Identifier
- St000684: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
[1,0]=>1
[1,0,1,0]=>2
[1,1,0,0]=>1
[1,0,1,0,1,0]=>3
[1,0,1,1,0,0]=>2
[1,1,0,0,1,0]=>2
[1,1,0,1,0,0]=>2
[1,1,1,0,0,0]=>1
[1,0,1,0,1,0,1,0]=>4
[1,0,1,0,1,1,0,0]=>3
[1,0,1,1,0,0,1,0]=>2
[1,0,1,1,0,1,0,0]=>3
[1,0,1,1,1,0,0,0]=>2
[1,1,0,0,1,0,1,0]=>3
[1,1,0,0,1,1,0,0]=>2
[1,1,0,1,0,0,1,0]=>3
[1,1,0,1,0,1,0,0]=>3
[1,1,0,1,1,0,0,0]=>2
[1,1,1,0,0,0,1,0]=>2
[1,1,1,0,0,1,0,0]=>2
[1,1,1,0,1,0,0,0]=>2
[1,1,1,1,0,0,0,0]=>1
[1,0,1,0,1,0,1,0,1,0]=>5
[1,0,1,0,1,0,1,1,0,0]=>4
[1,0,1,0,1,1,0,0,1,0]=>3
[1,0,1,0,1,1,0,1,0,0]=>4
[1,0,1,0,1,1,1,0,0,0]=>3
[1,0,1,1,0,0,1,0,1,0]=>3
[1,0,1,1,0,0,1,1,0,0]=>2
[1,0,1,1,0,1,0,0,1,0]=>4
[1,0,1,1,0,1,0,1,0,0]=>4
[1,0,1,1,0,1,1,0,0,0]=>3
[1,0,1,1,1,0,0,0,1,0]=>2
[1,0,1,1,1,0,0,1,0,0]=>2
[1,0,1,1,1,0,1,0,0,0]=>3
[1,0,1,1,1,1,0,0,0,0]=>2
[1,1,0,0,1,0,1,0,1,0]=>4
[1,1,0,0,1,0,1,1,0,0]=>3
[1,1,0,0,1,1,0,0,1,0]=>2
[1,1,0,0,1,1,0,1,0,0]=>3
[1,1,0,0,1,1,1,0,0,0]=>2
[1,1,0,1,0,0,1,0,1,0]=>4
[1,1,0,1,0,0,1,1,0,0]=>3
[1,1,0,1,0,1,0,0,1,0]=>4
[1,1,0,1,0,1,0,1,0,0]=>3
[1,1,0,1,0,1,1,0,0,0]=>3
[1,1,0,1,1,0,0,0,1,0]=>2
[1,1,0,1,1,0,0,1,0,0]=>3
[1,1,0,1,1,0,1,0,0,0]=>3
[1,1,0,1,1,1,0,0,0,0]=>2
[1,1,1,0,0,0,1,0,1,0]=>3
[1,1,1,0,0,0,1,1,0,0]=>2
[1,1,1,0,0,1,0,0,1,0]=>3
[1,1,1,0,0,1,0,1,0,0]=>3
[1,1,1,0,0,1,1,0,0,0]=>2
[1,1,1,0,1,0,0,0,1,0]=>3
[1,1,1,0,1,0,0,1,0,0]=>3
[1,1,1,0,1,0,1,0,0,0]=>3
[1,1,1,0,1,1,0,0,0,0]=>2
[1,1,1,1,0,0,0,0,1,0]=>2
[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]=>1
[1,0,1,0,1,0,1,0,1,0,1,0]=>6
[1,0,1,0,1,0,1,0,1,1,0,0]=>5
[1,0,1,0,1,0,1,1,0,0,1,0]=>4
[1,0,1,0,1,0,1,1,0,1,0,0]=>5
[1,0,1,0,1,0,1,1,1,0,0,0]=>4
[1,0,1,0,1,1,0,0,1,0,1,0]=>3
[1,0,1,0,1,1,0,0,1,1,0,0]=>3
[1,0,1,0,1,1,0,1,0,0,1,0]=>5
[1,0,1,0,1,1,0,1,0,1,0,0]=>5
[1,0,1,0,1,1,0,1,1,0,0,0]=>4
[1,0,1,0,1,1,1,0,0,0,1,0]=>3
[1,0,1,0,1,1,1,0,0,1,0,0]=>3
[1,0,1,0,1,1,1,0,1,0,0,0]=>4
[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]=>4
[1,0,1,1,0,0,1,0,1,1,0,0]=>3
[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]=>3
[1,0,1,1,0,0,1,1,1,0,0,0]=>2
[1,0,1,1,0,1,0,0,1,0,1,0]=>5
[1,0,1,1,0,1,0,0,1,1,0,0]=>4
[1,0,1,1,0,1,0,1,0,0,1,0]=>5
[1,0,1,1,0,1,0,1,0,1,0,0]=>4
[1,0,1,1,0,1,0,1,1,0,0,0]=>4
[1,0,1,1,0,1,1,0,0,0,1,0]=>3
[1,0,1,1,0,1,1,0,0,1,0,0]=>4
[1,0,1,1,0,1,1,0,1,0,0,0]=>4
[1,0,1,1,0,1,1,1,0,0,0,0]=>3
[1,0,1,1,1,0,0,0,1,0,1,0]=>3
[1,0,1,1,1,0,0,0,1,1,0,0]=>2
[1,0,1,1,1,0,0,1,0,0,1,0]=>3
[1,0,1,1,1,0,0,1,0,1,0,0]=>3
[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]=>4
[1,0,1,1,1,0,1,0,0,1,0,0]=>4
[1,0,1,1,1,0,1,0,1,0,0,0]=>4
[1,0,1,1,1,0,1,1,0,0,0,0]=>3
[1,0,1,1,1,1,0,0,0,0,1,0]=>2
[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]=>3
[1,0,1,1,1,1,1,0,0,0,0,0]=>2
[1,1,0,0,1,0,1,0,1,0,1,0]=>5
[1,1,0,0,1,0,1,0,1,1,0,0]=>4
[1,1,0,0,1,0,1,1,0,0,1,0]=>3
[1,1,0,0,1,0,1,1,0,1,0,0]=>4
[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]=>3
[1,1,0,0,1,1,0,0,1,1,0,0]=>2
[1,1,0,0,1,1,0,1,0,0,1,0]=>4
[1,1,0,0,1,1,0,1,0,1,0,0]=>4
[1,1,0,0,1,1,0,1,1,0,0,0]=>3
[1,1,0,0,1,1,1,0,0,0,1,0]=>2
[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]=>3
[1,1,0,0,1,1,1,1,0,0,0,0]=>2
[1,1,0,1,0,0,1,0,1,0,1,0]=>5
[1,1,0,1,0,0,1,0,1,1,0,0]=>4
[1,1,0,1,0,0,1,1,0,0,1,0]=>3
[1,1,0,1,0,0,1,1,0,1,0,0]=>4
[1,1,0,1,0,0,1,1,1,0,0,0]=>3
[1,1,0,1,0,1,0,0,1,0,1,0]=>5
[1,1,0,1,0,1,0,0,1,1,0,0]=>4
[1,1,0,1,0,1,0,1,0,0,1,0]=>4
[1,1,0,1,0,1,0,1,0,1,0,0]=>4
[1,1,0,1,0,1,0,1,1,0,0,0]=>3
[1,1,0,1,0,1,1,0,0,0,1,0]=>3
[1,1,0,1,0,1,1,0,0,1,0,0]=>4
[1,1,0,1,0,1,1,0,1,0,0,0]=>3
[1,1,0,1,0,1,1,1,0,0,0,0]=>3
[1,1,0,1,1,0,0,0,1,0,1,0]=>3
[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]=>4
[1,1,0,1,1,0,0,1,0,1,0,0]=>4
[1,1,0,1,1,0,0,1,1,0,0,0]=>3
[1,1,0,1,1,0,1,0,0,0,1,0]=>4
[1,1,0,1,1,0,1,0,0,1,0,0]=>4
[1,1,0,1,1,0,1,0,1,0,0,0]=>3
[1,1,0,1,1,0,1,1,0,0,0,0]=>3
[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]=>2
[1,1,0,1,1,1,0,0,1,0,0,0]=>3
[1,1,0,1,1,1,0,1,0,0,0,0]=>3
[1,1,0,1,1,1,1,0,0,0,0,0]=>2
[1,1,1,0,0,0,1,0,1,0,1,0]=>4
[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]=>2
[1,1,1,0,0,0,1,1,0,1,0,0]=>3
[1,1,1,0,0,0,1,1,1,0,0,0]=>2
[1,1,1,0,0,1,0,0,1,0,1,0]=>4
[1,1,1,0,0,1,0,0,1,1,0,0]=>3
[1,1,1,0,0,1,0,1,0,0,1,0]=>4
[1,1,1,0,0,1,0,1,0,1,0,0]=>3
[1,1,1,0,0,1,0,1,1,0,0,0]=>3
[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]=>3
[1,1,1,0,0,1,1,0,1,0,0,0]=>3
[1,1,1,0,0,1,1,1,0,0,0,0]=>2
[1,1,1,0,1,0,0,0,1,0,1,0]=>4
[1,1,1,0,1,0,0,0,1,1,0,0]=>3
[1,1,1,0,1,0,0,1,0,0,1,0]=>4
[1,1,1,0,1,0,0,1,0,1,0,0]=>3
[1,1,1,0,1,0,0,1,1,0,0,0]=>3
[1,1,1,0,1,0,1,0,0,0,1,0]=>4
[1,1,1,0,1,0,1,0,0,1,0,0]=>3
[1,1,1,0,1,0,1,0,1,0,0,0]=>3
[1,1,1,0,1,0,1,1,0,0,0,0]=>3
[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]=>3
[1,1,1,0,1,1,0,0,1,0,0,0]=>3
[1,1,1,0,1,1,0,1,0,0,0,0]=>3
[1,1,1,0,1,1,1,0,0,0,0,0]=>2
[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]=>2
[1,1,1,1,0,0,0,1,0,0,1,0]=>3
[1,1,1,1,0,0,0,1,0,1,0,0]=>3
[1,1,1,1,0,0,0,1,1,0,0,0]=>2
[1,1,1,1,0,0,1,0,0,0,1,0]=>3
[1,1,1,1,0,0,1,0,0,1,0,0]=>3
[1,1,1,1,0,0,1,0,1,0,0,0]=>3
[1,1,1,1,0,0,1,1,0,0,0,0]=>2
[1,1,1,1,0,1,0,0,0,0,1,0]=>3
[1,1,1,1,0,1,0,0,0,1,0,0]=>3
[1,1,1,1,0,1,0,0,1,0,0,0]=>3
[1,1,1,1,0,1,0,1,0,0,0,0]=>3
[1,1,1,1,0,1,1,0,0,0,0,0]=>2
[1,1,1,1,1,0,0,0,0,0,1,0]=>2
[1,1,1,1,1,0,0,0,0,1,0,0]=>2
[1,1,1,1,1,0,0,0,1,0,0,0]=>2
[1,1,1,1,1,0,0,1,0,0,0,0]=>2
[1,1,1,1,1,0,1,0,0,0,0,0]=>2
[1,1,1,1,1,1,0,0,0,0,0,0]=>1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]=>5
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>5
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>3
[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]=>6
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]=>4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]=>5
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]=>4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>3
[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]=>3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]=>3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]=>5
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>4
[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]=>3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]=>3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>4
[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]=>3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]=>3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>2
[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]=>3
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>6
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>5
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>4
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>5
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]=>4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>6
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>5
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>5
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]=>4
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]=>4
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]=>4
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]=>4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]=>3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]=>3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]=>5
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]=>4
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]=>5
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]=>4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]=>4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]=>3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]=>3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]=>4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]=>4
[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]=>4
[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]=>2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]=>3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]=>4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]=>3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]=>4
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]=>3
[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]=>3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]=>3
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]=>2
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]=>5
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]=>4
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]=>5
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>4
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]=>4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]=>5
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>4
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>4
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>4
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]=>3
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>4
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]=>4
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>4
[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]=>2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]=>3
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>3
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]=>2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]=>3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]=>3
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]=>3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]=>2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]=>4
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]=>4
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>4
[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]=>2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]=>2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>2
[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]=>2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>5
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>4
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>5
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>3
[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]=>5
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>5
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]=>4
[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]=>3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]=>4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>4
[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]=>2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>3
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>5
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>4
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>5
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>4
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]=>4
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]=>3
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]=>4
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]=>4
[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]=>2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]=>3
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]=>3
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]=>2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]=>4
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]=>4
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]=>4
[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]=>2
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]=>2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]=>2
[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]=>2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>6
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>5
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>4
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>5
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]=>4
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>3
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>3
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>5
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]=>4
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]=>3
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]=>3
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]=>4
[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]=>6
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>5
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>5
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]=>4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>5
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>5
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]=>4
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]=>3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]=>4
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]=>4
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]=>3
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]=>3
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]=>3
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]=>5
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]=>4
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]=>4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]=>4
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]=>4
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]=>3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]=>3
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]=>3
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]=>4
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]=>3
[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]=>4
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]=>3
[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]=>3
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]=>2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]=>5
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]=>4
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]=>5
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]=>4
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]=>4
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]=>3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>4
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]=>4
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]=>3
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]=>4
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]=>4
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]=>4
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]=>4
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]=>4
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]=>4
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]=>3
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]=>3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]=>4
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]=>4
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]=>3
[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]=>3
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]=>2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]=>3
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]=>3
[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]=>4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]=>4
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]=>4
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]=>3
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]=>4
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]=>4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]=>4
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]=>3
[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]=>2
[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]=>3
[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]=>2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>5
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>4
[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]=>4
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>3
[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]=>2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]=>4
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]=>4
[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]=>2
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]=>2
[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]=>2
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]=>5
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]=>4
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]=>3
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]=>4
[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]=>5
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]=>4
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]=>4
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>4
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]=>3
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]=>3
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]=>4
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]=>3
[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]=>3
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]=>2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]=>4
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]=>4
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>3
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]=>4
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]=>4
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]=>3
[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]=>2
[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]=>3
[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]=>2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]=>5
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]=>4
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]=>3
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]=>4
[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]=>5
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]=>4
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]=>4
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>4
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]=>3
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]=>3
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]=>4
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]=>3
[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]=>5
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]=>4
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]=>4
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>4
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]=>3
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]=>4
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>4
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>3
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>3
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]=>3
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>4
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]=>3
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>3
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>3
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]=>3
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]=>2
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]=>4
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>4
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]=>3
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]=>4
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]=>4
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]=>3
[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]=>4
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>4
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]=>3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>3
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]=>3
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]=>2
[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]=>3
[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]=>3
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>4
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>3
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>2
[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]=>2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]=>4
[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]=>4
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>3
[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]=>2
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]=>3
[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]=>2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]=>4
[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]=>4
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]=>3
[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]=>4
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]=>3
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]=>3
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]=>3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]=>2
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]=>3
[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]=>3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]=>2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]=>4
[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]=>4
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>3
[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]=>4
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]=>3
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]=>3
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]=>3
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]=>4
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>3
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]=>3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]=>3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]=>2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]=>3
[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]=>3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]=>3
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]=>2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>3
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>2
[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]=>2
[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]=>3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]=>2
[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]=>3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]=>2
[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]=>3
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]=>2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]=>2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>6
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>5
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>7
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>7
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]=>6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>5
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>5
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>6
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>5
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>7
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>7
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>6
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]=>6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]=>5
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]=>6
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]=>6
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]=>5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]=>6
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>6
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>6
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>5
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>7
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>6
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>5
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>6
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>7
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>6
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]=>5
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]=>6
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]=>6
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]=>6
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]=>6
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]=>6
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]=>5
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]=>6
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]=>6
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]=>6
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>5
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>5
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]=>5
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]=>5
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>5
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]=>5
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>6
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>5
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>5
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>5
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>5
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>5
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>7
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>6
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>5
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>6
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]=>5
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>6
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>6
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]=>5
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]=>5
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>7
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>6
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>5
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>6
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]=>5
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>5
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>6
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>6
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]=>5
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]=>5
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]=>6
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]=>6
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]=>5
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]=>5
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]=>5
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]=>6
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]=>5
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]=>6
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]=>5
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]=>6
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]=>6
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]=>5
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]=>5
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]=>5
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]=>5
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]=>5
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]=>5
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>5
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]=>5
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]=>5
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]=>6
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]=>5
[1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]=>5
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]=>6
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]=>5
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>5
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]=>5
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]=>6
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]=>5
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>5
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>5
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>5
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]=>5
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>5
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]=>5
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]=>5
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]=>5
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>5
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]=>5
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]=>5
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]=>5
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]=>5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>7
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>5
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]=>6
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>5
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]=>6
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]=>6
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]=>5
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]=>5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]=>6
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]=>5
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]=>6
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]=>5
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]=>5
[1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]=>5
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]=>5
[1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]=>5
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]=>5
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]=>6
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]=>5
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]=>5
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]=>6
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]=>5
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]=>5
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]=>5
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]=>5
[1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]=>5
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]=>5
[1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]=>7
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]=>6
[1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]=>5
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]=>6
[1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]=>5
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]=>6
[1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]=>6
[1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]=>5
[1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]=>5
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]=>6
[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]=>5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]=>6
[1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]=>5
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]=>5
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]=>5
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]=>5
[1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]=>5
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]=>7
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]=>6
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]=>5
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]=>6
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]=>5
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]=>6
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]=>6
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]=>5
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]=>5
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]=>6
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]=>5
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]=>6
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]=>5
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]=>6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]=>5
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]=>5
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]=>5
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]=>6
[1,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0]=>5
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]=>6
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]=>5
[1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]=>5
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]=>5
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]=>5
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]=>5
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]=>5
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]=>5
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]=>5
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]=>5
[1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]=>6
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]=>5
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]=>5
[1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]=>6
[1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]=>5
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]=>5
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]=>5
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]=>5
[1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]=>6
[1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]=>5
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]=>5
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]=>6
[1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]=>5
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]=>5
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]=>5
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]=>5
[1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]=>5
[1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]=>5
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]=>5
[1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]=>5
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]=>5
[1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]=>5
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>6
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>5
[1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]=>5
[1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]=>5
[1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]=>5
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]=>5
[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]=>6
[1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]=>5
[1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]=>5
[1,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]=>5
[1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]=>6
[1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]=>5
[1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]=>5
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]=>5
[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]=>5
[1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]=>5
[1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]=>6
[1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]=>5
[1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]=>5
[1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]=>5
[1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]=>6
[1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]=>5
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]=>5
[1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]=>5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]=>5
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]=>6
[1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]=>5
[1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]=>5
[1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]=>5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>5
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>5
[1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]=>5
[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>5
[1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]=>5
[1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]=>5
[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]=>5
[1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]=>5
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]=>5
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>5
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]=>5
[1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]=>5
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]=>5
[1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]=>5
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]=>5
[1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]=>5
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]=>5
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]=>5
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Description
The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
- For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
- For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
References
[1] Mare Global dimension and Dyck path combinatorics MathOverflow:259113
Code
# programm inspired by a programm from ahulpke in http://math.stackexchange.com/questions/1587593/finding-elements-of-cartesian-product-satisfying-a-set-of-constraints-in-gap. Programm builds all sequences corresponding to n-LNakayama algebras. All writen for the GAP package QPA.
BuildSequencesLNak:=function(n)
local all,range,len,new,seq,i,sel;
all:=[[1]];
range:=[2..n];
for len in [1..n-1] do
new:=[];
for seq in all do
sel:=Filtered(range, x->x<=1+seq[1]);
for i in sel do
Add(new,Concatenation([i],seq));
od;
od;
all:=new;
od;
return all;
end;
BuildSequencesLNak:=function(n)
local all,range,len,new,seq,i,sel;
all:=[[1]]; #last number is always 1
range:=[2..n]; except for the last entry the minimum is 2 and the maximum n (first entry)
for len in [1..n-1] do # this ken could also be named j. it only is needed in this code for the right number of repetitions of the loop.
new:=[];
for seq in all do
sel:=Filtered(range, x->x<=1+seq[1]);
for i in sel do
Add(new,Concatenation([i],seq));
od;
od;
all:=new;
od;
return all;
end;
#Next programm calculate the global dimension.
DeclareOperation("gldim", [IsList]);
InstallMethod(gldim, "for a representation of a quiver", [IsList],0,function(L)
local list, n, i, j, f, temp, temp2, temp3, u;
list:=L;
n:=Size(L);
f := function (x,y)
local c, z;
c:=(x+y) mod n; if c=0 then c:=n; fi;
z:=(x+1) mod n; if z=0 then z:=n; fi;
return([c,list[z]-y]);
end;
temp2:=[];
for i in [0..n-1] do
Append(temp2,[[i,1]]);
od;
temp:=[];
for i in [0..n-1] do
u:=temp2[i+1];
Append(temp,[[u]]);
od;
for i in [0..n-1] do
j:=1;
while j<(2*n+3) do
Append(temp[i+1],[f(temp[i+1][j][1],temp[i+1][j][2])]);
j:=j+1;
od;
od;
temp3:=[];
for i in [1..n] do
temp2:=[];
for j in [1..(2*n+3)] do
if temp[i][j][2]=0 then Append(temp2,[j]); fi;
od;
if Size(temp2)>0 then
u:=Minimum(temp2);
Append(temp3,[u]);
else
temp3:="inf"; break;
fi;
od;
if IsString(temp3)=false then
temp3:=(Maximum(temp3))-2;
fi;
return(temp3);
end
);
#Example:
gap> L:=[3,4,3,3,2,1];;gldim(L);
3
Created
Jan 10, 2017 at 12:41 by Rene Marczinzik
Updated
Jan 12, 2017 at 09:16 by Christian Stump
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