edit this statistic or download as text // json
Identifier
Values
=>
[1]=>0 [1,1]=>0 [2]=>0 [1,1,1]=>1 [1,2]=>0 [2,1]=>0 [3]=>1 [1,1,1,1]=>1 [1,1,2]=>0 [1,2,1]=>1 [1,3]=>0 [2,1,1]=>0 [2,2]=>1 [3,1]=>0 [4]=>1 [1,1,1,1,1]=>2 [1,1,1,2]=>1 [1,1,2,1]=>1 [1,1,3]=>1 [1,2,1,1]=>1 [1,2,2]=>0 [1,3,1]=>2 [1,4]=>0 [2,1,1,1]=>1 [2,1,2]=>2 [2,2,1]=>0 [2,3]=>0 [3,1,1]=>1 [3,2]=>0 [4,1]=>0 [5]=>2 [1,1,1,1,1,1]=>2 [1,1,1,1,2]=>1 [1,1,1,2,1]=>1 [1,1,1,3]=>1 [1,1,2,1,1]=>2 [1,1,2,2]=>1 [1,1,3,1]=>1 [1,1,4]=>0 [1,2,1,1,1]=>1 [1,2,1,2]=>0 [1,2,2,1]=>2 [1,2,3]=>0 [1,3,1,1]=>1 [1,3,2]=>0 [1,4,1]=>2 [1,5]=>0 [2,1,1,1,1]=>1 [2,1,1,2]=>2 [2,1,2,1]=>0 [2,1,3]=>0 [2,2,1,1]=>1 [2,2,2]=>2 [2,3,1]=>0 [2,4]=>1 [3,1,1,1]=>1 [3,1,2]=>0 [3,2,1]=>0 [3,3]=>2 [4,1,1]=>0 [4,2]=>1 [5,1]=>0 [6]=>2 [1,1,1,1,1,1,1]=>3 [1,1,1,1,1,2]=>2 [1,1,1,1,2,1]=>2 [1,1,1,1,3]=>2 [1,1,1,2,1,1]=>2 [1,1,1,2,2]=>1 [1,1,1,3,1]=>2 [1,1,1,4]=>1 [1,1,2,1,1,1]=>2 [1,1,2,1,2]=>1 [1,1,2,2,1]=>1 [1,1,2,3]=>1 [1,1,3,1,1]=>3 [1,1,3,2]=>2 [1,1,4,1]=>1 [1,1,5]=>1 [1,2,1,1,1,1]=>2 [1,2,1,1,2]=>1 [1,2,1,2,1]=>3 [1,2,1,3]=>1 [1,2,2,1,1]=>1 [1,2,2,2]=>0 [1,2,3,1]=>1 [1,2,4]=>2 [1,3,1,1,1]=>2 [1,3,1,2]=>1 [1,3,2,1]=>1 [1,3,3]=>1 [1,4,1,1]=>1 [1,4,2]=>0 [1,5,1]=>3 [1,6]=>0 [2,1,1,1,1,1]=>2 [2,1,1,1,2]=>3 [2,1,1,2,1]=>1 [2,1,1,3]=>1 [2,1,2,1,1]=>1 [2,1,2,2]=>2 [2,1,3,1]=>1 [2,1,4]=>0 [2,2,1,1,1]=>1 [2,2,1,2]=>2 [2,2,2,1]=>0 [2,2,3]=>0 [2,3,1,1]=>2 [2,3,2]=>3 [2,4,1]=>0 [2,5]=>0 [3,1,1,1,1]=>2 [3,1,1,2]=>1 [3,1,2,1]=>1 [3,1,3]=>3 [3,2,1,1]=>1 [3,2,2]=>0 [3,3,1]=>1 [3,4]=>0 [4,1,1,1]=>1 [4,1,2]=>0 [4,2,1]=>2 [4,3]=>0 [5,1,1]=>1 [5,2]=>0 [6,1]=>0 [7]=>3 [1,1,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,2]=>2 [1,1,1,1,1,2,1]=>2 [1,1,1,1,1,3]=>2 [1,1,1,1,2,1,1]=>2 [1,1,1,1,2,2]=>1 [1,1,1,1,3,1]=>2 [1,1,1,1,4]=>1 [1,1,1,2,1,1,1]=>3 [1,1,1,2,1,2]=>2 [1,1,1,2,2,1]=>2 [1,1,1,2,3]=>2 [1,1,1,3,1,1]=>2 [1,1,1,3,2]=>1 [1,1,1,4,1]=>1 [1,1,1,5]=>1 [1,1,2,1,1,1,1]=>2 [1,1,2,1,1,2]=>1 [1,1,2,1,2,1]=>1 [1,1,2,1,3]=>1 [1,1,2,2,1,1]=>3 [1,1,2,2,2]=>2 [1,1,2,3,1]=>1 [1,1,2,4]=>0 [1,1,3,1,1,1]=>2 [1,1,3,1,2]=>1 [1,1,3,2,1]=>1 [1,1,3,3]=>1 [1,1,4,1,1]=>3 [1,1,4,2]=>2 [1,1,5,1]=>1 [1,1,6]=>0 [1,2,1,1,1,1,1]=>2 [1,2,1,1,1,2]=>1 [1,2,1,1,2,1]=>3 [1,2,1,1,3]=>1 [1,2,1,2,1,1]=>1 [1,2,1,2,2]=>0 [1,2,1,3,1]=>1 [1,2,1,4]=>2 [1,2,2,1,1,1]=>2 [1,2,2,1,2]=>1 [1,2,2,2,1]=>3 [1,2,2,3]=>1 [1,2,3,1,1]=>1 [1,2,3,2]=>0 [1,2,4,1]=>2 [1,2,5]=>0 [1,3,1,1,1,1]=>2 [1,3,1,1,2]=>1 [1,3,1,2,1]=>1 [1,3,1,3]=>1 [1,3,2,1,1]=>1 [1,3,2,2]=>0 [1,3,3,1]=>3 [1,3,4]=>0 [1,4,1,1,1]=>1 [1,4,1,2]=>0 [1,4,2,1]=>2 [1,4,3]=>0 [1,5,1,1]=>1 [1,5,2]=>0 [1,6,1]=>3 [1,7]=>0 [2,1,1,1,1,1,1]=>2 [2,1,1,1,1,2]=>3 [2,1,1,1,2,1]=>1 [2,1,1,1,3]=>1 [2,1,1,2,1,1]=>1 [2,1,1,2,2]=>2 [2,1,1,3,1]=>1 [2,1,1,4]=>0 [2,1,2,1,1,1]=>2 [2,1,2,1,2]=>3 [2,1,2,2,1]=>1 [2,1,2,3]=>1 [2,1,3,1,1]=>1 [2,1,3,2]=>2 [2,1,4,1]=>0 [2,1,5]=>2 [2,2,1,1,1,1]=>1 [2,2,1,1,2]=>2 [2,2,1,2,1]=>0 [2,2,1,3]=>0 [2,2,2,1,1]=>2 [2,2,2,2]=>3 [2,2,3,1]=>0 [2,2,4]=>1 [2,3,1,1,1]=>1 [2,3,1,2]=>2 [2,3,2,1]=>0 [2,3,3]=>0 [2,4,1,1]=>2 [2,4,2]=>3 [2,5,1]=>0 [2,6]=>1 [3,1,1,1,1,1]=>2 [3,1,1,1,2]=>1 [3,1,1,2,1]=>1 [3,1,1,3]=>3 [3,1,2,1,1]=>1 [3,1,2,2]=>0 [3,1,3,1]=>1 [3,1,4]=>0 [3,2,1,1,1]=>2 [3,2,1,2]=>1 [3,2,2,1]=>1 [3,2,3]=>3 [3,3,1,1]=>1 [3,3,2]=>0 [3,4,1]=>0 [3,5]=>0 [4,1,1,1,1]=>1 [4,1,1,2]=>0 [4,1,2,1]=>2 [4,1,3]=>0 [4,2,1,1]=>0 [4,2,2]=>1 [4,3,1]=>0 [4,4]=>3 [5,1,1,1]=>1 [5,1,2]=>2 [5,2,1]=>0 [5,3]=>0 [6,1,1]=>0 [6,2]=>1 [7,1]=>0 [8]=>3 [1,1,1,1,1,1,1,1,1]=>4 [1,1,1,1,1,1,1,2]=>3 [1,1,1,1,1,1,2,1]=>3 [1,1,1,1,1,1,3]=>3 [1,1,1,1,1,2,1,1]=>3 [1,1,1,1,1,2,2]=>2 [1,1,1,1,1,3,1]=>3 [1,1,1,1,1,4]=>2 [1,1,1,1,2,1,1,1]=>3 [1,1,1,1,2,1,2]=>2 [1,1,1,1,2,2,1]=>2 [1,1,1,1,2,3]=>2 [1,1,1,1,3,1,1]=>3 [1,1,1,1,3,2]=>2 [1,1,1,1,4,1]=>2 [1,1,1,1,5]=>2 [1,1,1,2,1,1,1,1]=>3 [1,1,1,2,1,1,2]=>2 [1,1,1,2,1,2,1]=>2 [1,1,1,2,1,3]=>2 [1,1,1,2,2,1,1]=>2 [1,1,1,2,2,2]=>1 [1,1,1,2,3,1]=>2 [1,1,1,2,4]=>1 [1,1,1,3,1,1,1]=>4 [1,1,1,3,1,2]=>3 [1,1,1,3,2,1]=>3 [1,1,1,3,3]=>3 [1,1,1,4,1,1]=>2 [1,1,1,4,2]=>1 [1,1,1,5,1]=>2 [1,1,1,6]=>1 [1,1,2,1,1,1,1,1]=>3 [1,1,2,1,1,1,2]=>2 [1,1,2,1,1,2,1]=>2 [1,1,2,1,1,3]=>2 [1,1,2,1,2,1,1]=>4 [1,1,2,1,2,2]=>3 [1,1,2,1,3,1]=>2 [1,1,2,1,4]=>1 [1,1,2,2,1,1,1]=>2 [1,1,2,2,1,2]=>1 [1,1,2,2,2,1]=>1 [1,1,2,2,3]=>1 [1,1,2,3,1,1]=>2 [1,1,2,3,2]=>1 [1,1,2,4,1]=>3 [1,1,2,5]=>1 [1,1,3,1,1,1,1]=>3 [1,1,3,1,1,2]=>2 [1,1,3,1,2,1]=>2 [1,1,3,1,3]=>2 [1,1,3,2,1,1]=>2 [1,1,3,2,2]=>1 [1,1,3,3,1]=>2 [1,1,3,4]=>1 [1,1,4,1,1,1]=>2 [1,1,4,1,2]=>1 [1,1,4,2,1]=>1 [1,1,4,3]=>1 [1,1,5,1,1]=>4 [1,1,5,2]=>3 [1,1,6,1]=>1 [1,1,7]=>1 [1,2,1,1,1,1,1,1]=>3 [1,2,1,1,1,1,2]=>2 [1,2,1,1,1,2,1]=>4 [1,2,1,1,1,3]=>2 [1,2,1,1,2,1,1]=>2 [1,2,1,1,2,2]=>1 [1,2,1,1,3,1]=>2 [1,2,1,1,4]=>3 [1,2,1,2,1,1,1]=>2 [1,2,1,2,1,2]=>1 [1,2,1,2,2,1]=>3 [1,2,1,2,3]=>1 [1,2,1,3,1,1]=>2 [1,2,1,3,2]=>1 [1,2,1,4,1]=>1 [1,2,1,5]=>1 [1,2,2,1,1,1,1]=>2 [1,2,2,1,1,2]=>1 [1,2,2,1,2,1]=>3 [1,2,2,1,3]=>1 [1,2,2,2,1,1]=>1 [1,2,2,2,2]=>0 [1,2,2,3,1]=>1 [1,2,2,4]=>2 [1,2,3,1,1,1]=>3 [1,2,3,1,2]=>2 [1,2,3,2,1]=>4 [1,2,3,3]=>2 [1,2,4,1,1]=>1 [1,2,4,2]=>0 [1,2,5,1]=>1 [1,2,6]=>0 [1,3,1,1,1,1,1]=>3 [1,3,1,1,1,2]=>2 [1,3,1,1,2,1]=>2 [1,3,1,1,3]=>2 [1,3,1,2,1,1]=>2 [1,3,1,2,2]=>1 [1,3,1,3,1]=>4 [1,3,1,4]=>1 [1,3,2,1,1,1]=>2 [1,3,2,1,2]=>1 [1,3,2,2,1]=>1 [1,3,2,3]=>1 [1,3,3,1,1]=>2 [1,3,3,2]=>1 [1,3,4,1]=>1 [1,3,5]=>3 [1,4,1,1,1,1]=>2 [1,4,1,1,2]=>1 [1,4,1,2,1]=>1 [1,4,1,3]=>1 [1,4,2,1,1]=>3 [1,4,2,2]=>2 [1,4,3,1]=>1 [1,4,4]=>0 [1,5,1,1,1]=>2 [1,5,1,2]=>1 [1,5,2,1]=>1 [1,5,3]=>1 [1,6,1,1]=>1 [1,6,2]=>0 [1,7,1]=>4 [1,8]=>0 [2,1,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,2]=>4 [2,1,1,1,1,2,1]=>2 [2,1,1,1,1,3]=>2 [2,1,1,1,2,1,1]=>2 [2,1,1,1,2,2]=>3 [2,1,1,1,3,1]=>2 [2,1,1,1,4]=>1 [2,1,1,2,1,1,1]=>2 [2,1,1,2,1,2]=>3 [2,1,1,2,2,1]=>1 [2,1,1,2,3]=>1 [2,1,1,3,1,1]=>2 [2,1,1,3,2]=>3 [2,1,1,4,1]=>1 [2,1,1,5]=>1 [2,1,2,1,1,1,1]=>2 [2,1,2,1,1,2]=>3 [2,1,2,1,2,1]=>1 [2,1,2,1,3]=>1 [2,1,2,2,1,1]=>1 [2,1,2,2,2]=>2 [2,1,2,3,1]=>1 [2,1,2,4]=>0 [2,1,3,1,1,1]=>3 [2,1,3,1,2]=>4 [2,1,3,2,1]=>2 [2,1,3,3]=>2 [2,1,4,1,1]=>1 [2,1,4,2]=>2 [2,1,5,1]=>1 [2,1,6]=>0 [2,2,1,1,1,1,1]=>2 [2,2,1,1,1,2]=>3 [2,2,1,1,2,1]=>1 [2,2,1,1,3]=>1 [2,2,1,2,1,1]=>3 [2,2,1,2,2]=>4 [2,2,1,3,1]=>1 [2,2,1,4]=>2 [2,2,2,1,1,1]=>1 [2,2,2,1,2]=>2 [2,2,2,2,1]=>0 [2,2,2,3]=>0 [2,2,3,1,1]=>1 [2,2,3,2]=>2 [2,2,4,1]=>2 [2,2,5]=>0 [2,3,1,1,1,1]=>2 [2,3,1,1,2]=>3 [2,3,1,2,1]=>1 [2,3,1,3]=>1 [2,3,2,1,1]=>1 [2,3,2,2]=>2 [2,3,3,1]=>1 [2,3,4]=>0 [2,4,1,1,1]=>1 [2,4,1,2]=>2 [2,4,2,1]=>0 [2,4,3]=>0 [2,5,1,1]=>3 [2,5,2]=>4 [2,6,1]=>0 [2,7]=>0 [3,1,1,1,1,1,1]=>3 [3,1,1,1,1,2]=>2 [3,1,1,1,2,1]=>2 [3,1,1,1,3]=>4 [3,1,1,2,1,1]=>2 [3,1,1,2,2]=>1 [3,1,1,3,1]=>2 [3,1,1,4]=>1 [3,1,2,1,1,1]=>2 [3,1,2,1,2]=>1 [3,1,2,2,1]=>1 [3,1,2,3]=>3 [3,1,3,1,1]=>2 [3,1,3,2]=>1 [3,1,4,1]=>1 [3,1,5]=>1 [3,2,1,1,1,1]=>2 [3,2,1,1,2]=>1 [3,2,1,2,1]=>1 [3,2,1,3]=>3 [3,2,2,1,1]=>1 [3,2,2,2]=>0 [3,2,3,1]=>1 [3,2,4]=>0 [3,3,1,1,1]=>3 [3,3,1,2]=>2 [3,3,2,1]=>2 [3,3,3]=>4 [3,4,1,1]=>1 [3,4,2]=>0 [3,5,1]=>1 [3,6]=>2 [4,1,1,1,1,1]=>2 [4,1,1,1,2]=>1 [4,1,1,2,1]=>3 [4,1,1,3]=>1 [4,1,2,1,1]=>1 [4,1,2,2]=>2 [4,1,3,1]=>1 [4,1,4]=>4 [4,2,1,1,1]=>1 [4,2,1,2]=>0 [4,2,2,1]=>2 [4,2,3]=>0 [4,3,1,1]=>1 [4,3,2]=>0 [4,4,1]=>0 [4,5]=>0 [5,1,1,1,1]=>2 [5,1,1,2]=>1 [5,1,2,1]=>1 [5,1,3]=>1 [5,2,1,1]=>1 [5,2,2]=>0 [5,3,1]=>3 [5,4]=>0 [6,1,1,1]=>1 [6,1,2]=>0 [6,2,1]=>0 [6,3]=>2 [7,1,1]=>1 [7,2]=>0 [8,1]=>0 [9]=>4 [1,1,1,1,1,1,1,1,1,1]=>4 [1,1,1,1,1,1,1,1,2]=>3 [1,1,1,1,1,1,1,2,1]=>3 [1,1,1,1,1,1,1,3]=>3 [1,1,1,1,1,1,2,1,1]=>3 [1,1,1,1,1,1,2,2]=>2 [1,1,1,1,1,1,3,1]=>3 [1,1,1,1,1,1,4]=>2 [1,1,1,1,1,2,1,1,1]=>3 [1,1,1,1,1,2,1,2]=>2 [1,1,1,1,1,2,2,1]=>2 [1,1,1,1,1,2,3]=>2 [1,1,1,1,1,3,1,1]=>3 [1,1,1,1,1,3,2]=>2 [1,1,1,1,1,4,1]=>2 [1,1,1,1,1,5]=>2 [1,1,1,1,2,1,1,1,1]=>4 [1,1,1,1,2,1,1,2]=>3 [1,1,1,1,2,1,2,1]=>3 [1,1,1,1,2,1,3]=>3 [1,1,1,1,2,2,1,1]=>3 [1,1,1,1,2,2,2]=>2 [1,1,1,1,2,3,1]=>3 [1,1,1,1,2,4]=>2 [1,1,1,1,3,1,1,1]=>3 [1,1,1,1,3,1,2]=>2 [1,1,1,1,3,2,1]=>2 [1,1,1,1,3,3]=>2 [1,1,1,1,4,1,1]=>2 [1,1,1,1,4,2]=>1 [1,1,1,1,5,1]=>2 [1,1,1,1,6]=>1 [1,1,1,2,1,1,1,1,1]=>3 [1,1,1,2,1,1,1,2]=>2 [1,1,1,2,1,1,2,1]=>2 [1,1,1,2,1,1,3]=>2 [1,1,1,2,1,2,1,1]=>2 [1,1,1,2,1,2,2]=>1 [1,1,1,2,1,3,1]=>2 [1,1,1,2,1,4]=>1 [1,1,1,2,2,1,1,1]=>4 [1,1,1,2,2,1,2]=>3 [1,1,1,2,2,2,1]=>3 [1,1,1,2,2,3]=>3 [1,1,1,2,3,1,1]=>2 [1,1,1,2,3,2]=>1 [1,1,1,2,4,1]=>1 [1,1,1,2,5]=>1 [1,1,1,3,1,1,1,1]=>3 [1,1,1,3,1,1,2]=>2 [1,1,1,3,1,2,1]=>2 [1,1,1,3,1,3]=>2 [1,1,1,3,2,1,1]=>2 [1,1,1,3,2,2]=>1 [1,1,1,3,3,1]=>2 [1,1,1,3,4]=>1 [1,1,1,4,1,1,1]=>4 [1,1,1,4,1,2]=>3 [1,1,1,4,2,1]=>3 [1,1,1,4,3]=>3 [1,1,1,5,1,1]=>2 [1,1,1,5,2]=>1 [1,1,1,6,1]=>1 [1,1,1,7]=>1 [1,1,2,1,1,1,1,1,1]=>3 [1,1,2,1,1,1,1,2]=>2 [1,1,2,1,1,1,2,1]=>2 [1,1,2,1,1,1,3]=>2 [1,1,2,1,1,2,1,1]=>4 [1,1,2,1,1,2,2]=>3 [1,1,2,1,1,3,1]=>2 [1,1,2,1,1,4]=>1 [1,1,2,1,2,1,1,1]=>2 [1,1,2,1,2,1,2]=>1 [1,1,2,1,2,2,1]=>1 [1,1,2,1,2,3]=>1 [1,1,2,1,3,1,1]=>2 [1,1,2,1,3,2]=>1 [1,1,2,1,4,1]=>3 [1,1,2,1,5]=>1 [1,1,2,2,1,1,1,1]=>3 [1,1,2,2,1,1,2]=>2 [1,1,2,2,1,2,1]=>2 [1,1,2,2,1,3]=>2 [1,1,2,2,2,1,1]=>4 [1,1,2,2,2,2]=>3 [1,1,2,2,3,1]=>2 [1,1,2,2,4]=>1 [1,1,2,3,1,1,1]=>2 [1,1,2,3,1,2]=>1 [1,1,2,3,2,1]=>1 [1,1,2,3,3]=>1 [1,1,2,4,1,1]=>3 [1,1,2,4,2]=>2 [1,1,2,5,1]=>1 [1,1,2,6]=>2 [1,1,3,1,1,1,1,1]=>3 [1,1,3,1,1,1,2]=>2 [1,1,3,1,1,2,1]=>2 [1,1,3,1,1,3]=>2 [1,1,3,1,2,1,1]=>2 [1,1,3,1,2,2]=>1 [1,1,3,1,3,1]=>2 [1,1,3,1,4]=>1 [1,1,3,2,1,1,1]=>2 [1,1,3,2,1,2]=>1 [1,1,3,2,2,1]=>1 [1,1,3,2,3]=>1 [1,1,3,3,1,1]=>4 [1,1,3,3,2]=>3 [1,1,3,4,1]=>1 [1,1,3,5]=>1 [1,1,4,1,1,1,1]=>2 [1,1,4,1,1,2]=>1 [1,1,4,1,2,1]=>1 [1,1,4,1,3]=>1 [1,1,4,2,1,1]=>3 [1,1,4,2,2]=>2 [1,1,4,3,1]=>1 [1,1,4,4]=>0 [1,1,5,1,1,1]=>2 [1,1,5,1,2]=>1 [1,1,5,2,1]=>1 [1,1,5,3]=>1 [1,1,6,1,1]=>4 [1,1,6,2]=>3 [1,1,7,1]=>1 [1,1,8]=>0 [1,2,1,1,1,1,1,1,1]=>3 [1,2,1,1,1,1,1,2]=>2 [1,2,1,1,1,1,2,1]=>4 [1,2,1,1,1,1,3]=>2 [1,2,1,1,1,2,1,1]=>2 [1,2,1,1,1,2,2]=>1 [1,2,1,1,1,3,1]=>2 [1,2,1,1,1,4]=>3 [1,2,1,1,2,1,1,1]=>2 [1,2,1,1,2,1,2]=>1 [1,2,1,1,2,2,1]=>3 [1,2,1,1,2,3]=>1 [1,2,1,1,3,1,1]=>2 [1,2,1,1,3,2]=>1 [1,2,1,1,4,1]=>1 [1,2,1,1,5]=>1 [1,2,1,2,1,1,1,1]=>3 [1,2,1,2,1,1,2]=>2 [1,2,1,2,1,2,1]=>4 [1,2,1,2,1,3]=>2 [1,2,1,2,2,1,1]=>2 [1,2,1,2,2,2]=>1 [1,2,1,2,3,1]=>2 [1,2,1,2,4]=>3 [1,2,1,3,1,1,1]=>2 [1,2,1,3,1,2]=>1 [1,2,1,3,2,1]=>3 [1,2,1,3,3]=>1 [1,2,1,4,1,1]=>1 [1,2,1,4,2]=>0 [1,2,1,5,1]=>3 [1,2,1,6]=>0 [1,2,2,1,1,1,1,1]=>2 [1,2,2,1,1,1,2]=>1 [1,2,2,1,1,2,1]=>3 [1,2,2,1,1,3]=>1 [1,2,2,1,2,1,1]=>1 [1,2,2,1,2,2]=>0 [1,2,2,1,3,1]=>1 [1,2,2,1,4]=>2 [1,2,2,2,1,1,1]=>3 [1,2,2,2,1,2]=>2 [1,2,2,2,2,1]=>4 [1,2,2,2,3]=>2 [1,2,2,3,1,1]=>1 [1,2,2,3,2]=>0 [1,2,2,4,1]=>2 [1,2,2,5]=>0 [1,2,3,1,1,1,1]=>2 [1,2,3,1,1,2]=>1 [1,2,3,1,2,1]=>3 [1,2,3,1,3]=>1 [1,2,3,2,1,1]=>1 [1,2,3,2,2]=>0 [1,2,3,3,1]=>1 [1,2,3,4]=>2 [1,2,4,1,1,1]=>3 [1,2,4,1,2]=>2 [1,2,4,2,1]=>4 [1,2,4,3]=>2 [1,2,5,1,1]=>1 [1,2,5,2]=>0 [1,2,6,1]=>2 [1,2,7]=>2 [1,3,1,1,1,1,1,1]=>3 [1,3,1,1,1,1,2]=>2 [1,3,1,1,1,2,1]=>2 [1,3,1,1,1,3]=>2 [1,3,1,1,2,1,1]=>2 [1,3,1,1,2,2]=>1 [1,3,1,1,3,1]=>4 [1,3,1,1,4]=>1 [1,3,1,2,1,1,1]=>2 [1,3,1,2,1,2]=>1 [1,3,1,2,2,1]=>1 [1,3,1,2,3]=>1 [1,3,1,3,1,1]=>2 [1,3,1,3,2]=>1 [1,3,1,4,1]=>1 [1,3,1,5]=>3 [1,3,2,1,1,1,1]=>3 [1,3,2,1,1,2]=>2 [1,3,2,1,2,1]=>2 [1,3,2,1,3]=>2 [1,3,2,2,1,1]=>2 [1,3,2,2,2]=>1 [1,3,2,3,1]=>4 [1,3,2,4]=>1 [1,3,3,1,1,1]=>2 [1,3,3,1,2]=>1 [1,3,3,2,1]=>1 [1,3,3,3]=>1 [1,3,4,1,1]=>1 [1,3,4,2]=>0 [1,3,5,1]=>1 [1,3,6]=>0 [1,4,1,1,1,1,1]=>2 [1,4,1,1,1,2]=>1 [1,4,1,1,2,1]=>1 [1,4,1,1,3]=>1 [1,4,1,2,1,1]=>3 [1,4,1,2,2]=>2 [1,4,1,3,1]=>1 [1,4,1,4]=>0 [1,4,2,1,1,1]=>1 [1,4,2,1,2]=>0 [1,4,2,2,1]=>2 [1,4,2,3]=>0 [1,4,3,1,1]=>1 [1,4,3,2]=>0 [1,4,4,1]=>4 [1,4,5]=>0 [1,5,1,1,1,1]=>2 [1,5,1,1,2]=>1 [1,5,1,2,1]=>3 [1,5,1,3]=>1 [1,5,2,1,1]=>1 [1,5,2,2]=>0 [1,5,3,1]=>1 [1,5,4]=>2 [1,6,1,1,1]=>1 [1,6,1,2]=>0 [1,6,2,1]=>2 [1,6,3]=>0 [1,7,1,1]=>1 [1,7,2]=>0 [1,8,1]=>4 [1,9]=>0 [2,1,1,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,2]=>4 [2,1,1,1,1,1,2,1]=>2 [2,1,1,1,1,1,3]=>2 [2,1,1,1,1,2,1,1]=>2 [2,1,1,1,1,2,2]=>3 [2,1,1,1,1,3,1]=>2 [2,1,1,1,1,4]=>1 [2,1,1,1,2,1,1,1]=>2 [2,1,1,1,2,1,2]=>3 [2,1,1,1,2,2,1]=>1 [2,1,1,1,2,3]=>1 [2,1,1,1,3,1,1]=>2 [2,1,1,1,3,2]=>3 [2,1,1,1,4,1]=>1 [2,1,1,1,5]=>1 [2,1,1,2,1,1,1,1]=>3 [2,1,1,2,1,1,2]=>4 [2,1,1,2,1,2,1]=>2 [2,1,1,2,1,3]=>2 [2,1,1,2,2,1,1]=>2 [2,1,1,2,2,2]=>3 [2,1,1,2,3,1]=>2 [2,1,1,2,4]=>1 [2,1,1,3,1,1,1]=>2 [2,1,1,3,1,2]=>3 [2,1,1,3,2,1]=>1 [2,1,1,3,3]=>1 [2,1,1,4,1,1]=>1 [2,1,1,4,2]=>2 [2,1,1,5,1]=>1 [2,1,1,6]=>2 [2,1,2,1,1,1,1,1]=>2 [2,1,2,1,1,1,2]=>3 [2,1,2,1,1,2,1]=>1 [2,1,2,1,1,3]=>1 [2,1,2,1,2,1,1]=>1 [2,1,2,1,2,2]=>2 [2,1,2,1,3,1]=>1 [2,1,2,1,4]=>0 [2,1,2,2,1,1,1]=>3 [2,1,2,2,1,2]=>4 [2,1,2,2,2,1]=>2 [2,1,2,2,3]=>2 [2,1,2,3,1,1]=>1 [2,1,2,3,2]=>2 [2,1,2,4,1]=>0 [2,1,2,5]=>2 [2,1,3,1,1,1,1]=>2 [2,1,3,1,1,2]=>3 [2,1,3,1,2,1]=>1 [2,1,3,1,3]=>1 [2,1,3,2,1,1]=>1 [2,1,3,2,2]=>2 [2,1,3,3,1]=>1 [2,1,3,4]=>0 [2,1,4,1,1,1]=>3 [2,1,4,1,2]=>4 [2,1,4,2,1]=>2 [2,1,4,3]=>2 [2,1,5,1,1]=>1 [2,1,5,2]=>2 [2,1,6,1]=>0 [2,1,7]=>0 [2,2,1,1,1,1,1,1]=>2 [2,2,1,1,1,1,2]=>3 [2,2,1,1,1,2,1]=>1 [2,2,1,1,1,3]=>1 [2,2,1,1,2,1,1]=>3 [2,2,1,1,2,2]=>4 [2,2,1,1,3,1]=>1 [2,2,1,1,4]=>2 [2,2,1,2,1,1,1]=>1 [2,2,1,2,1,2]=>2 [2,2,1,2,2,1]=>0 [2,2,1,2,3]=>0 [2,2,1,3,1,1]=>1 [2,2,1,3,2]=>2 [2,2,1,4,1]=>2 [2,2,1,5]=>0 [2,2,2,1,1,1,1]=>2 [2,2,2,1,1,2]=>3 [2,2,2,1,2,1]=>1 [2,2,2,1,3]=>1 [2,2,2,2,1,1]=>3 [2,2,2,2,2]=>4 [2,2,2,3,1]=>1 [2,2,2,4]=>2 [2,2,3,1,1,1]=>1 [2,2,3,1,2]=>2 [2,2,3,2,1]=>0 [2,2,3,3]=>0 [2,2,4,1,1]=>2 [2,2,4,2]=>3 [2,2,5,1]=>0 [2,2,6]=>3 [2,3,1,1,1,1,1]=>2 [2,3,1,1,1,2]=>3 [2,3,1,1,2,1]=>1 [2,3,1,1,3]=>1 [2,3,1,2,1,1]=>1 [2,3,1,2,2]=>2 [2,3,1,3,1]=>1 [2,3,1,4]=>0 [2,3,2,1,1,1]=>1 [2,3,2,1,2]=>2 [2,3,2,2,1]=>0 [2,3,2,3]=>0 [2,3,3,1,1]=>3 [2,3,3,2]=>4 [2,3,4,1]=>0 [2,3,5]=>0 [2,4,1,1,1,1]=>1 [2,4,1,1,2]=>2 [2,4,1,2,1]=>0 [2,4,1,3]=>0 [2,4,2,1,1]=>2 [2,4,2,2]=>3 [2,4,3,1]=>0 [2,4,4]=>1 [2,5,1,1,1]=>1 [2,5,1,2]=>2 [2,5,2,1]=>0 [2,5,3]=>0 [2,6,1,1]=>3 [2,6,2]=>4 [2,7,1]=>0 [2,8]=>1 [3,1,1,1,1,1,1,1]=>3 [3,1,1,1,1,1,2]=>2 [3,1,1,1,1,2,1]=>2 [3,1,1,1,1,3]=>4 [3,1,1,1,2,1,1]=>2 [3,1,1,1,2,2]=>1 [3,1,1,1,3,1]=>2 [3,1,1,1,4]=>1 [3,1,1,2,1,1,1]=>2 [3,1,1,2,1,2]=>1 [3,1,1,2,2,1]=>1 [3,1,1,2,3]=>3 [3,1,1,3,1,1]=>2 [3,1,1,3,2]=>1 [3,1,1,4,1]=>1 [3,1,1,5]=>1 [3,1,2,1,1,1,1]=>3 [3,1,2,1,1,2]=>2 [3,1,2,1,2,1]=>2 [3,1,2,1,3]=>4 [3,1,2,2,1,1]=>2 [3,1,2,2,2]=>1 [3,1,2,3,1]=>2 [3,1,2,4]=>1 [3,1,3,1,1,1]=>2 [3,1,3,1,2]=>1 [3,1,3,2,1]=>1 [3,1,3,3]=>3 [3,1,4,1,1]=>1 [3,1,4,2]=>0 [3,1,5,1]=>1 [3,1,6]=>0 [3,2,1,1,1,1,1]=>2 [3,2,1,1,1,2]=>1 [3,2,1,1,2,1]=>1 [3,2,1,1,3]=>3 [3,2,1,2,1,1]=>1 [3,2,1,2,2]=>0 [3,2,1,3,1]=>1 [3,2,1,4]=>0 [3,2,2,1,1,1]=>3 [3,2,2,1,2]=>2 [3,2,2,2,1]=>2 [3,2,2,3]=>4 [3,2,3,1,1]=>1 [3,2,3,2]=>0 [3,2,4,1]=>0 [3,2,5]=>0 [3,3,1,1,1,1]=>2 [3,3,1,1,2]=>1 [3,3,1,2,1]=>1 [3,3,1,3]=>3 [3,3,2,1,1]=>1 [3,3,2,2]=>0 [3,3,3,1]=>1 [3,3,4]=>0 [3,4,1,1,1]=>3 [3,4,1,2]=>2 [3,4,2,1]=>2 [3,4,3]=>4 [3,5,1,1]=>1 [3,5,2]=>0 [3,6,1]=>0 [3,7]=>0 [4,1,1,1,1,1,1]=>2 [4,1,1,1,1,2]=>1 [4,1,1,1,2,1]=>3 [4,1,1,1,3]=>1 [4,1,1,2,1,1]=>1 [4,1,1,2,2]=>2 [4,1,1,3,1]=>1 [4,1,1,4]=>4 [4,1,2,1,1,1]=>1 [4,1,2,1,2]=>0 [4,1,2,2,1]=>2 [4,1,2,3]=>0 [4,1,3,1,1]=>1 [4,1,3,2]=>0 [4,1,4,1]=>0 [4,1,5]=>0 [4,2,1,1,1,1]=>2 [4,2,1,1,2]=>1 [4,2,1,2,1]=>3 [4,2,1,3]=>1 [4,2,2,1,1]=>1 [4,2,2,2]=>2 [4,2,3,1]=>1 [4,2,4]=>4 [4,3,1,1,1]=>1 [4,3,1,2]=>0 [4,3,2,1]=>2 [4,3,3]=>0 [4,4,1,1]=>0 [4,4,2]=>1 [4,5,1]=>2 [4,6]=>1 [5,1,1,1,1,1]=>2 [5,1,1,1,2]=>1 [5,1,1,2,1]=>1 [5,1,1,3]=>1 [5,1,2,1,1]=>1 [5,1,2,2]=>0 [5,1,3,1]=>3 [5,1,4]=>0 [5,2,1,1,1]=>1 [5,2,1,2]=>2 [5,2,2,1]=>0 [5,2,3]=>0 [5,3,1,1]=>1 [5,3,2]=>0 [5,4,1]=>0 [5,5]=>4 [6,1,1,1,1]=>1 [6,1,1,2]=>2 [6,1,2,1]=>0 [6,1,3]=>0 [6,2,1,1]=>2 [6,2,2]=>3 [6,3,1]=>0 [6,4]=>1 [7,1,1,1]=>1 [7,1,2]=>0 [7,2,1]=>2 [7,3]=>0 [8,1,1]=>0 [8,2]=>1 [9,1]=>0 [10]=>4
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Description
The index of the maximal parabolic seaweed algebra associated with the composition.
Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows:
* place $n$ dots on a horizontal line
* subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line
* subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line
By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is
$$ \operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1, $$
where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander.
This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.
References
[1] Dergachev, V., Kirillov, A. Index of Lie algebras of seaweed type MathSciNet:1774864
[2] Coll, V., Mayers, A., Mayers, N. Statistics on integer partitions arising from seaweed algebras arXiv:1809.09271
Code
def to_meander(la, mu):
    """
    Return the meander associated with two partitions as a graph.
    """
    la = Composition(la)
    mu = Composition(mu)
    n = la.size()
    assert mu.size() == n
    M = Graph(n)
    offset = 0
    for part in la:
        M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)])
        offset += part
    offset = 0
    for part in mu:
        M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)])
        offset += part
    return M

def seaweed_index(la, mu):
    M = to_meander(la, mu)
    index = -1
    for G in M.connected_components_subgraphs():
        if G.is_cycle():
            index += 2
        else:
            index += 1
    return index

def statistic(la):
    return seaweed_index(la, la.size())
Created
Sep 26, 2018 at 04:49 by Martin Rubey
Updated
Sep 26, 2018 at 10:46 by Martin Rubey