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
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>4
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>4
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>4
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>4
[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
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>4
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]=>4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]=>4
[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
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>4
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]=>4
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>5
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>5
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>4
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>4
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>4
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]=>3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]=>3
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]=>3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>3
[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
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]=>4
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]=>4
[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
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]=>4
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]=>4
[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
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]=>4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>4
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>3
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]=>3
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]=>4
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]=>3
[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 $n$-linear Nakayama algebra is a quiver algebra whose quiver is a directed line with $n$ vertices for $n\geq 2$, numbered from left to right by $0,1,\ldots,n-1$. It is uniquely determined by the dimensions $c_i$ of the indecomposable projective modules at vertex $i$, i.e., by its Kupisch series $[c_0,c_1,\ldots,c_{n-1}]$, which is subject to the conditions $c_{n-1}=1$ and $c_i-1\le c_{i+1}$ for all $i$. The number of such algebras is the $(n-1)$-st Catalan number $C_{n-1}$.
One can interpret these algebras via Dyck paths by associating to the algebra the top boundary of its Auslander–Reiten quiver, which is a Dyck path. For example, $[3,4,3,3,2,1]$ corresponds to the Dyck path $[1,1,0,1,1,0,0,1,0,0]$. The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
Conjecture: that there is an explicit bijection between $n$-linear Nakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
An $n$-linear Nakayama algebra is a quiver algebra whose quiver is a directed line with $n$ vertices for $n\geq 2$, numbered from left to right by $0,1,\ldots,n-1$. It is uniquely determined by the dimensions $c_i$ of the indecomposable projective modules at vertex $i$, i.e., by its Kupisch series $[c_0,c_1,\ldots,c_{n-1}]$, which is subject to the conditions $c_{n-1}=1$ and $c_i-1\le c_{i+1}$ for all $i$. The number of such algebras is the $(n-1)$-st Catalan number $C_{n-1}$.
One can interpret these algebras via Dyck paths by associating to the algebra the top boundary of its Auslander–Reiten quiver, which is a Dyck path. For example, $[3,4,3,3,2,1]$ corresponds to the Dyck path $[1,1,0,1,1,0,0,1,0,0]$. The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
Conjecture: that there is an explicit bijection between $n$-linear Nakayama 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
def kupisch(D):
H = D.heights()
return [1 + H[i] for i, s in enumerate(D) if s == 0] + [1]
def gldim_from_kupisch(L):
"""Global dimension of LNakayama algebra from Kupisch series."""
n = len(L)
def f(x, y):
c = (x + y) % n
if c == 0: c = n
z = (x + 1) % n
if z == 0: z = n
return (c, L[z - 1] - y)
temp = [[(i, 1)] for i in range(n)]
for i in range(n):
for _ in range(2 * n + 2):
temp[i].append(f(temp[i][-1][0], temp[i][-1][1]))
temp3 = []
for i in range(n):
zeros = [j + 1 for j in range(len(temp[i])) if temp[i][j][1] == 0]
if zeros:
temp3.append(min(zeros))
else:
return "inf"
return max(temp3) - 2
def statistic(D):
return gldim_from_kupisch(kupisch(D.reverse()))
Created
Jan 10, 2017 at 12:41 by Rene Marczinzik
Updated
Mar 12, 2026 at 18:07 by Nupur Jain
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