edit this statistic or download as text // json
Identifier
Values
=>
[1,2]=>1 [2,1]=>1 [1,2,3]=>1 [1,3,2]=>1 [2,1,3]=>1 [2,3,1]=>2 [3,1,2]=>2 [3,2,1]=>1 [1,2,3,4]=>1 [1,2,4,3]=>1 [1,3,2,4]=>1 [1,3,4,2]=>2 [1,4,2,3]=>2 [1,4,3,2]=>1 [2,1,3,4]=>1 [2,1,4,3]=>1 [2,3,1,4]=>2 [2,3,4,1]=>2 [2,4,1,3]=>2 [2,4,3,1]=>2 [3,1,2,4]=>2 [3,1,4,2]=>2 [3,2,1,4]=>1 [3,2,4,1]=>2 [3,4,1,2]=>1 [3,4,2,1]=>3 [4,1,2,3]=>2 [4,1,3,2]=>2 [4,2,1,3]=>2 [4,2,3,1]=>3 [4,3,1,2]=>3 [4,3,2,1]=>1 [1,2,3,4,5]=>1 [1,2,3,5,4]=>1 [1,2,4,3,5]=>1 [1,2,4,5,3]=>2 [1,2,5,3,4]=>2 [1,2,5,4,3]=>1 [1,3,2,4,5]=>1 [1,3,2,5,4]=>1 [1,3,4,2,5]=>2 [1,3,4,5,2]=>2 [1,3,5,2,4]=>2 [1,3,5,4,2]=>2 [1,4,2,3,5]=>2 [1,4,2,5,3]=>2 [1,4,3,2,5]=>1 [1,4,3,5,2]=>2 [1,4,5,2,3]=>1 [1,4,5,3,2]=>3 [1,5,2,3,4]=>2 [1,5,2,4,3]=>2 [1,5,3,2,4]=>2 [1,5,3,4,2]=>3 [1,5,4,2,3]=>3 [1,5,4,3,2]=>1 [2,1,3,4,5]=>1 [2,1,3,5,4]=>1 [2,1,4,3,5]=>1 [2,1,4,5,3]=>2 [2,1,5,3,4]=>2 [2,1,5,4,3]=>1 [2,3,1,4,5]=>2 [2,3,1,5,4]=>2 [2,3,4,1,5]=>2 [2,3,4,5,1]=>4 [2,3,5,1,4]=>2 [2,3,5,4,1]=>4 [2,4,1,3,5]=>2 [2,4,1,5,3]=>2 [2,4,3,1,5]=>2 [2,4,3,5,1]=>2 [2,4,5,1,3]=>4 [2,4,5,3,1]=>4 [2,5,1,3,4]=>2 [2,5,1,4,3]=>2 [2,5,3,1,4]=>4 [2,5,3,4,1]=>4 [2,5,4,1,3]=>2 [2,5,4,3,1]=>2 [3,1,2,4,5]=>2 [3,1,2,5,4]=>2 [3,1,4,2,5]=>2 [3,1,4,5,2]=>2 [3,1,5,2,4]=>2 [3,1,5,4,2]=>2 [3,2,1,4,5]=>1 [3,2,1,5,4]=>1 [3,2,4,1,5]=>2 [3,2,4,5,1]=>4 [3,2,5,1,4]=>2 [3,2,5,4,1]=>2 [3,4,1,2,5]=>1 [3,4,1,5,2]=>4 [3,4,2,1,5]=>3 [3,4,2,5,1]=>4 [3,4,5,1,2]=>6 [3,4,5,2,1]=>3 [3,5,1,2,4]=>4 [3,5,1,4,2]=>1 [3,5,2,1,4]=>2 [3,5,2,4,1]=>6 [3,5,4,1,2]=>2 [3,5,4,2,1]=>3 [4,1,2,3,5]=>2 [4,1,2,5,3]=>2 [4,1,3,2,5]=>2 [4,1,3,5,2]=>4 [4,1,5,2,3]=>4 [4,1,5,3,2]=>2 [4,2,1,3,5]=>2 [4,2,1,5,3]=>2 [4,2,3,1,5]=>3 [4,2,3,5,1]=>4 [4,2,5,1,3]=>1 [4,2,5,3,1]=>6 [4,3,1,2,5]=>3 [4,3,1,5,2]=>2 [4,3,2,1,5]=>1 [4,3,2,5,1]=>2 [4,3,5,1,2]=>2 [4,3,5,2,1]=>3 [4,5,1,2,3]=>6 [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]=>4 [5,1,2,4,3]=>4 [5,1,3,2,4]=>2 [5,1,3,4,2]=>4 [5,1,4,2,3]=>4 [5,1,4,3,2]=>2 [5,2,1,3,4]=>4 [5,2,1,4,3]=>2 [5,2,3,1,4]=>4 [5,2,3,4,1]=>3 [5,2,4,1,3]=>6 [5,2,4,3,1]=>3 [5,3,1,2,4]=>4 [5,3,1,4,2]=>6 [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]=>3 [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]=>1 [1,2,3,4,5,6]=>1 [1,2,3,4,6,5]=>1 [1,2,3,5,4,6]=>1 [1,2,3,5,6,4]=>2 [1,2,3,6,4,5]=>2 [1,2,3,6,5,4]=>1 [1,2,4,3,5,6]=>1 [1,2,4,3,6,5]=>1 [1,2,4,5,3,6]=>2 [1,2,4,5,6,3]=>2 [1,2,4,6,3,5]=>2 [1,2,4,6,5,3]=>2 [1,2,5,3,4,6]=>2 [1,2,5,3,6,4]=>2 [1,2,5,4,3,6]=>1 [1,2,5,4,6,3]=>2 [1,2,5,6,3,4]=>1 [1,2,5,6,4,3]=>3 [1,2,6,3,4,5]=>2 [1,2,6,3,5,4]=>2 [1,2,6,4,3,5]=>2 [1,2,6,4,5,3]=>3 [1,2,6,5,3,4]=>3 [1,2,6,5,4,3]=>1 [1,3,2,4,5,6]=>1 [1,3,2,4,6,5]=>1 [1,3,2,5,4,6]=>1 [1,3,2,5,6,4]=>2 [1,3,2,6,4,5]=>2 [1,3,2,6,5,4]=>1 [1,3,4,2,5,6]=>2 [1,3,4,2,6,5]=>2 [1,3,4,5,2,6]=>2 [1,3,4,5,6,2]=>4 [1,3,4,6,2,5]=>2 [1,3,4,6,5,2]=>4 [1,3,5,2,4,6]=>2 [1,3,5,2,6,4]=>2 [1,3,5,4,2,6]=>2 [1,3,5,4,6,2]=>2 [1,3,5,6,2,4]=>4 [1,3,5,6,4,2]=>4 [1,3,6,2,4,5]=>2 [1,3,6,2,5,4]=>2 [1,3,6,4,2,5]=>4 [1,3,6,4,5,2]=>4 [1,3,6,5,2,4]=>2 [1,3,6,5,4,2]=>2 [1,4,2,3,5,6]=>2 [1,4,2,3,6,5]=>2 [1,4,2,5,3,6]=>2 [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]=>1 [1,4,3,2,6,5]=>1 [1,4,3,5,2,6]=>2 [1,4,3,5,6,2]=>4 [1,4,3,6,2,5]=>2 [1,4,3,6,5,2]=>2 [1,4,5,2,3,6]=>1 [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]=>6 [1,4,5,6,3,2]=>3 [1,4,6,2,3,5]=>4 [1,4,6,2,5,3]=>1 [1,4,6,3,2,5]=>2 [1,4,6,3,5,2]=>6 [1,4,6,5,2,3]=>2 [1,4,6,5,3,2]=>3 [1,5,2,3,4,6]=>2 [1,5,2,3,6,4]=>2 [1,5,2,4,3,6]=>2 [1,5,2,4,6,3]=>4 [1,5,2,6,3,4]=>4 [1,5,2,6,4,3]=>2 [1,5,3,2,4,6]=>2 [1,5,3,2,6,4]=>2 [1,5,3,4,2,6]=>3 [1,5,3,4,6,2]=>4 [1,5,3,6,2,4]=>1 [1,5,3,6,4,2]=>6 [1,5,4,2,3,6]=>3 [1,5,4,2,6,3]=>2 [1,5,4,3,2,6]=>1 [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]=>6 [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]=>4 [1,6,2,3,5,4]=>4 [1,6,2,4,3,5]=>2 [1,6,2,4,5,3]=>4 [1,6,2,5,3,4]=>4 [1,6,2,5,4,3]=>2 [1,6,3,2,4,5]=>4 [1,6,3,2,5,4]=>2 [1,6,3,4,2,5]=>4 [1,6,3,4,5,2]=>3 [1,6,3,5,2,4]=>6 [1,6,3,5,4,2]=>3 [1,6,4,2,3,5]=>4 [1,6,4,2,5,3]=>6 [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]=>3 [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]=>1 [2,1,3,4,5,6]=>1 [2,1,3,4,6,5]=>1 [2,1,3,5,4,6]=>1 [2,1,3,5,6,4]=>2 [2,1,3,6,4,5]=>2 [2,1,3,6,5,4]=>1 [2,1,4,3,5,6]=>1 [2,1,4,3,6,5]=>1 [2,1,4,5,3,6]=>2 [2,1,4,5,6,3]=>2 [2,1,4,6,3,5]=>2 [2,1,4,6,5,3]=>2 [2,1,5,3,4,6]=>2 [2,1,5,3,6,4]=>2 [2,1,5,4,3,6]=>1 [2,1,5,4,6,3]=>2 [2,1,5,6,3,4]=>1 [2,1,5,6,4,3]=>3 [2,1,6,3,4,5]=>2 [2,1,6,3,5,4]=>2 [2,1,6,4,3,5]=>2 [2,1,6,4,5,3]=>3 [2,1,6,5,3,4]=>3 [2,1,6,5,4,3]=>1 [2,3,1,4,5,6]=>2 [2,3,1,4,6,5]=>2 [2,3,1,5,4,6]=>2 [2,3,1,5,6,4]=>4 [2,3,1,6,4,5]=>4 [2,3,1,6,5,4]=>2 [2,3,4,1,5,6]=>2 [2,3,4,1,6,5]=>2 [2,3,4,5,1,6]=>4 [2,3,4,5,6,1]=>4 [2,3,4,6,1,5]=>4 [2,3,4,6,5,1]=>2 [2,3,5,1,4,6]=>2 [2,3,5,1,6,4]=>2 [2,3,5,4,1,6]=>4 [2,3,5,4,6,1]=>4 [2,3,5,6,1,4]=>4 [2,3,5,6,4,1]=>8 [2,3,6,1,4,5]=>2 [2,3,6,1,5,4]=>2 [2,3,6,4,1,5]=>8 [2,3,6,4,5,1]=>8 [2,3,6,5,1,4]=>4 [2,3,6,5,4,1]=>4 [2,4,1,3,5,6]=>2 [2,4,1,3,6,5]=>2 [2,4,1,5,3,6]=>2 [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]=>2 [2,4,3,5,6,1]=>4 [2,4,3,6,1,5]=>2 [2,4,3,6,5,1]=>4 [2,4,5,1,3,6]=>4 [2,4,5,1,6,3]=>8 [2,4,5,3,1,6]=>4 [2,4,5,3,6,1]=>6 [2,4,5,6,1,3]=>4 [2,4,5,6,3,1]=>6 [2,4,6,1,3,5]=>8 [2,4,6,1,5,3]=>4 [2,4,6,3,1,5]=>4 [2,4,6,3,5,1]=>8 [2,4,6,5,1,3]=>2 [2,4,6,5,3,1]=>8 [2,5,1,3,4,6]=>2 [2,5,1,3,6,4]=>2 [2,5,1,4,3,6]=>2 [2,5,1,4,6,3]=>4 [2,5,1,6,3,4]=>4 [2,5,1,6,4,3]=>2 [2,5,3,1,4,6]=>4 [2,5,3,1,6,4]=>4 [2,5,3,4,1,6]=>4 [2,5,3,4,6,1]=>6 [2,5,3,6,1,4]=>4 [2,5,3,6,4,1]=>8 [2,5,4,1,3,6]=>2 [2,5,4,1,6,3]=>4 [2,5,4,3,1,6]=>2 [2,5,4,3,6,1]=>2 [2,5,4,6,1,3]=>8 [2,5,4,6,3,1]=>4 [2,5,6,1,3,4]=>8 [2,5,6,1,4,3]=>4 [2,5,6,3,1,4]=>8 [2,5,6,3,4,1]=>2 [2,5,6,4,1,3]=>8 [2,5,6,4,3,1]=>6 [2,6,1,3,4,5]=>4 [2,6,1,3,5,4]=>4 [2,6,1,4,3,5]=>2 [2,6,1,4,5,3]=>4 [2,6,1,5,3,4]=>4 [2,6,1,5,4,3]=>2 [2,6,3,1,4,5]=>8 [2,6,3,1,5,4]=>4 [2,6,3,4,1,5]=>2 [2,6,3,4,5,1]=>6 [2,6,3,5,1,4]=>4 [2,6,3,5,4,1]=>8 [2,6,4,1,3,5]=>4 [2,6,4,1,5,3]=>8 [2,6,4,3,1,5]=>6 [2,6,4,3,5,1]=>4 [2,6,4,5,1,3]=>8 [2,6,4,5,3,1]=>6 [2,6,5,1,3,4]=>2 [2,6,5,1,4,3]=>2 [2,6,5,3,1,4]=>6 [2,6,5,3,4,1]=>6 [2,6,5,4,1,3]=>2 [2,6,5,4,3,1]=>2 [3,1,2,4,5,6]=>2 [3,1,2,4,6,5]=>2 [3,1,2,5,4,6]=>2 [3,1,2,5,6,4]=>4 [3,1,2,6,4,5]=>4 [3,1,2,6,5,4]=>2 [3,1,4,2,5,6]=>2 [3,1,4,2,6,5]=>2 [3,1,4,5,2,6]=>2 [3,1,4,5,6,2]=>4 [3,1,4,6,2,5]=>2 [3,1,4,6,5,2]=>4 [3,1,5,2,4,6]=>2 [3,1,5,2,6,4]=>2 [3,1,5,4,2,6]=>2 [3,1,5,4,6,2]=>2 [3,1,5,6,2,4]=>4 [3,1,5,6,4,2]=>4 [3,1,6,2,4,5]=>2 [3,1,6,2,5,4]=>2 [3,1,6,4,2,5]=>4 [3,1,6,4,5,2]=>4 [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]=>2 [3,2,1,6,4,5]=>2 [3,2,1,6,5,4]=>1 [3,2,4,1,5,6]=>2 [3,2,4,1,6,5]=>2 [3,2,4,5,1,6]=>4 [3,2,4,5,6,1]=>2 [3,2,4,6,1,5]=>4 [3,2,4,6,5,1]=>3 [3,2,5,1,4,6]=>2 [3,2,5,1,6,4]=>2 [3,2,5,4,1,6]=>2 [3,2,5,4,6,1]=>4 [3,2,5,6,1,4]=>4 [3,2,5,6,4,1]=>6 [3,2,6,1,4,5]=>2 [3,2,6,1,5,4]=>2 [3,2,6,4,1,5]=>4 [3,2,6,4,5,1]=>6 [3,2,6,5,1,4]=>2 [3,2,6,5,4,1]=>2 [3,4,1,2,5,6]=>1 [3,4,1,2,6,5]=>1 [3,4,1,5,2,6]=>4 [3,4,1,5,6,2]=>4 [3,4,1,6,2,5]=>4 [3,4,1,6,5,2]=>4 [3,4,2,1,5,6]=>3 [3,4,2,1,6,5]=>3 [3,4,2,5,1,6]=>4 [3,4,2,5,6,1]=>8 [3,4,2,6,1,5]=>4 [3,4,2,6,5,1]=>6 [3,4,5,1,2,6]=>6 [3,4,5,1,6,2]=>4 [3,4,5,2,1,6]=>3 [3,4,5,2,6,1]=>6 [3,4,5,6,1,2]=>4 [3,4,5,6,2,1]=>9 [3,4,6,1,2,5]=>8 [3,4,6,1,5,2]=>6 [3,4,6,2,1,5]=>2 [3,4,6,2,5,1]=>8 [3,4,6,5,1,2]=>3 [3,4,6,5,2,1]=>7 [3,5,1,2,4,6]=>4 [3,5,1,2,6,4]=>4 [3,5,1,4,2,6]=>1 [3,5,1,4,6,2]=>4 [3,5,1,6,2,4]=>3 [3,5,1,6,4,2]=>6 [3,5,2,1,4,6]=>2 [3,5,2,1,6,4]=>2 [3,5,2,4,1,6]=>6 [3,5,2,4,6,1]=>8 [3,5,2,6,1,4]=>6 [3,5,2,6,4,1]=>6 [3,5,4,1,2,6]=>2 [3,5,4,1,6,2]=>8 [3,5,4,2,1,6]=>3 [3,5,4,2,6,1]=>4 [3,5,4,6,1,2]=>9 [3,5,4,6,2,1]=>3 [3,5,6,1,2,4]=>6 [3,5,6,1,4,2]=>6 [3,5,6,2,1,4]=>8 [3,5,6,2,4,1]=>7 [3,5,6,4,1,2]=>8 [3,5,6,4,2,1]=>6 [3,6,1,2,4,5]=>4 [3,6,1,2,5,4]=>4 [3,6,1,4,2,5]=>4 [3,6,1,4,5,2]=>1 [3,6,1,5,2,4]=>6 [3,6,1,5,4,2]=>1 [3,6,2,1,4,5]=>4 [3,6,2,1,5,4]=>2 [3,6,2,4,1,5]=>4 [3,6,2,4,5,1]=>8 [3,6,2,5,1,4]=>6 [3,6,2,5,4,1]=>6 [3,6,4,1,2,5]=>8 [3,6,4,1,5,2]=>8 [3,6,4,2,1,5]=>6 [3,6,4,2,5,1]=>9 [3,6,4,5,1,2]=>8 [3,6,4,5,2,1]=>6 [3,6,5,1,2,4]=>8 [3,6,5,1,4,2]=>2 [3,6,5,2,1,4]=>4 [3,6,5,2,4,1]=>8 [3,6,5,4,1,2]=>2 [3,6,5,4,2,1]=>3 [4,1,2,3,5,6]=>2 [4,1,2,3,6,5]=>2 [4,1,2,5,3,6]=>2 [4,1,2,5,6,3]=>2 [4,1,2,6,3,5]=>2 [4,1,2,6,5,3]=>2 [4,1,3,2,5,6]=>2 [4,1,3,2,6,5]=>2 [4,1,3,5,2,6]=>4 [4,1,3,5,6,2]=>8 [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]=>8 [4,1,5,3,2,6]=>2 [4,1,5,3,6,2]=>4 [4,1,5,6,2,3]=>8 [4,1,5,6,3,2]=>2 [4,1,6,2,3,5]=>8 [4,1,6,2,5,3]=>4 [4,1,6,3,2,5]=>4 [4,1,6,3,5,2]=>8 [4,1,6,5,2,3]=>4 [4,1,6,5,3,2]=>2 [4,2,1,3,5,6]=>2 [4,2,1,3,6,5]=>2 [4,2,1,5,3,6]=>2 [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]=>3 [4,2,3,1,6,5]=>3 [4,2,3,5,1,6]=>4 [4,2,3,5,6,1]=>8 [4,2,3,6,1,5]=>4 [4,2,3,6,5,1]=>6 [4,2,5,1,3,6]=>1 [4,2,5,1,6,3]=>4 [4,2,5,3,1,6]=>6 [4,2,5,3,6,1]=>8 [4,2,5,6,1,3]=>6 [4,2,5,6,3,1]=>8 [4,2,6,1,3,5]=>4 [4,2,6,1,5,3]=>1 [4,2,6,3,1,5]=>8 [4,2,6,3,5,1]=>6 [4,2,6,5,1,3]=>2 [4,2,6,5,3,1]=>6 [4,3,1,2,5,6]=>3 [4,3,1,2,6,5]=>3 [4,3,1,5,2,6]=>2 [4,3,1,5,6,2]=>4 [4,3,1,6,2,5]=>2 [4,3,1,6,5,2]=>2 [4,3,2,1,5,6]=>1 [4,3,2,1,6,5]=>1 [4,3,2,5,1,6]=>2 [4,3,2,5,6,1]=>4 [4,3,2,6,1,5]=>2 [4,3,2,6,5,1]=>2 [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]=>8 [4,3,5,6,1,2]=>3 [4,3,5,6,2,1]=>7 [4,3,6,1,2,5]=>4 [4,3,6,1,5,2]=>2 [4,3,6,2,1,5]=>2 [4,3,6,2,5,1]=>6 [4,3,6,5,1,2]=>4 [4,3,6,5,2,1]=>4 [4,5,1,2,3,6]=>6 [4,5,1,2,6,3]=>8 [4,5,1,3,2,6]=>2 [4,5,1,3,6,2]=>8 [4,5,1,6,2,3]=>6 [4,5,1,6,3,2]=>8 [4,5,2,1,3,6]=>2 [4,5,2,1,6,3]=>4 [4,5,2,3,1,6]=>3 [4,5,2,3,6,1]=>2 [4,5,2,6,1,3]=>6 [4,5,2,6,3,1]=>7 [4,5,3,1,2,6]=>3 [4,5,3,1,6,2]=>8 [4,5,3,2,1,6]=>4 [4,5,3,2,6,1]=>6 [4,5,3,6,1,2]=>8 [4,5,3,6,2,1]=>6 [4,5,6,1,2,3]=>4 [4,5,6,1,3,2]=>4 [4,5,6,2,1,3]=>4 [4,5,6,2,3,1]=>12 [4,5,6,3,1,2]=>12 [4,5,6,3,2,1]=>4 [4,6,1,2,3,5]=>4 [4,6,1,2,5,3]=>6 [4,6,1,3,2,5]=>8 [4,6,1,3,5,2]=>8 [4,6,1,5,2,3]=>6 [4,6,1,5,3,2]=>2 [4,6,2,1,3,5]=>2 [4,6,2,1,5,3]=>2 [4,6,2,3,1,5]=>8 [4,6,2,3,5,1]=>7 [4,6,2,5,1,3]=>4 [4,6,2,5,3,1]=>4 [4,6,3,1,2,5]=>8 [4,6,3,1,5,2]=>3 [4,6,3,2,1,5]=>2 [4,6,3,2,5,1]=>8 [4,6,3,5,1,2]=>6 [4,6,3,5,2,1]=>12 [4,6,5,1,2,3]=>4 [4,6,5,1,3,2]=>1 [4,6,5,2,1,3]=>6 [4,6,5,2,3,1]=>6 [4,6,5,3,1,2]=>6 [4,6,5,3,2,1]=>4 [5,1,2,3,4,6]=>4 [5,1,2,3,6,4]=>4 [5,1,2,4,3,6]=>4 [5,1,2,4,6,3]=>8 [5,1,2,6,3,4]=>4 [5,1,2,6,4,3]=>4 [5,1,3,2,4,6]=>2 [5,1,3,2,6,4]=>2 [5,1,3,4,2,6]=>4 [5,1,3,4,6,2]=>2 [5,1,3,6,2,4]=>4 [5,1,3,6,4,2]=>4 [5,1,4,2,3,6]=>4 [5,1,4,2,6,3]=>4 [5,1,4,3,2,6]=>2 [5,1,4,3,6,2]=>6 [5,1,4,6,2,3]=>8 [5,1,4,6,3,2]=>6 [5,1,6,2,3,4]=>4 [5,1,6,2,4,3]=>2 [5,1,6,3,2,4]=>8 [5,1,6,3,4,2]=>8 [5,1,6,4,2,3]=>8 [5,1,6,4,3,2]=>2 [5,2,1,3,4,6]=>4 [5,2,1,3,6,4]=>4 [5,2,1,4,3,6]=>2 [5,2,1,4,6,3]=>4 [5,2,1,6,3,4]=>4 [5,2,1,6,4,3]=>2 [5,2,3,1,4,6]=>4 [5,2,3,1,6,4]=>4 [5,2,3,4,1,6]=>3 [5,2,3,4,6,1]=>6 [5,2,3,6,1,4]=>1 [5,2,3,6,4,1]=>8 [5,2,4,1,3,6]=>6 [5,2,4,1,6,3]=>8 [5,2,4,3,1,6]=>3 [5,2,4,3,6,1]=>4 [5,2,4,6,1,3]=>8 [5,2,4,6,3,1]=>9 [5,2,6,1,3,4]=>6 [5,2,6,1,4,3]=>2 [5,2,6,3,1,4]=>8 [5,2,6,3,4,1]=>7 [5,2,6,4,1,3]=>3 [5,2,6,4,3,1]=>8 [5,3,1,2,4,6]=>4 [5,3,1,2,6,4]=>4 [5,3,1,4,2,6]=>6 [5,3,1,4,6,2]=>4 [5,3,1,6,2,4]=>6 [5,3,1,6,4,2]=>6 [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]=>8 [5,3,2,6,1,4]=>1 [5,3,2,6,4,1]=>6 [5,3,4,1,2,6]=>3 [5,3,4,1,6,2]=>8 [5,3,4,2,1,6]=>4 [5,3,4,2,6,1]=>6 [5,3,4,6,1,2]=>8 [5,3,4,6,2,1]=>6 [5,3,6,1,2,4]=>6 [5,3,6,1,4,2]=>4 [5,3,6,2,1,4]=>2 [5,3,6,2,4,1]=>4 [5,3,6,4,1,2]=>6 [5,3,6,4,2,1]=>12 [5,4,1,2,3,6]=>3 [5,4,1,2,6,3]=>2 [5,4,1,3,2,6]=>3 [5,4,1,3,6,2]=>6 [5,4,1,6,2,3]=>8 [5,4,1,6,3,2]=>4 [5,4,2,1,3,6]=>3 [5,4,2,1,6,3]=>2 [5,4,2,3,1,6]=>4 [5,4,2,3,6,1]=>6 [5,4,2,6,1,3]=>2 [5,4,2,6,3,1]=>8 [5,4,3,1,2,6]=>4 [5,4,3,1,6,2]=>2 [5,4,3,2,1,6]=>1 [5,4,3,2,6,1]=>2 [5,4,3,6,1,2]=>2 [5,4,3,6,2,1]=>3 [5,4,6,1,2,3]=>4 [5,4,6,1,3,2]=>6 [5,4,6,2,1,3]=>1 [5,4,6,2,3,1]=>6 [5,4,6,3,1,2]=>6 [5,4,6,3,2,1]=>4 [5,6,1,2,3,4]=>4 [5,6,1,2,4,3]=>3 [5,6,1,3,2,4]=>9 [5,6,1,3,4,2]=>8 [5,6,1,4,2,3]=>8 [5,6,1,4,3,2]=>2 [5,6,2,1,3,4]=>3 [5,6,2,1,4,3]=>4 [5,6,2,3,1,4]=>8 [5,6,2,3,4,1]=>12 [5,6,2,4,1,3]=>6 [5,6,2,4,3,1]=>6 [5,6,3,1,2,4]=>8 [5,6,3,1,4,2]=>6 [5,6,3,2,1,4]=>2 [5,6,3,2,4,1]=>6 [5,6,3,4,1,2]=>1 [5,6,3,4,2,1]=>6 [5,6,4,1,2,3]=>12 [5,6,4,1,3,2]=>6 [5,6,4,2,1,3]=>6 [5,6,4,2,3,1]=>6 [5,6,4,3,1,2]=>6 [5,6,4,3,2,1]=>5 [6,1,2,3,4,5]=>4 [6,1,2,3,5,4]=>2 [6,1,2,4,3,5]=>4 [6,1,2,4,5,3]=>8 [6,1,2,5,3,4]=>8 [6,1,2,5,4,3]=>4 [6,1,3,2,4,5]=>4 [6,1,3,2,5,4]=>4 [6,1,3,4,2,5]=>6 [6,1,3,4,5,2]=>6 [6,1,3,5,2,4]=>8 [6,1,3,5,4,2]=>8 [6,1,4,2,3,5]=>6 [6,1,4,2,5,3]=>8 [6,1,4,3,2,5]=>2 [6,1,4,3,5,2]=>4 [6,1,4,5,2,3]=>2 [6,1,4,5,3,2]=>6 [6,1,5,2,3,4]=>6 [6,1,5,2,4,3]=>8 [6,1,5,3,2,4]=>4 [6,1,5,3,4,2]=>6 [6,1,5,4,2,3]=>6 [6,1,5,4,3,2]=>2 [6,2,1,3,4,5]=>2 [6,2,1,3,5,4]=>3 [6,2,1,4,3,5]=>4 [6,2,1,4,5,3]=>6 [6,2,1,5,3,4]=>6 [6,2,1,5,4,3]=>2 [6,2,3,1,4,5]=>8 [6,2,3,1,5,4]=>6 [6,2,3,4,1,5]=>6 [6,2,3,4,5,1]=>9 [6,2,3,5,1,4]=>8 [6,2,3,5,4,1]=>7 [6,2,4,1,3,5]=>8 [6,2,4,1,5,3]=>6 [6,2,4,3,1,5]=>4 [6,2,4,3,5,1]=>3 [6,2,4,5,1,3]=>7 [6,2,4,5,3,1]=>6 [6,2,5,1,3,4]=>8 [6,2,5,1,4,3]=>6 [6,2,5,3,1,4]=>9 [6,2,5,3,4,1]=>6 [6,2,5,4,1,3]=>8 [6,2,5,4,3,1]=>3 [6,3,1,2,4,5]=>8 [6,3,1,2,5,4]=>6 [6,3,1,4,2,5]=>8 [6,3,1,4,5,2]=>8 [6,3,1,5,2,4]=>6 [6,3,1,5,4,2]=>6 [6,3,2,1,4,5]=>4 [6,3,2,1,5,4]=>2 [6,3,2,4,1,5]=>8 [6,3,2,4,5,1]=>7 [6,3,2,5,1,4]=>6 [6,3,2,5,4,1]=>4 [6,3,4,1,2,5]=>2 [6,3,4,1,5,2]=>7 [6,3,4,2,1,5]=>6 [6,3,4,2,5,1]=>6 [6,3,4,5,1,2]=>12 [6,3,4,5,2,1]=>4 [6,3,5,1,2,4]=>7 [6,3,5,1,4,2]=>4 [6,3,5,2,1,4]=>8 [6,3,5,2,4,1]=>12 [6,3,5,4,1,2]=>6 [6,3,5,4,2,1]=>4 [6,4,1,2,3,5]=>6 [6,4,1,2,5,3]=>8 [6,4,1,3,2,5]=>4 [6,4,1,3,5,2]=>9 [6,4,1,5,2,3]=>7 [6,4,1,5,3,2]=>8 [6,4,2,1,3,5]=>8 [6,4,2,1,5,3]=>6 [6,4,2,3,1,5]=>6 [6,4,2,3,5,1]=>6 [6,4,2,5,1,3]=>4 [6,4,2,5,3,1]=>12 [6,4,3,1,2,5]=>6 [6,4,3,1,5,2]=>8 [6,4,3,2,1,5]=>2 [6,4,3,2,5,1]=>3 [6,4,3,5,1,2]=>6 [6,4,3,5,2,1]=>4 [6,4,5,1,2,3]=>12 [6,4,5,1,3,2]=>6 [6,4,5,2,1,3]=>6 [6,4,5,2,3,1]=>6 [6,4,5,3,1,2]=>6 [6,4,5,3,2,1]=>5 [6,5,1,2,3,4]=>9 [6,5,1,2,4,3]=>7 [6,5,1,3,2,4]=>3 [6,5,1,3,4,2]=>6 [6,5,1,4,2,3]=>6 [6,5,1,4,3,2]=>3 [6,5,2,1,3,4]=>7 [6,5,2,1,4,3]=>4 [6,5,2,3,1,4]=>6 [6,5,2,3,4,1]=>4 [6,5,2,4,1,3]=>12 [6,5,2,4,3,1]=>4 [6,5,3,1,2,4]=>6 [6,5,3,1,4,2]=>12 [6,5,3,2,1,4]=>3 [6,5,3,2,4,1]=>4 [6,5,3,4,1,2]=>6 [6,5,3,4,2,1]=>5 [6,5,4,1,2,3]=>4 [6,5,4,1,3,2]=>4 [6,5,4,2,1,3]=>4 [6,5,4,2,3,1]=>5 [6,5,4,3,1,2]=>5 [6,5,4,3,2,1]=>1 [1,2,3,4,5,6,7]=>1 [1,2,3,4,5,7,6]=>1 [1,2,3,4,6,5,7]=>1 [1,2,3,4,6,7,5]=>2 [1,2,3,4,7,5,6]=>2 [1,2,3,4,7,6,5]=>1 [1,2,3,5,4,6,7]=>1 [1,2,3,5,4,7,6]=>1 [1,2,3,5,6,4,7]=>2 [1,2,3,5,6,7,4]=>2 [1,2,3,5,7,4,6]=>2 [1,2,3,5,7,6,4]=>2 [1,2,3,6,4,5,7]=>2 [1,2,3,6,4,7,5]=>2 [1,2,3,6,5,4,7]=>1 [1,2,3,6,5,7,4]=>2 [1,2,3,6,7,4,5]=>1 [1,2,3,6,7,5,4]=>3 [1,2,3,7,4,5,6]=>2 [1,2,3,7,4,6,5]=>2 [1,2,3,7,5,4,6]=>2 [1,2,3,7,5,6,4]=>3 [1,2,3,7,6,4,5]=>3 [1,2,3,7,6,5,4]=>1 [1,2,4,3,5,6,7]=>1 [1,2,4,3,5,7,6]=>1 [1,2,4,3,6,5,7]=>1 [1,2,4,3,6,7,5]=>2 [1,2,4,3,7,5,6]=>2 [1,2,4,3,7,6,5]=>1 [1,2,4,5,3,6,7]=>2 [1,2,4,5,3,7,6]=>2 [1,2,4,5,6,3,7]=>2 [1,2,4,5,6,7,3]=>4 [1,2,4,5,7,3,6]=>2 [1,2,4,5,7,6,3]=>4 [1,2,4,6,3,5,7]=>2 [1,2,4,6,3,7,5]=>2 [1,2,4,6,5,3,7]=>2 [1,2,4,6,5,7,3]=>2 [1,2,4,6,7,3,5]=>4 [1,2,4,6,7,5,3]=>4 [1,2,4,7,3,5,6]=>2 [1,2,4,7,3,6,5]=>2 [1,2,4,7,5,3,6]=>4 [1,2,4,7,5,6,3]=>4 [1,2,4,7,6,3,5]=>2 [1,2,4,7,6,5,3]=>2 [1,2,5,3,4,6,7]=>2 [1,2,5,3,4,7,6]=>2 [1,2,5,3,6,4,7]=>2 [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]=>1 [1,2,5,4,3,7,6]=>1 [1,2,5,4,6,3,7]=>2 [1,2,5,4,6,7,3]=>4 [1,2,5,4,7,3,6]=>2 [1,2,5,4,7,6,3]=>2 [1,2,5,6,3,4,7]=>1 [1,2,5,6,3,7,4]=>4 [1,2,5,6,4,3,7]=>3 [1,2,5,6,4,7,3]=>4 [1,2,5,6,7,3,4]=>6 [1,2,5,6,7,4,3]=>3 [1,2,5,7,3,4,6]=>4 [1,2,5,7,3,6,4]=>1 [1,2,5,7,4,3,6]=>2 [1,2,5,7,4,6,3]=>6 [1,2,5,7,6,3,4]=>2 [1,2,5,7,6,4,3]=>3 [1,2,6,3,4,5,7]=>2 [1,2,6,3,4,7,5]=>2 [1,2,6,3,5,4,7]=>2 [1,2,6,3,5,7,4]=>4 [1,2,6,3,7,4,5]=>4 [1,2,6,3,7,5,4]=>2 [1,2,6,4,3,5,7]=>2 [1,2,6,4,3,7,5]=>2 [1,2,6,4,5,3,7]=>3 [1,2,6,4,5,7,3]=>4 [1,2,6,4,7,3,5]=>1 [1,2,6,4,7,5,3]=>6 [1,2,6,5,3,4,7]=>3 [1,2,6,5,3,7,4]=>2 [1,2,6,5,4,3,7]=>1 [1,2,6,5,4,7,3]=>2 [1,2,6,5,7,3,4]=>2 [1,2,6,5,7,4,3]=>3 [1,2,6,7,3,4,5]=>6 [1,2,6,7,3,5,4]=>2 [1,2,6,7,4,3,5]=>2 [1,2,6,7,4,5,3]=>3 [1,2,6,7,5,3,4]=>3 [1,2,6,7,5,4,3]=>4 [1,2,7,3,4,5,6]=>4 [1,2,7,3,4,6,5]=>4 [1,2,7,3,5,4,6]=>2 [1,2,7,3,5,6,4]=>4 [1,2,7,3,6,4,5]=>4 [1,2,7,3,6,5,4]=>2 [1,2,7,4,3,5,6]=>4 [1,2,7,4,3,6,5]=>2 [1,2,7,4,5,3,6]=>4 [1,2,7,4,5,6,3]=>3 [1,2,7,4,6,3,5]=>6 [1,2,7,4,6,5,3]=>3 [1,2,7,5,3,4,6]=>4 [1,2,7,5,3,6,4]=>6 [1,2,7,5,4,3,6]=>2 [1,2,7,5,4,6,3]=>3 [1,2,7,5,6,3,4]=>3 [1,2,7,5,6,4,3]=>4 [1,2,7,6,3,4,5]=>3 [1,2,7,6,3,5,4]=>3 [1,2,7,6,4,3,5]=>3 [1,2,7,6,4,5,3]=>4 [1,2,7,6,5,3,4]=>4 [1,2,7,6,5,4,3]=>1 [1,3,2,4,5,6,7]=>1 [1,3,2,4,5,7,6]=>1 [1,3,2,4,6,5,7]=>1 [1,3,2,4,6,7,5]=>2 [1,3,2,4,7,5,6]=>2 [1,3,2,4,7,6,5]=>1 [1,3,2,5,4,6,7]=>1 [1,3,2,5,4,7,6]=>1 [1,3,2,5,6,4,7]=>2 [1,3,2,5,6,7,4]=>2 [1,3,2,5,7,4,6]=>2 [1,3,2,5,7,6,4]=>2 [1,3,2,6,4,5,7]=>2 [1,3,2,6,4,7,5]=>2 [1,3,2,6,5,4,7]=>1 [1,3,2,6,5,7,4]=>2 [1,3,2,6,7,4,5]=>1 [1,3,2,6,7,5,4]=>3 [1,3,2,7,4,5,6]=>2 [1,3,2,7,4,6,5]=>2 [1,3,2,7,5,4,6]=>2 [1,3,2,7,5,6,4]=>3 [1,3,2,7,6,4,5]=>3 [1,3,2,7,6,5,4]=>1 [1,3,4,2,5,6,7]=>2 [1,3,4,2,5,7,6]=>2 [1,3,4,2,6,5,7]=>2 [1,3,4,2,6,7,5]=>4 [1,3,4,2,7,5,6]=>4 [1,3,4,2,7,6,5]=>2 [1,3,4,5,2,6,7]=>2 [1,3,4,5,2,7,6]=>2 [1,3,4,5,6,2,7]=>4 [1,3,4,5,6,7,2]=>4 [1,3,4,5,7,2,6]=>4 [1,3,4,5,7,6,2]=>2 [1,3,4,6,2,5,7]=>2 [1,3,4,6,2,7,5]=>2 [1,3,4,6,5,2,7]=>4 [1,3,4,6,5,7,2]=>4 [1,3,4,6,7,2,5]=>4 [1,3,4,6,7,5,2]=>8 [1,3,4,7,2,5,6]=>2 [1,3,4,7,2,6,5]=>2 [1,3,4,7,5,2,6]=>8 [1,3,4,7,5,6,2]=>8 [1,3,4,7,6,2,5]=>4 [1,3,4,7,6,5,2]=>4 [1,3,5,2,4,6,7]=>2 [1,3,5,2,4,7,6]=>2 [1,3,5,2,6,4,7]=>2 [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]=>2 [1,3,5,4,6,7,2]=>4 [1,3,5,4,7,2,6]=>2 [1,3,5,4,7,6,2]=>4 [1,3,5,6,2,4,7]=>4 [1,3,5,6,2,7,4]=>8 [1,3,5,6,4,2,7]=>4 [1,3,5,6,4,7,2]=>6 [1,3,5,6,7,2,4]=>4 [1,3,5,6,7,4,2]=>6 [1,3,5,7,2,4,6]=>8 [1,3,5,7,2,6,4]=>4 [1,3,5,7,4,2,6]=>4 [1,3,5,7,4,6,2]=>8 [1,3,5,7,6,2,4]=>2 [1,3,5,7,6,4,2]=>8 [1,3,6,2,4,5,7]=>2 [1,3,6,2,4,7,5]=>2 [1,3,6,2,5,4,7]=>2 [1,3,6,2,5,7,4]=>4 [1,3,6,2,7,4,5]=>4 [1,3,6,2,7,5,4]=>2 [1,3,6,4,2,5,7]=>4 [1,3,6,4,2,7,5]=>4 [1,3,6,4,5,2,7]=>4 [1,3,6,4,5,7,2]=>6 [1,3,6,4,7,2,5]=>4 [1,3,6,4,7,5,2]=>8 [1,3,6,5,2,4,7]=>2 [1,3,6,5,2,7,4]=>4 [1,3,6,5,4,2,7]=>2 [1,3,6,5,4,7,2]=>2 [1,3,6,5,7,2,4]=>8 [1,3,6,5,7,4,2]=>4 [1,3,6,7,2,4,5]=>8 [1,3,6,7,2,5,4]=>4 [1,3,6,7,4,2,5]=>8 [1,3,6,7,4,5,2]=>2 [1,3,6,7,5,2,4]=>8 [1,3,6,7,5,4,2]=>6 [1,3,7,2,4,5,6]=>4 [1,3,7,2,4,6,5]=>4 [1,3,7,2,5,4,6]=>2 [1,3,7,2,5,6,4]=>4 [1,3,7,2,6,4,5]=>4 [1,3,7,2,6,5,4]=>2 [1,3,7,4,2,5,6]=>8 [1,3,7,4,2,6,5]=>4 [1,3,7,4,5,2,6]=>2 [1,3,7,4,5,6,2]=>6 [1,3,7,4,6,2,5]=>4 [1,3,7,4,6,5,2]=>8 [1,3,7,5,2,4,6]=>4 [1,3,7,5,2,6,4]=>8 [1,3,7,5,4,2,6]=>6 [1,3,7,5,4,6,2]=>4 [1,3,7,5,6,2,4]=>8 [1,3,7,5,6,4,2]=>6 [1,3,7,6,2,4,5]=>2 [1,3,7,6,2,5,4]=>2 [1,3,7,6,4,2,5]=>6 [1,3,7,6,4,5,2]=>6 [1,3,7,6,5,2,4]=>2 [1,3,7,6,5,4,2]=>2 [1,4,2,3,5,6,7]=>2 [1,4,2,3,5,7,6]=>2 [1,4,2,3,6,5,7]=>2 [1,4,2,3,6,7,5]=>4 [1,4,2,3,7,5,6]=>4 [1,4,2,3,7,6,5]=>2 [1,4,2,5,3,6,7]=>2 [1,4,2,5,3,7,6]=>2 [1,4,2,5,6,3,7]=>2 [1,4,2,5,6,7,3]=>4 [1,4,2,5,7,3,6]=>2 [1,4,2,5,7,6,3]=>4 [1,4,2,6,3,5,7]=>2 [1,4,2,6,3,7,5]=>2 [1,4,2,6,5,3,7]=>2 [1,4,2,6,5,7,3]=>2 [1,4,2,6,7,3,5]=>4 [1,4,2,6,7,5,3]=>4 [1,4,2,7,3,5,6]=>2 [1,4,2,7,3,6,5]=>2 [1,4,2,7,5,3,6]=>4 [1,4,2,7,5,6,3]=>4 [1,4,2,7,6,3,5]=>2 [1,4,2,7,6,5,3]=>2 [1,4,3,2,5,6,7]=>1 [1,4,3,2,5,7,6]=>1 [1,4,3,2,6,5,7]=>1 [1,4,3,2,6,7,5]=>2 [1,4,3,2,7,5,6]=>2 [1,4,3,2,7,6,5]=>1 [1,4,3,5,2,6,7]=>2 [1,4,3,5,2,7,6]=>2 [1,4,3,5,6,2,7]=>4 [1,4,3,5,6,7,2]=>2 [1,4,3,5,7,2,6]=>4 [1,4,3,5,7,6,2]=>3 [1,4,3,6,2,5,7]=>2 [1,4,3,6,2,7,5]=>2 [1,4,3,6,5,2,7]=>2 [1,4,3,6,5,7,2]=>4 [1,4,3,6,7,2,5]=>4 [1,4,3,6,7,5,2]=>6 [1,4,3,7,2,5,6]=>2 [1,4,3,7,2,6,5]=>2 [1,4,3,7,5,2,6]=>4 [1,4,3,7,5,6,2]=>6 [1,4,3,7,6,2,5]=>2 [1,4,3,7,6,5,2]=>2 [1,4,5,2,3,6,7]=>1 [1,4,5,2,3,7,6]=>1 [1,4,5,2,6,3,7]=>4 [1,4,5,2,6,7,3]=>4 [1,4,5,2,7,3,6]=>4 [1,4,5,2,7,6,3]=>4 [1,4,5,3,2,6,7]=>3 [1,4,5,3,2,7,6]=>3 [1,4,5,3,6,2,7]=>4 [1,4,5,3,6,7,2]=>8 [1,4,5,3,7,2,6]=>4 [1,4,5,3,7,6,2]=>6 [1,4,5,6,2,3,7]=>6 [1,4,5,6,2,7,3]=>4 [1,4,5,6,3,2,7]=>3 [1,4,5,6,3,7,2]=>6 [1,4,5,6,7,2,3]=>4 [1,4,5,6,7,3,2]=>9 [1,4,5,7,2,3,6]=>8 [1,4,5,7,2,6,3]=>6 [1,4,5,7,3,2,6]=>2 [1,4,5,7,3,6,2]=>8 [1,4,5,7,6,2,3]=>3 [1,4,5,7,6,3,2]=>7 [1,4,6,2,3,5,7]=>4 [1,4,6,2,3,7,5]=>4 [1,4,6,2,5,3,7]=>1 [1,4,6,2,5,7,3]=>4 [1,4,6,2,7,3,5]=>3 [1,4,6,2,7,5,3]=>6 [1,4,6,3,2,5,7]=>2 [1,4,6,3,2,7,5]=>2 [1,4,6,3,5,2,7]=>6 [1,4,6,3,5,7,2]=>8 [1,4,6,3,7,2,5]=>6 [1,4,6,3,7,5,2]=>6 [1,4,6,5,2,3,7]=>2 [1,4,6,5,2,7,3]=>8 [1,4,6,5,3,2,7]=>3 [1,4,6,5,3,7,2]=>4
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Description
The number of permutations with the same antidiagonal sums.
The X-ray of a permutation $\pi$ is the vector of the sums of the antidiagonals of the permutation matrix of $\pi$, read from left to right. For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is
$$\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right),$$
so its X-ray is $(0, 1, 1, 1, 0, 0, 0, 2, 0)$.
This statistic records the number of permutations having the same X-ray as the given permutation. In [1] this is called the degeneracy of the X-ray of the permutation.
By [prop.1, 1], the number of different X-rays of permutations of size $n$ equals the number of nondecreasing differences of permutations of size $n$, [2].
References
[1] Bebeacua, C., Mansour, T., Postnikov, A., Severini, S. On the X-rays of permutations arXiv:math/0506334
[2] Number of nondecreasing sequences that are differences of two permutations of 1,2,...,n. OEIS:A019589
Code
def X_ray(pi):
    P = Permutation(pi).to_matrix()
    n = P.nrows()
    return tuple(sum(P[k-1-j][j] for j in range(max(0, k-n), min(k,n)))
                 for k in range(1,2*n))

@cached_function
def X_rays(n):
    return sorted(X_ray(pi) for pi in Permutations(n))
    
def statistic(pi):
    return X_rays(pi.size()).count(X_ray(pi))
Created
Jul 14, 2017 at 09:26 by Martin Rubey
Updated
Jul 14, 2017 at 11:53 by Martin Rubey