [6,4,3,1,5,2]=>4 [6,4,3,2,1,5]=>3 [6,4,3,2,5,1]=>4 [6,4,3,5,1,2]=>5 [6,4,3,5,2,1]=>4 [6,4,5,1,2,3]=>5 [6,4,5,1,3,2]=>6 [6,4,5,2,1,3]=>5 [6,4,5,2,3,1]=>6 [6,4,5,3,1,2]=>5 [6,4,5,3,2,1]=>4 [6,5,1,2,3,4]=>3 [6,5,1,2,4,3]=>4 [6,5,1,3,2,4]=>5 [6,5,1,3,4,2]=>4 [6,5,1,4,2,3]=>5 [6,5,1,4,3,2]=>4 [6,5,2,1,3,4]=>3 [6,5,2,1,4,3]=>4 [6,5,2,3,1,4]=>5 [6,5,2,3,4,1]=>4 [6,5,2,4,1,3]=>5 [6,5,2,4,3,1]=>4 [6,5,3,1,2,4]=>3 [6,5,3,1,4,2]=>4 [6,5,3,2,1,4]=>3 [6,5,3,2,4,1]=>4 [6,5,3,4,1,2]=>5 [6,5,3,4,2,1]=>4 [6,5,4,1,2,3]=>3 [6,5,4,1,3,2]=>4 [6,5,4,2,1,3]=>3 [6,5,4,2,3,1]=>4 [6,5,4,3,1,2]=>3 [6,5,4,3,2,1]=>2
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Description
The number of up-down runs of a permutation.
An up-down run of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$.
For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and
$643$.
References
[1] Zhuang, Y. Counting permutations by runs MathSciNet:3499494
[2] Zhuang, Y. Eulerian polynomials and descent statistics arXiv:1610.07218
Code
def statistic(x):
   if len(x)>1 and x[0]>x[1]:
        return x.number_of_peaks() + x.complement().number_of_peaks() + 2
   elif len(x)>0:
        return x.number_of_peaks() + x.complement().number_of_peaks() + 1
   else:
        return 0
Created
Oct 27, 2016 at 21:11 by Yan Zhuang
Updated
Oct 27, 2016 at 21:11 by Yan Zhuang