edit this statistic or download as text // json
Identifier
Values
=>
[1]=>0 [1,2]=>0 [2,1]=>1 [1,2,3]=>0 [1,3,2]=>2 [2,1,3]=>1 [2,3,1]=>3 [3,1,2]=>1 [3,2,1]=>2 [1,2,3,4]=>0 [1,2,4,3]=>3 [1,3,2,4]=>2 [1,3,4,2]=>5 [1,4,2,3]=>2 [1,4,3,2]=>3 [2,1,3,4]=>1 [2,1,4,3]=>4 [2,3,1,4]=>3 [2,3,4,1]=>6 [2,4,1,3]=>3 [2,4,3,1]=>4 [3,1,2,4]=>1 [3,1,4,2]=>4 [3,2,1,4]=>2 [3,2,4,1]=>5 [3,4,1,2]=>3 [3,4,2,1]=>4 [4,1,2,3]=>1 [4,1,3,2]=>2 [4,2,1,3]=>2 [4,2,3,1]=>3 [4,3,1,2]=>4 [4,3,2,1]=>5 [1,2,3,4,5]=>0 [1,2,3,5,4]=>4 [1,2,4,3,5]=>3 [1,2,4,5,3]=>7 [1,2,5,3,4]=>3 [1,2,5,4,3]=>4 [1,3,2,4,5]=>2 [1,3,2,5,4]=>6 [1,3,4,2,5]=>5 [1,3,4,5,2]=>9 [1,3,5,2,4]=>5 [1,3,5,4,2]=>6 [1,4,2,3,5]=>2 [1,4,2,5,3]=>6 [1,4,3,2,5]=>3 [1,4,3,5,2]=>7 [1,4,5,2,3]=>5 [1,4,5,3,2]=>6 [1,5,2,3,4]=>2 [1,5,2,4,3]=>3 [1,5,3,2,4]=>3 [1,5,3,4,2]=>4 [1,5,4,2,3]=>6 [1,5,4,3,2]=>7 [2,1,3,4,5]=>1 [2,1,3,5,4]=>5 [2,1,4,3,5]=>4 [2,1,4,5,3]=>8 [2,1,5,3,4]=>4 [2,1,5,4,3]=>5 [2,3,1,4,5]=>3 [2,3,1,5,4]=>7 [2,3,4,1,5]=>6 [2,3,4,5,1]=>10 [2,3,5,1,4]=>6 [2,3,5,4,1]=>7 [2,4,1,3,5]=>3 [2,4,1,5,3]=>7 [2,4,3,1,5]=>4 [2,4,3,5,1]=>8 [2,4,5,1,3]=>6 [2,4,5,3,1]=>7 [2,5,1,3,4]=>3 [2,5,1,4,3]=>4 [2,5,3,1,4]=>4 [2,5,3,4,1]=>5 [2,5,4,1,3]=>7 [2,5,4,3,1]=>8 [3,1,2,4,5]=>1 [3,1,2,5,4]=>5 [3,1,4,2,5]=>4 [3,1,4,5,2]=>8 [3,1,5,2,4]=>4 [3,1,5,4,2]=>5 [3,2,1,4,5]=>2 [3,2,1,5,4]=>6 [3,2,4,1,5]=>5 [3,2,4,5,1]=>9 [3,2,5,1,4]=>5 [3,2,5,4,1]=>6 [3,4,1,2,5]=>3 [3,4,1,5,2]=>7 [3,4,2,1,5]=>4 [3,4,2,5,1]=>8 [3,4,5,1,2]=>6 [3,4,5,2,1]=>7 [3,5,1,2,4]=>3 [3,5,1,4,2]=>4 [3,5,2,1,4]=>4 [3,5,2,4,1]=>5 [3,5,4,1,2]=>7 [3,5,4,2,1]=>8 [4,1,2,3,5]=>1 [4,1,2,5,3]=>5 [4,1,3,2,5]=>2 [4,1,3,5,2]=>6 [4,1,5,2,3]=>4 [4,1,5,3,2]=>5 [4,2,1,3,5]=>2 [4,2,1,5,3]=>6 [4,2,3,1,5]=>3 [4,2,3,5,1]=>7 [4,2,5,1,3]=>5 [4,2,5,3,1]=>6 [4,3,1,2,5]=>4 [4,3,1,5,2]=>8 [4,3,2,1,5]=>5 [4,3,2,5,1]=>9 [4,3,5,1,2]=>7 [4,3,5,2,1]=>8 [4,5,1,2,3]=>3 [4,5,1,3,2]=>4 [4,5,2,1,3]=>4 [4,5,2,3,1]=>5 [4,5,3,1,2]=>5 [4,5,3,2,1]=>6 [5,1,2,3,4]=>1 [5,1,2,4,3]=>2 [5,1,3,2,4]=>2 [5,1,3,4,2]=>3 [5,1,4,2,3]=>5 [5,1,4,3,2]=>6 [5,2,1,3,4]=>2 [5,2,1,4,3]=>3 [5,2,3,1,4]=>3 [5,2,3,4,1]=>4 [5,2,4,1,3]=>6 [5,2,4,3,1]=>7 [5,3,1,2,4]=>4 [5,3,1,4,2]=>5 [5,3,2,1,4]=>5 [5,3,2,4,1]=>6 [5,3,4,1,2]=>8 [5,3,4,2,1]=>9 [5,4,1,2,3]=>4 [5,4,1,3,2]=>5 [5,4,2,1,3]=>5 [5,4,2,3,1]=>6 [5,4,3,1,2]=>6 [5,4,3,2,1]=>7 [1,2,3,4,5,6]=>0 [1,2,3,4,6,5]=>5 [1,2,3,5,4,6]=>4 [1,2,3,5,6,4]=>9 [1,2,3,6,4,5]=>4 [1,2,3,6,5,4]=>5 [1,2,4,3,5,6]=>3 [1,2,4,3,6,5]=>8 [1,2,4,5,3,6]=>7 [1,2,4,5,6,3]=>12 [1,2,4,6,3,5]=>7 [1,2,4,6,5,3]=>8 [1,2,5,3,4,6]=>3 [1,2,5,3,6,4]=>8 [1,2,5,4,3,6]=>4 [1,2,5,4,6,3]=>9 [1,2,5,6,3,4]=>7 [1,2,5,6,4,3]=>8 [1,2,6,3,4,5]=>3 [1,2,6,3,5,4]=>4 [1,2,6,4,3,5]=>4 [1,2,6,4,5,3]=>5 [1,2,6,5,3,4]=>8 [1,2,6,5,4,3]=>9 [1,3,2,4,5,6]=>2 [1,3,2,4,6,5]=>7 [1,3,2,5,4,6]=>6 [1,3,2,5,6,4]=>11 [1,3,2,6,4,5]=>6 [1,3,2,6,5,4]=>7 [1,3,4,2,5,6]=>5 [1,3,4,2,6,5]=>10 [1,3,4,5,2,6]=>9 [1,3,4,5,6,2]=>14 [1,3,4,6,2,5]=>9 [1,3,4,6,5,2]=>10 [1,3,5,2,4,6]=>5 [1,3,5,2,6,4]=>10 [1,3,5,4,2,6]=>6 [1,3,5,4,6,2]=>11 [1,3,5,6,2,4]=>9 [1,3,5,6,4,2]=>10 [1,3,6,2,4,5]=>5 [1,3,6,2,5,4]=>6 [1,3,6,4,2,5]=>6 [1,3,6,4,5,2]=>7 [1,3,6,5,2,4]=>10 [1,3,6,5,4,2]=>11 [1,4,2,3,5,6]=>2 [1,4,2,3,6,5]=>7 [1,4,2,5,3,6]=>6 [1,4,2,5,6,3]=>11 [1,4,2,6,3,5]=>6 [1,4,2,6,5,3]=>7 [1,4,3,2,5,6]=>3 [1,4,3,2,6,5]=>8 [1,4,3,5,2,6]=>7 [1,4,3,5,6,2]=>12 [1,4,3,6,2,5]=>7 [1,4,3,6,5,2]=>8 [1,4,5,2,3,6]=>5 [1,4,5,2,6,3]=>10 [1,4,5,3,2,6]=>6 [1,4,5,3,6,2]=>11 [1,4,5,6,2,3]=>9 [1,4,5,6,3,2]=>10 [1,4,6,2,3,5]=>5 [1,4,6,2,5,3]=>6 [1,4,6,3,2,5]=>6 [1,4,6,3,5,2]=>7 [1,4,6,5,2,3]=>10 [1,4,6,5,3,2]=>11 [1,5,2,3,4,6]=>2 [1,5,2,3,6,4]=>7 [1,5,2,4,3,6]=>3 [1,5,2,4,6,3]=>8 [1,5,2,6,3,4]=>6 [1,5,2,6,4,3]=>7 [1,5,3,2,4,6]=>3 [1,5,3,2,6,4]=>8 [1,5,3,4,2,6]=>4 [1,5,3,4,6,2]=>9 [1,5,3,6,2,4]=>7 [1,5,3,6,4,2]=>8 [1,5,4,2,3,6]=>6 [1,5,4,2,6,3]=>11 [1,5,4,3,2,6]=>7 [1,5,4,3,6,2]=>12 [1,5,4,6,2,3]=>10 [1,5,4,6,3,2]=>11 [1,5,6,2,3,4]=>5 [1,5,6,2,4,3]=>6 [1,5,6,3,2,4]=>6 [1,5,6,3,4,2]=>7 [1,5,6,4,2,3]=>7 [1,5,6,4,3,2]=>8 [1,6,2,3,4,5]=>2 [1,6,2,3,5,4]=>3 [1,6,2,4,3,5]=>3 [1,6,2,4,5,3]=>4 [1,6,2,5,3,4]=>7 [1,6,2,5,4,3]=>8 [1,6,3,2,4,5]=>3 [1,6,3,2,5,4]=>4 [1,6,3,4,2,5]=>4 [1,6,3,4,5,2]=>5 [1,6,3,5,2,4]=>8 [1,6,3,5,4,2]=>9 [1,6,4,2,3,5]=>6 [1,6,4,2,5,3]=>7 [1,6,4,3,2,5]=>7 [1,6,4,3,5,2]=>8 [1,6,4,5,2,3]=>11 [1,6,4,5,3,2]=>12 [1,6,5,2,3,4]=>6 [1,6,5,2,4,3]=>7 [1,6,5,3,2,4]=>7 [1,6,5,3,4,2]=>8 [1,6,5,4,2,3]=>8 [1,6,5,4,3,2]=>9 [2,1,3,4,5,6]=>1 [2,1,3,4,6,5]=>6 [2,1,3,5,4,6]=>5 [2,1,3,5,6,4]=>10 [2,1,3,6,4,5]=>5 [2,1,3,6,5,4]=>6 [2,1,4,3,5,6]=>4 [2,1,4,3,6,5]=>9 [2,1,4,5,3,6]=>8 [2,1,4,5,6,3]=>13 [2,1,4,6,3,5]=>8 [2,1,4,6,5,3]=>9 [2,1,5,3,4,6]=>4 [2,1,5,3,6,4]=>9 [2,1,5,4,3,6]=>5 [2,1,5,4,6,3]=>10 [2,1,5,6,3,4]=>8 [2,1,5,6,4,3]=>9 [2,1,6,3,4,5]=>4 [2,1,6,3,5,4]=>5 [2,1,6,4,3,5]=>5 [2,1,6,4,5,3]=>6 [2,1,6,5,3,4]=>9 [2,1,6,5,4,3]=>10 [2,3,1,4,5,6]=>3 [2,3,1,4,6,5]=>8 [2,3,1,5,4,6]=>7 [2,3,1,5,6,4]=>12 [2,3,1,6,4,5]=>7 [2,3,1,6,5,4]=>8 [2,3,4,1,5,6]=>6 [2,3,4,1,6,5]=>11 [2,3,4,5,1,6]=>10 [2,3,4,5,6,1]=>15 [2,3,4,6,1,5]=>10 [2,3,4,6,5,1]=>11 [2,3,5,1,4,6]=>6 [2,3,5,1,6,4]=>11 [2,3,5,4,1,6]=>7 [2,3,5,4,6,1]=>12 [2,3,5,6,1,4]=>10 [2,3,5,6,4,1]=>11 [2,3,6,1,4,5]=>6 [2,3,6,1,5,4]=>7 [2,3,6,4,1,5]=>7 [2,3,6,4,5,1]=>8 [2,3,6,5,1,4]=>11 [2,3,6,5,4,1]=>12 [2,4,1,3,5,6]=>3 [2,4,1,3,6,5]=>8 [2,4,1,5,3,6]=>7 [2,4,1,5,6,3]=>12 [2,4,1,6,3,5]=>7 [2,4,1,6,5,3]=>8 [2,4,3,1,5,6]=>4 [2,4,3,1,6,5]=>9 [2,4,3,5,1,6]=>8 [2,4,3,5,6,1]=>13 [2,4,3,6,1,5]=>8 [2,4,3,6,5,1]=>9 [2,4,5,1,3,6]=>6 [2,4,5,1,6,3]=>11 [2,4,5,3,1,6]=>7 [2,4,5,3,6,1]=>12 [2,4,5,6,1,3]=>10 [2,4,5,6,3,1]=>11 [2,4,6,1,3,5]=>6 [2,4,6,1,5,3]=>7 [2,4,6,3,1,5]=>7 [2,4,6,3,5,1]=>8 [2,4,6,5,1,3]=>11 [2,4,6,5,3,1]=>12 [2,5,1,3,4,6]=>3 [2,5,1,3,6,4]=>8 [2,5,1,4,3,6]=>4 [2,5,1,4,6,3]=>9 [2,5,1,6,3,4]=>7 [2,5,1,6,4,3]=>8 [2,5,3,1,4,6]=>4 [2,5,3,1,6,4]=>9 [2,5,3,4,1,6]=>5 [2,5,3,4,6,1]=>10 [2,5,3,6,1,4]=>8 [2,5,3,6,4,1]=>9 [2,5,4,1,3,6]=>7 [2,5,4,1,6,3]=>12 [2,5,4,3,1,6]=>8 [2,5,4,3,6,1]=>13 [2,5,4,6,1,3]=>11 [2,5,4,6,3,1]=>12 [2,5,6,1,3,4]=>6 [2,5,6,1,4,3]=>7 [2,5,6,3,1,4]=>7 [2,5,6,3,4,1]=>8 [2,5,6,4,1,3]=>8 [2,5,6,4,3,1]=>9 [2,6,1,3,4,5]=>3 [2,6,1,3,5,4]=>4 [2,6,1,4,3,5]=>4 [2,6,1,4,5,3]=>5 [2,6,1,5,3,4]=>8 [2,6,1,5,4,3]=>9 [2,6,3,1,4,5]=>4 [2,6,3,1,5,4]=>5 [2,6,3,4,1,5]=>5 [2,6,3,4,5,1]=>6 [2,6,3,5,1,4]=>9 [2,6,3,5,4,1]=>10 [2,6,4,1,3,5]=>7 [2,6,4,1,5,3]=>8 [2,6,4,3,1,5]=>8 [2,6,4,3,5,1]=>9 [2,6,4,5,1,3]=>12 [2,6,4,5,3,1]=>13 [2,6,5,1,3,4]=>7 [2,6,5,1,4,3]=>8 [2,6,5,3,1,4]=>8 [2,6,5,3,4,1]=>9 [2,6,5,4,1,3]=>9 [2,6,5,4,3,1]=>10 [3,1,2,4,5,6]=>1 [3,1,2,4,6,5]=>6 [3,1,2,5,4,6]=>5 [3,1,2,5,6,4]=>10 [3,1,2,6,4,5]=>5 [3,1,2,6,5,4]=>6 [3,1,4,2,5,6]=>4 [3,1,4,2,6,5]=>9 [3,1,4,5,2,6]=>8 [3,1,4,5,6,2]=>13 [3,1,4,6,2,5]=>8 [3,1,4,6,5,2]=>9 [3,1,5,2,4,6]=>4 [3,1,5,2,6,4]=>9 [3,1,5,4,2,6]=>5 [3,1,5,4,6,2]=>10 [3,1,5,6,2,4]=>8 [3,1,5,6,4,2]=>9 [3,1,6,2,4,5]=>4 [3,1,6,2,5,4]=>5 [3,1,6,4,2,5]=>5 [3,1,6,4,5,2]=>6 [3,1,6,5,2,4]=>9 [3,1,6,5,4,2]=>10 [3,2,1,4,5,6]=>2 [3,2,1,4,6,5]=>7 [3,2,1,5,4,6]=>6 [3,2,1,5,6,4]=>11 [3,2,1,6,4,5]=>6 [3,2,1,6,5,4]=>7 [3,2,4,1,5,6]=>5 [3,2,4,1,6,5]=>10 [3,2,4,5,1,6]=>9 [3,2,4,5,6,1]=>14 [3,2,4,6,1,5]=>9 [3,2,4,6,5,1]=>10 [3,2,5,1,4,6]=>5 [3,2,5,1,6,4]=>10 [3,2,5,4,1,6]=>6 [3,2,5,4,6,1]=>11 [3,2,5,6,1,4]=>9 [3,2,5,6,4,1]=>10 [3,2,6,1,4,5]=>5 [3,2,6,1,5,4]=>6 [3,2,6,4,1,5]=>6 [3,2,6,4,5,1]=>7 [3,2,6,5,1,4]=>10 [3,2,6,5,4,1]=>11 [3,4,1,2,5,6]=>3 [3,4,1,2,6,5]=>8 [3,4,1,5,2,6]=>7 [3,4,1,5,6,2]=>12 [3,4,1,6,2,5]=>7 [3,4,1,6,5,2]=>8 [3,4,2,1,5,6]=>4 [3,4,2,1,6,5]=>9 [3,4,2,5,1,6]=>8 [3,4,2,5,6,1]=>13 [3,4,2,6,1,5]=>8 [3,4,2,6,5,1]=>9 [3,4,5,1,2,6]=>6 [3,4,5,1,6,2]=>11 [3,4,5,2,1,6]=>7 [3,4,5,2,6,1]=>12 [3,4,5,6,1,2]=>10 [3,4,5,6,2,1]=>11 [3,4,6,1,2,5]=>6 [3,4,6,1,5,2]=>7 [3,4,6,2,1,5]=>7 [3,4,6,2,5,1]=>8 [3,4,6,5,1,2]=>11 [3,4,6,5,2,1]=>12 [3,5,1,2,4,6]=>3 [3,5,1,2,6,4]=>8 [3,5,1,4,2,6]=>4 [3,5,1,4,6,2]=>9 [3,5,1,6,2,4]=>7 [3,5,1,6,4,2]=>8 [3,5,2,1,4,6]=>4 [3,5,2,1,6,4]=>9 [3,5,2,4,1,6]=>5 [3,5,2,4,6,1]=>10 [3,5,2,6,1,4]=>8 [3,5,2,6,4,1]=>9 [3,5,4,1,2,6]=>7 [3,5,4,1,6,2]=>12 [3,5,4,2,1,6]=>8 [3,5,4,2,6,1]=>13 [3,5,4,6,1,2]=>11 [3,5,4,6,2,1]=>12 [3,5,6,1,2,4]=>6 [3,5,6,1,4,2]=>7 [3,5,6,2,1,4]=>7 [3,5,6,2,4,1]=>8 [3,5,6,4,1,2]=>8 [3,5,6,4,2,1]=>9 [3,6,1,2,4,5]=>3 [3,6,1,2,5,4]=>4 [3,6,1,4,2,5]=>4 [3,6,1,4,5,2]=>5 [3,6,1,5,2,4]=>8 [3,6,1,5,4,2]=>9 [3,6,2,1,4,5]=>4 [3,6,2,1,5,4]=>5 [3,6,2,4,1,5]=>5 [3,6,2,4,5,1]=>6 [3,6,2,5,1,4]=>9 [3,6,2,5,4,1]=>10 [3,6,4,1,2,5]=>7 [3,6,4,1,5,2]=>8 [3,6,4,2,1,5]=>8 [3,6,4,2,5,1]=>9 [3,6,4,5,1,2]=>12 [3,6,4,5,2,1]=>13 [3,6,5,1,2,4]=>7 [3,6,5,1,4,2]=>8 [3,6,5,2,1,4]=>8 [3,6,5,2,4,1]=>9 [3,6,5,4,1,2]=>9 [3,6,5,4,2,1]=>10 [4,1,2,3,5,6]=>1 [4,1,2,3,6,5]=>6 [4,1,2,5,3,6]=>5 [4,1,2,5,6,3]=>10 [4,1,2,6,3,5]=>5 [4,1,2,6,5,3]=>6 [4,1,3,2,5,6]=>2 [4,1,3,2,6,5]=>7 [4,1,3,5,2,6]=>6 [4,1,3,5,6,2]=>11 [4,1,3,6,2,5]=>6 [4,1,3,6,5,2]=>7 [4,1,5,2,3,6]=>4 [4,1,5,2,6,3]=>9 [4,1,5,3,2,6]=>5 [4,1,5,3,6,2]=>10 [4,1,5,6,2,3]=>8 [4,1,5,6,3,2]=>9 [4,1,6,2,3,5]=>4 [4,1,6,2,5,3]=>5 [4,1,6,3,2,5]=>5 [4,1,6,3,5,2]=>6 [4,1,6,5,2,3]=>9 [4,1,6,5,3,2]=>10 [4,2,1,3,5,6]=>2 [4,2,1,3,6,5]=>7 [4,2,1,5,3,6]=>6 [4,2,1,5,6,3]=>11 [4,2,1,6,3,5]=>6 [4,2,1,6,5,3]=>7 [4,2,3,1,5,6]=>3 [4,2,3,1,6,5]=>8 [4,2,3,5,1,6]=>7 [4,2,3,5,6,1]=>12 [4,2,3,6,1,5]=>7 [4,2,3,6,5,1]=>8 [4,2,5,1,3,6]=>5 [4,2,5,1,6,3]=>10 [4,2,5,3,1,6]=>6 [4,2,5,3,6,1]=>11 [4,2,5,6,1,3]=>9 [4,2,5,6,3,1]=>10 [4,2,6,1,3,5]=>5 [4,2,6,1,5,3]=>6 [4,2,6,3,1,5]=>6 [4,2,6,3,5,1]=>7 [4,2,6,5,1,3]=>10 [4,2,6,5,3,1]=>11 [4,3,1,2,5,6]=>4 [4,3,1,2,6,5]=>9 [4,3,1,5,2,6]=>8 [4,3,1,5,6,2]=>13 [4,3,1,6,2,5]=>8 [4,3,1,6,5,2]=>9 [4,3,2,1,5,6]=>5 [4,3,2,1,6,5]=>10 [4,3,2,5,1,6]=>9 [4,3,2,5,6,1]=>14 [4,3,2,6,1,5]=>9 [4,3,2,6,5,1]=>10 [4,3,5,1,2,6]=>7 [4,3,5,1,6,2]=>12 [4,3,5,2,1,6]=>8 [4,3,5,2,6,1]=>13 [4,3,5,6,1,2]=>11 [4,3,5,6,2,1]=>12 [4,3,6,1,2,5]=>7 [4,3,6,1,5,2]=>8 [4,3,6,2,1,5]=>8 [4,3,6,2,5,1]=>9 [4,3,6,5,1,2]=>12 [4,3,6,5,2,1]=>13 [4,5,1,2,3,6]=>3 [4,5,1,2,6,3]=>8 [4,5,1,3,2,6]=>4 [4,5,1,3,6,2]=>9 [4,5,1,6,2,3]=>7 [4,5,1,6,3,2]=>8 [4,5,2,1,3,6]=>4 [4,5,2,1,6,3]=>9 [4,5,2,3,1,6]=>5 [4,5,2,3,6,1]=>10 [4,5,2,6,1,3]=>8 [4,5,2,6,3,1]=>9 [4,5,3,1,2,6]=>5 [4,5,3,1,6,2]=>10 [4,5,3,2,1,6]=>6 [4,5,3,2,6,1]=>11 [4,5,3,6,1,2]=>9 [4,5,3,6,2,1]=>10 [4,5,6,1,2,3]=>6 [4,5,6,1,3,2]=>7 [4,5,6,2,1,3]=>7 [4,5,6,2,3,1]=>8 [4,5,6,3,1,2]=>8 [4,5,6,3,2,1]=>9 [4,6,1,2,3,5]=>3 [4,6,1,2,5,3]=>4 [4,6,1,3,2,5]=>4 [4,6,1,3,5,2]=>5 [4,6,1,5,2,3]=>8 [4,6,1,5,3,2]=>9 [4,6,2,1,3,5]=>4 [4,6,2,1,5,3]=>5 [4,6,2,3,1,5]=>5 [4,6,2,3,5,1]=>6 [4,6,2,5,1,3]=>9 [4,6,2,5,3,1]=>10 [4,6,3,1,2,5]=>5 [4,6,3,1,5,2]=>6 [4,6,3,2,1,5]=>6 [4,6,3,2,5,1]=>7 [4,6,3,5,1,2]=>10 [4,6,3,5,2,1]=>11 [4,6,5,1,2,3]=>7 [4,6,5,1,3,2]=>8 [4,6,5,2,1,3]=>8 [4,6,5,2,3,1]=>9 [4,6,5,3,1,2]=>9 [4,6,5,3,2,1]=>10 [5,1,2,3,4,6]=>1 [5,1,2,3,6,4]=>6 [5,1,2,4,3,6]=>2 [5,1,2,4,6,3]=>7 [5,1,2,6,3,4]=>5 [5,1,2,6,4,3]=>6 [5,1,3,2,4,6]=>2 [5,1,3,2,6,4]=>7 [5,1,3,4,2,6]=>3 [5,1,3,4,6,2]=>8 [5,1,3,6,2,4]=>6 [5,1,3,6,4,2]=>7 [5,1,4,2,3,6]=>5 [5,1,4,2,6,3]=>10 [5,1,4,3,2,6]=>6 [5,1,4,3,6,2]=>11 [5,1,4,6,2,3]=>9 [5,1,4,6,3,2]=>10 [5,1,6,2,3,4]=>4 [5,1,6,2,4,3]=>5 [5,1,6,3,2,4]=>5 [5,1,6,3,4,2]=>6 [5,1,6,4,2,3]=>6 [5,1,6,4,3,2]=>7 [5,2,1,3,4,6]=>2 [5,2,1,3,6,4]=>7 [5,2,1,4,3,6]=>3 [5,2,1,4,6,3]=>8 [5,2,1,6,3,4]=>6 [5,2,1,6,4,3]=>7 [5,2,3,1,4,6]=>3 [5,2,3,1,6,4]=>8 [5,2,3,4,1,6]=>4 [5,2,3,4,6,1]=>9 [5,2,3,6,1,4]=>7 [5,2,3,6,4,1]=>8 [5,2,4,1,3,6]=>6 [5,2,4,1,6,3]=>11 [5,2,4,3,1,6]=>7 [5,2,4,3,6,1]=>12 [5,2,4,6,1,3]=>10 [5,2,4,6,3,1]=>11 [5,2,6,1,3,4]=>5 [5,2,6,1,4,3]=>6 [5,2,6,3,1,4]=>6 [5,2,6,3,4,1]=>7 [5,2,6,4,1,3]=>7 [5,2,6,4,3,1]=>8 [5,3,1,2,4,6]=>4 [5,3,1,2,6,4]=>9 [5,3,1,4,2,6]=>5 [5,3,1,4,6,2]=>10 [5,3,1,6,2,4]=>8 [5,3,1,6,4,2]=>9 [5,3,2,1,4,6]=>5 [5,3,2,1,6,4]=>10 [5,3,2,4,1,6]=>6 [5,3,2,4,6,1]=>11 [5,3,2,6,1,4]=>9 [5,3,2,6,4,1]=>10 [5,3,4,1,2,6]=>8 [5,3,4,1,6,2]=>13 [5,3,4,2,1,6]=>9 [5,3,4,2,6,1]=>14 [5,3,4,6,1,2]=>12 [5,3,4,6,2,1]=>13 [5,3,6,1,2,4]=>7 [5,3,6,1,4,2]=>8 [5,3,6,2,1,4]=>8 [5,3,6,2,4,1]=>9 [5,3,6,4,1,2]=>9 [5,3,6,4,2,1]=>10 [5,4,1,2,3,6]=>4 [5,4,1,2,6,3]=>9 [5,4,1,3,2,6]=>5 [5,4,1,3,6,2]=>10 [5,4,1,6,2,3]=>8 [5,4,1,6,3,2]=>9 [5,4,2,1,3,6]=>5 [5,4,2,1,6,3]=>10 [5,4,2,3,1,6]=>6 [5,4,2,3,6,1]=>11 [5,4,2,6,1,3]=>9 [5,4,2,6,3,1]=>10 [5,4,3,1,2,6]=>6 [5,4,3,1,6,2]=>11 [5,4,3,2,1,6]=>7 [5,4,3,2,6,1]=>12 [5,4,3,6,1,2]=>10 [5,4,3,6,2,1]=>11 [5,4,6,1,2,3]=>7 [5,4,6,1,3,2]=>8 [5,4,6,2,1,3]=>8 [5,4,6,2,3,1]=>9 [5,4,6,3,1,2]=>9 [5,4,6,3,2,1]=>10 [5,6,1,2,3,4]=>3 [5,6,1,2,4,3]=>4 [5,6,1,3,2,4]=>4 [5,6,1,3,4,2]=>5 [5,6,1,4,2,3]=>5 [5,6,1,4,3,2]=>6 [5,6,2,1,3,4]=>4 [5,6,2,1,4,3]=>5 [5,6,2,3,1,4]=>5 [5,6,2,3,4,1]=>6 [5,6,2,4,1,3]=>6 [5,6,2,4,3,1]=>7 [5,6,3,1,2,4]=>5 [5,6,3,1,4,2]=>6 [5,6,3,2,1,4]=>6 [5,6,3,2,4,1]=>7 [5,6,3,4,1,2]=>7 [5,6,3,4,2,1]=>8 [5,6,4,1,2,3]=>8 [5,6,4,1,3,2]=>9 [5,6,4,2,1,3]=>9 [5,6,4,2,3,1]=>10 [5,6,4,3,1,2]=>10 [5,6,4,3,2,1]=>11 [6,1,2,3,4,5]=>1 [6,1,2,3,5,4]=>2 [6,1,2,4,3,5]=>2 [6,1,2,4,5,3]=>3 [6,1,2,5,3,4]=>6 [6,1,2,5,4,3]=>7 [6,1,3,2,4,5]=>2 [6,1,3,2,5,4]=>3 [6,1,3,4,2,5]=>3 [6,1,3,4,5,2]=>4 [6,1,3,5,2,4]=>7 [6,1,3,5,4,2]=>8 [6,1,4,2,3,5]=>5 [6,1,4,2,5,3]=>6 [6,1,4,3,2,5]=>6 [6,1,4,3,5,2]=>7 [6,1,4,5,2,3]=>10 [6,1,4,5,3,2]=>11 [6,1,5,2,3,4]=>5 [6,1,5,2,4,3]=>6 [6,1,5,3,2,4]=>6 [6,1,5,3,4,2]=>7 [6,1,5,4,2,3]=>7 [6,1,5,4,3,2]=>8 [6,2,1,3,4,5]=>2 [6,2,1,3,5,4]=>3 [6,2,1,4,3,5]=>3 [6,2,1,4,5,3]=>4 [6,2,1,5,3,4]=>7 [6,2,1,5,4,3]=>8 [6,2,3,1,4,5]=>3 [6,2,3,1,5,4]=>4 [6,2,3,4,1,5]=>4 [6,2,3,4,5,1]=>5 [6,2,3,5,1,4]=>8 [6,2,3,5,4,1]=>9 [6,2,4,1,3,5]=>6 [6,2,4,1,5,3]=>7 [6,2,4,3,1,5]=>7 [6,2,4,3,5,1]=>8 [6,2,4,5,1,3]=>11 [6,2,4,5,3,1]=>12 [6,2,5,1,3,4]=>6 [6,2,5,1,4,3]=>7 [6,2,5,3,1,4]=>7 [6,2,5,3,4,1]=>8 [6,2,5,4,1,3]=>8 [6,2,5,4,3,1]=>9 [6,3,1,2,4,5]=>4 [6,3,1,2,5,4]=>5 [6,3,1,4,2,5]=>5 [6,3,1,4,5,2]=>6 [6,3,1,5,2,4]=>9 [6,3,1,5,4,2]=>10 [6,3,2,1,4,5]=>5 [6,3,2,1,5,4]=>6 [6,3,2,4,1,5]=>6 [6,3,2,4,5,1]=>7 [6,3,2,5,1,4]=>10 [6,3,2,5,4,1]=>11 [6,3,4,1,2,5]=>8 [6,3,4,1,5,2]=>9 [6,3,4,2,1,5]=>9 [6,3,4,2,5,1]=>10 [6,3,4,5,1,2]=>13 [6,3,4,5,2,1]=>14 [6,3,5,1,2,4]=>8 [6,3,5,1,4,2]=>9 [6,3,5,2,1,4]=>9 [6,3,5,2,4,1]=>10 [6,3,5,4,1,2]=>10 [6,3,5,4,2,1]=>11 [6,4,1,2,3,5]=>4 [6,4,1,2,5,3]=>5 [6,4,1,3,2,5]=>5 [6,4,1,3,5,2]=>6 [6,4,1,5,2,3]=>9 [6,4,1,5,3,2]=>10 [6,4,2,1,3,5]=>5 [6,4,2,1,5,3]=>6 [6,4,2,3,1,5]=>6 [6,4,2,3,5,1]=>7 [6,4,2,5,1,3]=>10 [6,4,2,5,3,1]=>11 [6,4,3,1,2,5]=>6 [6,4,3,1,5,2]=>7 [6,4,3,2,1,5]=>7 [6,4,3,2,5,1]=>8 [6,4,3,5,1,2]=>11 [6,4,3,5,2,1]=>12 [6,4,5,1,2,3]=>8 [6,4,5,1,3,2]=>9 [6,4,5,2,1,3]=>9 [6,4,5,2,3,1]=>10 [6,4,5,3,1,2]=>10 [6,4,5,3,2,1]=>11 [6,5,1,2,3,4]=>4 [6,5,1,2,4,3]=>5 [6,5,1,3,2,4]=>5 [6,5,1,3,4,2]=>6 [6,5,1,4,2,3]=>6 [6,5,1,4,3,2]=>7 [6,5,2,1,3,4]=>5 [6,5,2,1,4,3]=>6 [6,5,2,3,1,4]=>6 [6,5,2,3,4,1]=>7 [6,5,2,4,1,3]=>7 [6,5,2,4,3,1]=>8 [6,5,3,1,2,4]=>6 [6,5,3,1,4,2]=>7 [6,5,3,2,1,4]=>7 [6,5,3,2,4,1]=>8 [6,5,3,4,1,2]=>8 [6,5,3,4,2,1]=>9 [6,5,4,1,2,3]=>9 [6,5,4,1,3,2]=>10 [6,5,4,2,1,3]=>10 [6,5,4,2,3,1]=>11 [6,5,4,3,1,2]=>11 [6,5,4,3,2,1]=>12
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Description
The Denert index of a permutation.
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
References
[1] Denert, M. The genus zeta function of hereditary orders in central simple algebras over global fields MathSciNet:0993928
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian MathSciNet:1061147
[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). OEIS:A008302
Code
def statistic(pi):
    n = pi.size()
    A = sum( 1 for l,k in Subsets(range(n),2) if pi[k] <  pi[l] <= k+1   )
    B = sum( 1 for l,k in Subsets(range(n),2) if pi[l] <= k+1   <  pi[k] )
    C = sum( 1 for l,k in Subsets(range(n),2) if k+1   <  pi[k] <  pi[l] )
    return A+B+C
Created
Jul 24, 2013 at 12:25 by Christian Stump
Updated
May 31, 2015 at 21:39 by Christian Stump