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Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>0 [1,0,1,0]=>0 [1,1,0,0]=>1 [1,0,1,0,1,0]=>0 [1,0,1,1,0,0]=>1 [1,1,0,0,1,0]=>1 [1,1,0,1,0,0]=>0 [1,1,1,0,0,0]=>2 [1,0,1,0,1,0,1,0]=>0 [1,0,1,0,1,1,0,0]=>1 [1,0,1,1,0,0,1,0]=>1 [1,0,1,1,0,1,0,0]=>0 [1,0,1,1,1,0,0,0]=>2 [1,1,0,0,1,0,1,0]=>1 [1,1,0,0,1,1,0,0]=>2 [1,1,0,1,0,0,1,0]=>0 [1,1,0,1,0,1,0,0]=>0 [1,1,0,1,1,0,0,0]=>1 [1,1,1,0,0,0,1,0]=>2 [1,1,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,0,0]=>1 [1,1,1,1,0,0,0,0]=>3 [1,0,1,0,1,0,1,0,1,0]=>0 [1,0,1,0,1,0,1,1,0,0]=>1 [1,0,1,0,1,1,0,0,1,0]=>1 [1,0,1,0,1,1,0,1,0,0]=>0 [1,0,1,0,1,1,1,0,0,0]=>2 [1,0,1,1,0,0,1,0,1,0]=>1 [1,0,1,1,0,0,1,1,0,0]=>2 [1,0,1,1,0,1,0,0,1,0]=>0 [1,0,1,1,0,1,0,1,0,0]=>0 [1,0,1,1,0,1,1,0,0,0]=>1 [1,0,1,1,1,0,0,0,1,0]=>2 [1,0,1,1,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,0,0,0]=>3 [1,1,0,0,1,0,1,0,1,0]=>1 [1,1,0,0,1,0,1,1,0,0]=>2 [1,1,0,0,1,1,0,0,1,0]=>2 [1,1,0,0,1,1,0,1,0,0]=>1 [1,1,0,0,1,1,1,0,0,0]=>3 [1,1,0,1,0,0,1,0,1,0]=>0 [1,1,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,0,1,0]=>0 [1,1,0,1,0,1,0,1,0,0]=>0 [1,1,0,1,0,1,1,0,0,0]=>1 [1,1,0,1,1,0,0,0,1,0]=>1 [1,1,0,1,1,0,0,1,0,0]=>0 [1,1,0,1,1,0,1,0,0,0]=>1 [1,1,0,1,1,1,0,0,0,0]=>2 [1,1,1,0,0,0,1,0,1,0]=>2 [1,1,1,0,0,0,1,1,0,0]=>3 [1,1,1,0,0,1,0,0,1,0]=>1 [1,1,1,0,0,1,0,1,0,0]=>1 [1,1,1,0,0,1,1,0,0,0]=>2 [1,1,1,0,1,0,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,1,0,0,0]=>0 [1,1,1,0,1,1,0,0,0,0]=>2 [1,1,1,1,0,0,0,0,1,0]=>3 [1,1,1,1,0,0,0,1,0,0]=>2 [1,1,1,1,0,0,1,0,0,0]=>2 [1,1,1,1,0,1,0,0,0,0]=>2 [1,1,1,1,1,0,0,0,0,0]=>4 [1,0,1,0,1,0,1,0,1,0,1,0]=>0 [1,0,1,0,1,0,1,0,1,1,0,0]=>1 [1,0,1,0,1,0,1,1,0,0,1,0]=>1 [1,0,1,0,1,0,1,1,0,1,0,0]=>0 [1,0,1,0,1,0,1,1,1,0,0,0]=>2 [1,0,1,0,1,1,0,0,1,0,1,0]=>1 [1,0,1,0,1,1,0,0,1,1,0,0]=>2 [1,0,1,0,1,1,0,1,0,0,1,0]=>0 [1,0,1,0,1,1,0,1,0,1,0,0]=>0 [1,0,1,0,1,1,0,1,1,0,0,0]=>1 [1,0,1,0,1,1,1,0,0,0,1,0]=>2 [1,0,1,0,1,1,1,0,0,1,0,0]=>1 [1,0,1,0,1,1,1,0,1,0,0,0]=>1 [1,0,1,0,1,1,1,1,0,0,0,0]=>3 [1,0,1,1,0,0,1,0,1,0,1,0]=>1 [1,0,1,1,0,0,1,0,1,1,0,0]=>2 [1,0,1,1,0,0,1,1,0,0,1,0]=>2 [1,0,1,1,0,0,1,1,0,1,0,0]=>1 [1,0,1,1,0,0,1,1,1,0,0,0]=>3 [1,0,1,1,0,1,0,0,1,0,1,0]=>0 [1,0,1,1,0,1,0,0,1,1,0,0]=>1 [1,0,1,1,0,1,0,1,0,0,1,0]=>0 [1,0,1,1,0,1,0,1,0,1,0,0]=>0 [1,0,1,1,0,1,0,1,1,0,0,0]=>1 [1,0,1,1,0,1,1,0,0,0,1,0]=>1 [1,0,1,1,0,1,1,0,0,1,0,0]=>0 [1,0,1,1,0,1,1,0,1,0,0,0]=>1 [1,0,1,1,0,1,1,1,0,0,0,0]=>2 [1,0,1,1,1,0,0,0,1,0,1,0]=>2 [1,0,1,1,1,0,0,0,1,1,0,0]=>3 [1,0,1,1,1,0,0,1,0,0,1,0]=>1 [1,0,1,1,1,0,0,1,0,1,0,0]=>1 [1,0,1,1,1,0,0,1,1,0,0,0]=>2 [1,0,1,1,1,0,1,0,0,0,1,0]=>1 [1,0,1,1,1,0,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,1,0,0,0]=>0 [1,0,1,1,1,0,1,1,0,0,0,0]=>2 [1,0,1,1,1,1,0,0,0,0,1,0]=>3 [1,0,1,1,1,1,0,0,0,1,0,0]=>2 [1,0,1,1,1,1,0,0,1,0,0,0]=>2 [1,0,1,1,1,1,0,1,0,0,0,0]=>2 [1,0,1,1,1,1,1,0,0,0,0,0]=>4 [1,1,0,0,1,0,1,0,1,0,1,0]=>1 [1,1,0,0,1,0,1,0,1,1,0,0]=>2 [1,1,0,0,1,0,1,1,0,0,1,0]=>2 [1,1,0,0,1,0,1,1,0,1,0,0]=>1 [1,1,0,0,1,0,1,1,1,0,0,0]=>3 [1,1,0,0,1,1,0,0,1,0,1,0]=>2 [1,1,0,0,1,1,0,0,1,1,0,0]=>3 [1,1,0,0,1,1,0,1,0,0,1,0]=>1 [1,1,0,0,1,1,0,1,0,1,0,0]=>1 [1,1,0,0,1,1,0,1,1,0,0,0]=>2 [1,1,0,0,1,1,1,0,0,0,1,0]=>3 [1,1,0,0,1,1,1,0,0,1,0,0]=>2 [1,1,0,0,1,1,1,0,1,0,0,0]=>2 [1,1,0,0,1,1,1,1,0,0,0,0]=>4 [1,1,0,1,0,0,1,0,1,0,1,0]=>0 [1,1,0,1,0,0,1,0,1,1,0,0]=>1 [1,1,0,1,0,0,1,1,0,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,1,0,0]=>0 [1,1,0,1,0,0,1,1,1,0,0,0]=>2 [1,1,0,1,0,1,0,0,1,0,1,0]=>0 [1,1,0,1,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,1,0,0,1,0]=>0 [1,1,0,1,0,1,0,1,0,1,0,0]=>0 [1,1,0,1,0,1,0,1,1,0,0,0]=>1 [1,1,0,1,0,1,1,0,0,0,1,0]=>1 [1,1,0,1,0,1,1,0,0,1,0,0]=>0 [1,1,0,1,0,1,1,0,1,0,0,0]=>1 [1,1,0,1,0,1,1,1,0,0,0,0]=>2 [1,1,0,1,1,0,0,0,1,0,1,0]=>1 [1,1,0,1,1,0,0,0,1,1,0,0]=>2 [1,1,0,1,1,0,0,1,0,0,1,0]=>0 [1,1,0,1,1,0,0,1,0,1,0,0]=>0 [1,1,0,1,1,0,0,1,1,0,0,0]=>1 [1,1,0,1,1,0,1,0,0,0,1,0]=>1 [1,1,0,1,1,0,1,0,0,1,0,0]=>1 [1,1,0,1,1,0,1,0,1,0,0,0]=>0 [1,1,0,1,1,0,1,1,0,0,0,0]=>2 [1,1,0,1,1,1,0,0,0,0,1,0]=>2 [1,1,0,1,1,1,0,0,0,1,0,0]=>1 [1,1,0,1,1,1,0,0,1,0,0,0]=>1 [1,1,0,1,1,1,0,1,0,0,0,0]=>2 [1,1,0,1,1,1,1,0,0,0,0,0]=>3 [1,1,1,0,0,0,1,0,1,0,1,0]=>2 [1,1,1,0,0,0,1,0,1,1,0,0]=>3 [1,1,1,0,0,0,1,1,0,0,1,0]=>3 [1,1,1,0,0,0,1,1,0,1,0,0]=>2 [1,1,1,0,0,0,1,1,1,0,0,0]=>4 [1,1,1,0,0,1,0,0,1,0,1,0]=>1 [1,1,1,0,0,1,0,0,1,1,0,0]=>2 [1,1,1,0,0,1,0,1,0,0,1,0]=>1 [1,1,1,0,0,1,0,1,0,1,0,0]=>1 [1,1,1,0,0,1,0,1,1,0,0,0]=>2 [1,1,1,0,0,1,1,0,0,0,1,0]=>2 [1,1,1,0,0,1,1,0,0,1,0,0]=>1 [1,1,1,0,0,1,1,0,1,0,0,0]=>2 [1,1,1,0,0,1,1,1,0,0,0,0]=>3 [1,1,1,0,1,0,0,0,1,0,1,0]=>1 [1,1,1,0,1,0,0,0,1,1,0,0]=>2 [1,1,1,0,1,0,0,1,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,1,0,0]=>1 [1,1,1,0,1,0,0,1,1,0,0,0]=>2 [1,1,1,0,1,0,1,0,0,0,1,0]=>0 [1,1,1,0,1,0,1,0,0,1,0,0]=>0 [1,1,1,0,1,0,1,0,1,0,0,0]=>0 [1,1,1,0,1,0,1,1,0,0,0,0]=>1 [1,1,1,0,1,1,0,0,0,0,1,0]=>2 [1,1,1,0,1,1,0,0,0,1,0,0]=>1 [1,1,1,0,1,1,0,0,1,0,0,0]=>2 [1,1,1,0,1,1,0,1,0,0,0,0]=>1 [1,1,1,0,1,1,1,0,0,0,0,0]=>3 [1,1,1,1,0,0,0,0,1,0,1,0]=>3 [1,1,1,1,0,0,0,0,1,1,0,0]=>4 [1,1,1,1,0,0,0,1,0,0,1,0]=>2 [1,1,1,1,0,0,0,1,0,1,0,0]=>2 [1,1,1,1,0,0,0,1,1,0,0,0]=>3 [1,1,1,1,0,0,1,0,0,0,1,0]=>2 [1,1,1,1,0,0,1,0,0,1,0,0]=>2 [1,1,1,1,0,0,1,0,1,0,0,0]=>1 [1,1,1,1,0,0,1,1,0,0,0,0]=>3 [1,1,1,1,0,1,0,0,0,0,1,0]=>2 [1,1,1,1,0,1,0,0,0,1,0,0]=>2 [1,1,1,1,0,1,0,0,1,0,0,0]=>1 [1,1,1,1,0,1,0,1,0,0,0,0]=>1 [1,1,1,1,0,1,1,0,0,0,0,0]=>3 [1,1,1,1,1,0,0,0,0,0,1,0]=>4 [1,1,1,1,1,0,0,0,0,1,0,0]=>3 [1,1,1,1,1,0,0,0,1,0,0,0]=>3 [1,1,1,1,1,0,0,1,0,0,0,0]=>3 [1,1,1,1,1,0,1,0,0,0,0,0]=>3 [1,1,1,1,1,1,0,0,0,0,0,0]=>5 [1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>0 [1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>1 [1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>1 [1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>0 [1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>2 [1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>1 [1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>2 [1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>0 [1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>0 [1,0,1,0,1,0,1,1,0,1,1,0,0,0]=>1 [1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>2 [1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>1 [1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>1 [1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>3 [1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>1 [1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>2 [1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>2 [1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>1 [1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>3 [1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>0 [1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>1 [1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>0 [1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>0 [1,0,1,0,1,1,0,1,0,1,1,0,0,0]=>1 [1,0,1,0,1,1,0,1,1,0,0,0,1,0]=>1 [1,0,1,0,1,1,0,1,1,0,0,1,0,0]=>0 [1,0,1,0,1,1,0,1,1,0,1,0,0,0]=>1 [1,0,1,0,1,1,0,1,1,1,0,0,0,0]=>2 [1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>2 [1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>3 [1,0,1,0,1,1,1,0,0,1,0,0,1,0]=>1 [1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>1 [1,0,1,0,1,1,1,0,0,1,1,0,0,0]=>2 [1,0,1,0,1,1,1,0,1,0,0,0,1,0]=>1 [1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>1 [1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>0 [1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>2 [1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>3 [1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>2 [1,0,1,0,1,1,1,1,0,0,1,0,0,0]=>2 [1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>2 [1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>4 [1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>1 [1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>2 [1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>2 [1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>1 [1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>3 [1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>2 [1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>3 [1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>1 [1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>1 [1,0,1,1,0,0,1,1,0,1,1,0,0,0]=>2 [1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>3 [1,0,1,1,0,0,1,1,1,0,0,1,0,0]=>2 [1,0,1,1,0,0,1,1,1,0,1,0,0,0]=>2 [1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>4 [1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>0 [1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>1 [1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>1 [1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>0 [1,0,1,1,0,1,0,0,1,1,1,0,0,0]=>2 [1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>0 [1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>1 [1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>0 [1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>0 [1,0,1,1,0,1,0,1,0,1,1,0,0,0]=>1 [1,0,1,1,0,1,0,1,1,0,0,0,1,0]=>1 [1,0,1,1,0,1,0,1,1,0,0,1,0,0]=>0 [1,0,1,1,0,1,0,1,1,0,1,0,0,0]=>1 [1,0,1,1,0,1,0,1,1,1,0,0,0,0]=>2 [1,0,1,1,0,1,1,0,0,0,1,0,1,0]=>1 [1,0,1,1,0,1,1,0,0,0,1,1,0,0]=>2 [1,0,1,1,0,1,1,0,0,1,0,0,1,0]=>0 [1,0,1,1,0,1,1,0,0,1,0,1,0,0]=>0 [1,0,1,1,0,1,1,0,0,1,1,0,0,0]=>1 [1,0,1,1,0,1,1,0,1,0,0,0,1,0]=>1 [1,0,1,1,0,1,1,0,1,0,0,1,0,0]=>1 [1,0,1,1,0,1,1,0,1,0,1,0,0,0]=>0 [1,0,1,1,0,1,1,0,1,1,0,0,0,0]=>2 [1,0,1,1,0,1,1,1,0,0,0,0,1,0]=>2 [1,0,1,1,0,1,1,1,0,0,0,1,0,0]=>1 [1,0,1,1,0,1,1,1,0,0,1,0,0,0]=>1 [1,0,1,1,0,1,1,1,0,1,0,0,0,0]=>2 [1,0,1,1,0,1,1,1,1,0,0,0,0,0]=>3 [1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>2 [1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>3 [1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>3 [1,0,1,1,1,0,0,0,1,1,0,1,0,0]=>2 [1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>4 [1,0,1,1,1,0,0,1,0,0,1,0,1,0]=>1 [1,0,1,1,1,0,0,1,0,0,1,1,0,0]=>2 [1,0,1,1,1,0,0,1,0,1,0,0,1,0]=>1 [1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>1 [1,0,1,1,1,0,0,1,0,1,1,0,0,0]=>2 [1,0,1,1,1,0,0,1,1,0,0,0,1,0]=>2 [1,0,1,1,1,0,0,1,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,0,1,1,0,1,0,0,0]=>2 [1,0,1,1,1,0,0,1,1,1,0,0,0,0]=>3 [1,0,1,1,1,0,1,0,0,0,1,0,1,0]=>1 [1,0,1,1,1,0,1,0,0,0,1,1,0,0]=>2 [1,0,1,1,1,0,1,0,0,1,0,0,1,0]=>1 [1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,0,1,1,0,0,0]=>2 [1,0,1,1,1,0,1,0,1,0,0,0,1,0]=>0 [1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>0 [1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>0 [1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>1 [1,0,1,1,1,0,1,1,0,0,0,0,1,0]=>2 [1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,1,0,0,1,0,0,0]=>2 [1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>1 [1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>3 [1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>3 [1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>4 [1,0,1,1,1,1,0,0,0,1,0,0,1,0]=>2 [1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>2 [1,0,1,1,1,1,0,0,0,1,1,0,0,0]=>3 [1,0,1,1,1,1,0,0,1,0,0,0,1,0]=>2 [1,0,1,1,1,1,0,0,1,0,0,1,0,0]=>2 [1,0,1,1,1,1,0,0,1,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,0,1,1,0,0,0,0]=>3 [1,0,1,1,1,1,0,1,0,0,0,0,1,0]=>2 [1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>2 [1,0,1,1,1,1,0,1,0,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>1 [1,0,1,1,1,1,0,1,1,0,0,0,0,0]=>3 [1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>4 [1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>3 [1,0,1,1,1,1,1,0,0,0,1,0,0,0]=>3 [1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>3 [1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>3 [1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>5 [1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>1 [1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>2 [1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>2 [1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>1 [1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>3 [1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>2 [1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>3 [1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>1 [1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>1 [1,1,0,0,1,0,1,1,0,1,1,0,0,0]=>2 [1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>3 [1,1,0,0,1,0,1,1,1,0,0,1,0,0]=>2 [1,1,0,0,1,0,1,1,1,0,1,0,0,0]=>2 [1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>4 [1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>2 [1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>3 [1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>3 [1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>2 [1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>4 [1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>1 [1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>2 [1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>1 [1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>1 [1,1,0,0,1,1,0,1,0,1,1,0,0,0]=>2 [1,1,0,0,1,1,0,1,1,0,0,0,1,0]=>2 [1,1,0,0,1,1,0,1,1,0,0,1,0,0]=>1 [1,1,0,0,1,1,0,1,1,0,1,0,0,0]=>2 [1,1,0,0,1,1,0,1,1,1,0,0,0,0]=>3 [1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>3 [1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>4 [1,1,0,0,1,1,1,0,0,1,0,0,1,0]=>2 [1,1,0,0,1,1,1,0,0,1,0,1,0,0]=>2 [1,1,0,0,1,1,1,0,0,1,1,0,0,0]=>3 [1,1,0,0,1,1,1,0,1,0,0,0,1,0]=>2 [1,1,0,0,1,1,1,0,1,0,0,1,0,0]=>2 [1,1,0,0,1,1,1,0,1,0,1,0,0,0]=>1 [1,1,0,0,1,1,1,0,1,1,0,0,0,0]=>3 [1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>4 [1,1,0,0,1,1,1,1,0,0,0,1,0,0]=>3 [1,1,0,0,1,1,1,1,0,0,1,0,0,0]=>3 [1,1,0,0,1,1,1,1,0,1,0,0,0,0]=>3 [1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>5 [1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>0 [1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>1 [1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>1 [1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>0 [1,1,0,1,0,0,1,0,1,1,1,0,0,0]=>2 [1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>2 [1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>0 [1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>0 [1,1,0,1,0,0,1,1,0,1,1,0,0,0]=>1 [1,1,0,1,0,0,1,1,1,0,0,0,1,0]=>2 [1,1,0,1,0,0,1,1,1,0,0,1,0,0]=>1 [1,1,0,1,0,0,1,1,1,0,1,0,0,0]=>1 [1,1,0,1,0,0,1,1,1,1,0,0,0,0]=>3 [1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>0 [1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>1 [1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>0 [1,1,0,1,0,1,0,0,1,1,1,0,0,0]=>2 [1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>0 [1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>0 [1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>0 [1,1,0,1,0,1,0,1,0,1,1,0,0,0]=>1 [1,1,0,1,0,1,0,1,1,0,0,0,1,0]=>1 [1,1,0,1,0,1,0,1,1,0,0,1,0,0]=>0 [1,1,0,1,0,1,0,1,1,0,1,0,0,0]=>1 [1,1,0,1,0,1,0,1,1,1,0,0,0,0]=>2 [1,1,0,1,0,1,1,0,0,0,1,0,1,0]=>1 [1,1,0,1,0,1,1,0,0,0,1,1,0,0]=>2 [1,1,0,1,0,1,1,0,0,1,0,0,1,0]=>0 [1,1,0,1,0,1,1,0,0,1,0,1,0,0]=>0 [1,1,0,1,0,1,1,0,0,1,1,0,0,0]=>1 [1,1,0,1,0,1,1,0,1,0,0,0,1,0]=>1 [1,1,0,1,0,1,1,0,1,0,0,1,0,0]=>1 [1,1,0,1,0,1,1,0,1,0,1,0,0,0]=>0 [1,1,0,1,0,1,1,0,1,1,0,0,0,0]=>2 [1,1,0,1,0,1,1,1,0,0,0,0,1,0]=>2 [1,1,0,1,0,1,1,1,0,0,0,1,0,0]=>1 [1,1,0,1,0,1,1,1,0,0,1,0,0,0]=>1 [1,1,0,1,0,1,1,1,0,1,0,0,0,0]=>2 [1,1,0,1,0,1,1,1,1,0,0,0,0,0]=>3 [1,1,0,1,1,0,0,0,1,0,1,0,1,0]=>1 [1,1,0,1,1,0,0,0,1,0,1,1,0,0]=>2 [1,1,0,1,1,0,0,0,1,1,0,0,1,0]=>2 [1,1,0,1,1,0,0,0,1,1,0,1,0,0]=>1 [1,1,0,1,1,0,0,0,1,1,1,0,0,0]=>3 [1,1,0,1,1,0,0,1,0,0,1,0,1,0]=>0 [1,1,0,1,1,0,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,1,0,0,1,0,1,0,0,1,0]=>0 [1,1,0,1,1,0,0,1,0,1,0,1,0,0]=>0 [1,1,0,1,1,0,0,1,0,1,1,0,0,0]=>1 [1,1,0,1,1,0,0,1,1,0,0,0,1,0]=>1 [1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>0 [1,1,0,1,1,0,0,1,1,0,1,0,0,0]=>1 [1,1,0,1,1,0,0,1,1,1,0,0,0,0]=>2 [1,1,0,1,1,0,1,0,0,0,1,0,1,0]=>1 [1,1,0,1,1,0,1,0,0,0,1,1,0,0]=>2 [1,1,0,1,1,0,1,0,0,1,0,0,1,0]=>1 [1,1,0,1,1,0,1,0,0,1,0,1,0,0]=>1 [1,1,0,1,1,0,1,0,0,1,1,0,0,0]=>2 [1,1,0,1,1,0,1,0,1,0,0,0,1,0]=>0 [1,1,0,1,1,0,1,0,1,0,0,1,0,0]=>0 [1,1,0,1,1,0,1,0,1,0,1,0,0,0]=>0 [1,1,0,1,1,0,1,0,1,1,0,0,0,0]=>1 [1,1,0,1,1,0,1,1,0,0,0,0,1,0]=>2 [1,1,0,1,1,0,1,1,0,0,0,1,0,0]=>1 [1,1,0,1,1,0,1,1,0,0,1,0,0,0]=>2 [1,1,0,1,1,0,1,1,0,1,0,0,0,0]=>1 [1,1,0,1,1,0,1,1,1,0,0,0,0,0]=>3 [1,1,0,1,1,1,0,0,0,0,1,0,1,0]=>2 [1,1,0,1,1,1,0,0,0,0,1,1,0,0]=>3 [1,1,0,1,1,1,0,0,0,1,0,0,1,0]=>1 [1,1,0,1,1,1,0,0,0,1,0,1,0,0]=>1 [1,1,0,1,1,1,0,0,0,1,1,0,0,0]=>2 [1,1,0,1,1,1,0,0,1,0,0,0,1,0]=>1 [1,1,0,1,1,1,0,0,1,0,0,1,0,0]=>1 [1,1,0,1,1,1,0,0,1,0,1,0,0,0]=>0 [1,1,0,1,1,1,0,0,1,1,0,0,0,0]=>2 [1,1,0,1,1,1,0,1,0,0,0,0,1,0]=>2 [1,1,0,1,1,1,0,1,0,0,0,1,0,0]=>2 [1,1,0,1,1,1,0,1,0,0,1,0,0,0]=>1 [1,1,0,1,1,1,0,1,0,1,0,0,0,0]=>1 [1,1,0,1,1,1,0,1,1,0,0,0,0,0]=>3 [1,1,0,1,1,1,1,0,0,0,0,0,1,0]=>3 [1,1,0,1,1,1,1,0,0,0,0,1,0,0]=>2 [1,1,0,1,1,1,1,0,0,0,1,0,0,0]=>2 [1,1,0,1,1,1,1,0,0,1,0,0,0,0]=>2 [1,1,0,1,1,1,1,0,1,0,0,0,0,0]=>3 [1,1,0,1,1,1,1,1,0,0,0,0,0,0]=>4 [1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>2 [1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>3 [1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>3 [1,1,1,0,0,0,1,0,1,1,0,1,0,0]=>2 [1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>4 [1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>3 [1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>4 [1,1,1,0,0,0,1,1,0,1,0,0,1,0]=>2 [1,1,1,0,0,0,1,1,0,1,0,1,0,0]=>2 [1,1,1,0,0,0,1,1,0,1,1,0,0,0]=>3 [1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>4 [1,1,1,0,0,0,1,1,1,0,0,1,0,0]=>3 [1,1,1,0,0,0,1,1,1,0,1,0,0,0]=>3 [1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>5 [1,1,1,0,0,1,0,0,1,0,1,0,1,0]=>1 [1,1,1,0,0,1,0,0,1,0,1,1,0,0]=>2 [1,1,1,0,0,1,0,0,1,1,0,0,1,0]=>2 [1,1,1,0,0,1,0,0,1,1,0,1,0,0]=>1 [1,1,1,0,0,1,0,0,1,1,1,0,0,0]=>3 [1,1,1,0,0,1,0,1,0,0,1,0,1,0]=>1 [1,1,1,0,0,1,0,1,0,0,1,1,0,0]=>2 [1,1,1,0,0,1,0,1,0,1,0,0,1,0]=>1 [1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>1 [1,1,1,0,0,1,0,1,0,1,1,0,0,0]=>2 [1,1,1,0,0,1,0,1,1,0,0,0,1,0]=>2 [1,1,1,0,0,1,0,1,1,0,0,1,0,0]=>1 [1,1,1,0,0,1,0,1,1,0,1,0,0,0]=>2 [1,1,1,0,0,1,0,1,1,1,0,0,0,0]=>3 [1,1,1,0,0,1,1,0,0,0,1,0,1,0]=>2 [1,1,1,0,0,1,1,0,0,0,1,1,0,0]=>3 [1,1,1,0,0,1,1,0,0,1,0,0,1,0]=>1 [1,1,1,0,0,1,1,0,0,1,0,1,0,0]=>1 [1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>2 [1,1,1,0,0,1,1,0,1,0,0,0,1,0]=>2 [1,1,1,0,0,1,1,0,1,0,0,1,0,0]=>2 [1,1,1,0,0,1,1,0,1,0,1,0,0,0]=>1 [1,1,1,0,0,1,1,0,1,1,0,0,0,0]=>3 [1,1,1,0,0,1,1,1,0,0,0,0,1,0]=>3 [1,1,1,0,0,1,1,1,0,0,0,1,0,0]=>2 [1,1,1,0,0,1,1,1,0,0,1,0,0,0]=>2 [1,1,1,0,0,1,1,1,0,1,0,0,0,0]=>3 [1,1,1,0,0,1,1,1,1,0,0,0,0,0]=>4 [1,1,1,0,1,0,0,0,1,0,1,0,1,0]=>1 [1,1,1,0,1,0,0,0,1,0,1,1,0,0]=>2 [1,1,1,0,1,0,0,0,1,1,0,0,1,0]=>2 [1,1,1,0,1,0,0,0,1,1,0,1,0,0]=>1 [1,1,1,0,1,0,0,0,1,1,1,0,0,0]=>3 [1,1,1,0,1,0,0,1,0,0,1,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,0,1,1,0,0]=>2 [1,1,1,0,1,0,0,1,0,1,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>1 [1,1,1,0,1,0,0,1,0,1,1,0,0,0]=>2 [1,1,1,0,1,0,0,1,1,0,0,0,1,0]=>2 [1,1,1,0,1,0,0,1,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,0,1,1,0,1,0,0,0]=>2 [1,1,1,0,1,0,0,1,1,1,0,0,0,0]=>3 [1,1,1,0,1,0,1,0,0,0,1,0,1,0]=>0 [1,1,1,0,1,0,1,0,0,0,1,1,0,0]=>1 [1,1,1,0,1,0,1,0,0,1,0,0,1,0]=>0 [1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>0 [1,1,1,0,1,0,1,0,0,1,1,0,0,0]=>1 [1,1,1,0,1,0,1,0,1,0,0,0,1,0]=>0 [1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>0 [1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>0 [1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>1 [1,1,1,0,1,0,1,1,0,0,0,0,1,0]=>1 [1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>0 [1,1,1,0,1,0,1,1,0,0,1,0,0,0]=>1 [1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>1 [1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>2 [1,1,1,0,1,1,0,0,0,0,1,0,1,0]=>2 [1,1,1,0,1,1,0,0,0,0,1,1,0,0]=>3 [1,1,1,0,1,1,0,0,0,1,0,0,1,0]=>1 [1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>1 [1,1,1,0,1,1,0,0,0,1,1,0,0,0]=>2 [1,1,1,0,1,1,0,0,1,0,0,0,1,0]=>2 [1,1,1,0,1,1,0,0,1,0,0,1,0,0]=>2 [1,1,1,0,1,1,0,0,1,0,1,0,0,0]=>1 [1,1,1,0,1,1,0,0,1,1,0,0,0,0]=>3 [1,1,1,0,1,1,0,1,0,0,0,0,1,0]=>1 [1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>1 [1,1,1,0,1,1,0,1,0,0,1,0,0,0]=>0 [1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>1 [1,1,1,0,1,1,0,1,1,0,0,0,0,0]=>2 [1,1,1,0,1,1,1,0,0,0,0,0,1,0]=>3 [1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>2 [1,1,1,0,1,1,1,0,0,0,1,0,0,0]=>2 [1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>3 [1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>2 [1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>4 [1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>3 [1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>4 [1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>4 [1,1,1,1,0,0,0,0,1,1,0,1,0,0]=>3 [1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>5 [1,1,1,1,0,0,0,1,0,0,1,0,1,0]=>2 [1,1,1,1,0,0,0,1,0,0,1,1,0,0]=>3 [1,1,1,1,0,0,0,1,0,1,0,0,1,0]=>2 [1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>2 [1,1,1,1,0,0,0,1,0,1,1,0,0,0]=>3 [1,1,1,1,0,0,0,1,1,0,0,0,1,0]=>3 [1,1,1,1,0,0,0,1,1,0,0,1,0,0]=>2 [1,1,1,1,0,0,0,1,1,0,1,0,0,0]=>3 [1,1,1,1,0,0,0,1,1,1,0,0,0,0]=>4 [1,1,1,1,0,0,1,0,0,0,1,0,1,0]=>2 [1,1,1,1,0,0,1,0,0,0,1,1,0,0]=>3 [1,1,1,1,0,0,1,0,0,1,0,0,1,0]=>2 [1,1,1,1,0,0,1,0,0,1,0,1,0,0]=>2 [1,1,1,1,0,0,1,0,0,1,1,0,0,0]=>3 [1,1,1,1,0,0,1,0,1,0,0,0,1,0]=>1 [1,1,1,1,0,0,1,0,1,0,0,1,0,0]=>1 [1,1,1,1,0,0,1,0,1,0,1,0,0,0]=>1 [1,1,1,1,0,0,1,0,1,1,0,0,0,0]=>2 [1,1,1,1,0,0,1,1,0,0,0,0,1,0]=>3 [1,1,1,1,0,0,1,1,0,0,0,1,0,0]=>2 [1,1,1,1,0,0,1,1,0,0,1,0,0,0]=>3 [1,1,1,1,0,0,1,1,0,1,0,0,0,0]=>2 [1,1,1,1,0,0,1,1,1,0,0,0,0,0]=>4 [1,1,1,1,0,1,0,0,0,0,1,0,1,0]=>2 [1,1,1,1,0,1,0,0,0,0,1,1,0,0]=>3 [1,1,1,1,0,1,0,0,0,1,0,0,1,0]=>2 [1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>2 [1,1,1,1,0,1,0,0,0,1,1,0,0,0]=>3 [1,1,1,1,0,1,0,0,1,0,0,0,1,0]=>1 [1,1,1,1,0,1,0,0,1,0,0,1,0,0]=>1 [1,1,1,1,0,1,0,0,1,0,1,0,0,0]=>1 [1,1,1,1,0,1,0,0,1,1,0,0,0,0]=>2 [1,1,1,1,0,1,0,1,0,0,0,0,1,0]=>1 [1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>1 [1,1,1,1,0,1,0,1,0,0,1,0,0,0]=>1 [1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>0 [1,1,1,1,0,1,0,1,1,0,0,0,0,0]=>2 [1,1,1,1,0,1,1,0,0,0,0,0,1,0]=>3 [1,1,1,1,0,1,1,0,0,0,0,1,0,0]=>2 [1,1,1,1,0,1,1,0,0,0,1,0,0,0]=>3 [1,1,1,1,0,1,1,0,0,1,0,0,0,0]=>2 [1,1,1,1,0,1,1,0,1,0,0,0,0,0]=>2 [1,1,1,1,0,1,1,1,0,0,0,0,0,0]=>4 [1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>4 [1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>5 [1,1,1,1,1,0,0,0,0,1,0,0,1,0]=>3 [1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>3 [1,1,1,1,1,0,0,0,0,1,1,0,0,0]=>4 [1,1,1,1,1,0,0,0,1,0,0,0,1,0]=>3 [1,1,1,1,1,0,0,0,1,0,0,1,0,0]=>3 [1,1,1,1,1,0,0,0,1,0,1,0,0,0]=>2 [1,1,1,1,1,0,0,0,1,1,0,0,0,0]=>4 [1,1,1,1,1,0,0,1,0,0,0,0,1,0]=>3 [1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>3 [1,1,1,1,1,0,0,1,0,0,1,0,0,0]=>2 [1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>2 [1,1,1,1,1,0,0,1,1,0,0,0,0,0]=>4 [1,1,1,1,1,0,1,0,0,0,0,0,1,0]=>3 [1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>3 [1,1,1,1,1,0,1,0,0,0,1,0,0,0]=>2 [1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>2 [1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>2 [1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>4 [1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>5 [1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>4 [1,1,1,1,1,1,0,0,0,0,1,0,0,0]=>4 [1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>4 [1,1,1,1,1,1,0,0,1,0,0,0,0,0]=>4 [1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>4 [1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>6
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Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
This is also the number of twin vertices in the unit interval graph corresponding to the Dyck path.
A Dyck path of size $n$ determines a unit interval graph on vertex set $[n]$, see [1]. Two vertices ${i,i+1}$ are twins if they have the same closed neighbourhood in the unit interval graph.
Put differently, this is the number of indices $i$ such that the $i$-th and $(i+1)$-st up steps are consecutive, and the $i$-th and $(i+1)$-st down steps are consecutive.
References
[1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles MathSciNet:3862644
Code
# Alexandersson
def statistic(D):
    a = D.to_area_sequence()
    b = list(reversed(D.reverse().to_area_sequence()))
    return sum(1 for i in range(len(a) - 1) if (a[i] + 1 == a[i+1]) and (b[i] - 1 == b[i+1]))

# Marczinzik

gap('LoadPackage("QPA");')

def kupisch(D):
    H = D.heights()
    return [1 + H[i] for i, s in enumerate(D) if s == 0] + [1]

def statistic_alternative(D):
    K = kupisch(DyckWord(D))
    A = gap.NakayamaAlgebra(K, gap.GF(3))
    simples = A.SimpleModules()
    L = [x for x in simples
         if (not gap.DominantDimensionOfModule(x, 30).sage()
             and not gap.DominantDimensionOfModule(gap.DualOfModule(x), 30).sage())]
    return len(L)
Created
May 10, 2018 at 11:24 by Rene Marczinzik
Updated
Oct 10, 2025 at 09:56 by Martin Rubey