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
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
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 $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.
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
[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
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