Identifier
- St000491: Set partitions ⟶ ℤ
Values
=>
Cc0009;cc-rep
{{1,2}}=>0
{{1},{2}}=>0
{{1,2,3}}=>0
{{1,2},{3}}=>0
{{1,3},{2}}=>1
{{1},{2,3}}=>0
{{1},{2},{3}}=>0
{{1,2,3,4}}=>0
{{1,2,3},{4}}=>0
{{1,2,4},{3}}=>1
{{1,2},{3,4}}=>0
{{1,2},{3},{4}}=>0
{{1,3,4},{2}}=>2
{{1,3},{2,4}}=>1
{{1,3},{2},{4}}=>1
{{1,4},{2,3}}=>1
{{1},{2,3,4}}=>0
{{1},{2,3},{4}}=>0
{{1,4},{2},{3}}=>2
{{1},{2,4},{3}}=>1
{{1},{2},{3,4}}=>0
{{1},{2},{3},{4}}=>0
{{1,2,3,4,5}}=>0
{{1,2,3,4},{5}}=>0
{{1,2,3,5},{4}}=>1
{{1,2,3},{4,5}}=>0
{{1,2,3},{4},{5}}=>0
{{1,2,4,5},{3}}=>2
{{1,2,4},{3,5}}=>1
{{1,2,4},{3},{5}}=>1
{{1,2,5},{3,4}}=>1
{{1,2},{3,4,5}}=>0
{{1,2},{3,4},{5}}=>0
{{1,2,5},{3},{4}}=>2
{{1,2},{3,5},{4}}=>1
{{1,2},{3},{4,5}}=>0
{{1,2},{3},{4},{5}}=>0
{{1,3,4,5},{2}}=>3
{{1,3,4},{2,5}}=>2
{{1,3,4},{2},{5}}=>2
{{1,3,5},{2,4}}=>2
{{1,3},{2,4,5}}=>1
{{1,3},{2,4},{5}}=>1
{{1,3,5},{2},{4}}=>3
{{1,3},{2,5},{4}}=>2
{{1,3},{2},{4,5}}=>1
{{1,3},{2},{4},{5}}=>1
{{1,4,5},{2,3}}=>2
{{1,4},{2,3,5}}=>1
{{1,4},{2,3},{5}}=>1
{{1,5},{2,3,4}}=>1
{{1},{2,3,4,5}}=>0
{{1},{2,3,4},{5}}=>0
{{1,5},{2,3},{4}}=>2
{{1},{2,3,5},{4}}=>1
{{1},{2,3},{4,5}}=>0
{{1},{2,3},{4},{5}}=>0
{{1,4,5},{2},{3}}=>4
{{1,4},{2,5},{3}}=>3
{{1,4},{2},{3,5}}=>2
{{1,4},{2},{3},{5}}=>2
{{1,5},{2,4},{3}}=>3
{{1},{2,4,5},{3}}=>2
{{1},{2,4},{3,5}}=>1
{{1},{2,4},{3},{5}}=>1
{{1,5},{2},{3,4}}=>2
{{1},{2,5},{3,4}}=>1
{{1},{2},{3,4,5}}=>0
{{1},{2},{3,4},{5}}=>0
{{1,5},{2},{3},{4}}=>3
{{1},{2,5},{3},{4}}=>2
{{1},{2},{3,5},{4}}=>1
{{1},{2},{3},{4,5}}=>0
{{1},{2},{3},{4},{5}}=>0
{{1,2,3,4,5,6}}=>0
{{1,2,3,4,5},{6}}=>0
{{1,2,3,4,6},{5}}=>1
{{1,2,3,4},{5,6}}=>0
{{1,2,3,4},{5},{6}}=>0
{{1,2,3,5,6},{4}}=>2
{{1,2,3,5},{4,6}}=>1
{{1,2,3,5},{4},{6}}=>1
{{1,2,3,6},{4,5}}=>1
{{1,2,3},{4,5,6}}=>0
{{1,2,3},{4,5},{6}}=>0
{{1,2,3,6},{4},{5}}=>2
{{1,2,3},{4,6},{5}}=>1
{{1,2,3},{4},{5,6}}=>0
{{1,2,3},{4},{5},{6}}=>0
{{1,2,4,5,6},{3}}=>3
{{1,2,4,5},{3,6}}=>2
{{1,2,4,5},{3},{6}}=>2
{{1,2,4,6},{3,5}}=>2
{{1,2,4},{3,5,6}}=>1
{{1,2,4},{3,5},{6}}=>1
{{1,2,4,6},{3},{5}}=>3
{{1,2,4},{3,6},{5}}=>2
{{1,2,4},{3},{5,6}}=>1
{{1,2,4},{3},{5},{6}}=>1
{{1,2,5,6},{3,4}}=>2
{{1,2,5},{3,4,6}}=>1
{{1,2,5},{3,4},{6}}=>1
{{1,2,6},{3,4,5}}=>1
{{1,2},{3,4,5,6}}=>0
{{1,2},{3,4,5},{6}}=>0
{{1,2,6},{3,4},{5}}=>2
{{1,2},{3,4,6},{5}}=>1
{{1,2},{3,4},{5,6}}=>0
{{1,2},{3,4},{5},{6}}=>0
{{1,2,5,6},{3},{4}}=>4
{{1,2,5},{3,6},{4}}=>3
{{1,2,5},{3},{4,6}}=>2
{{1,2,5},{3},{4},{6}}=>2
{{1,2,6},{3,5},{4}}=>3
{{1,2},{3,5,6},{4}}=>2
{{1,2},{3,5},{4,6}}=>1
{{1,2},{3,5},{4},{6}}=>1
{{1,2,6},{3},{4,5}}=>2
{{1,2},{3,6},{4,5}}=>1
{{1,2},{3},{4,5,6}}=>0
{{1,2},{3},{4,5},{6}}=>0
{{1,2,6},{3},{4},{5}}=>3
{{1,2},{3,6},{4},{5}}=>2
{{1,2},{3},{4,6},{5}}=>1
{{1,2},{3},{4},{5,6}}=>0
{{1,2},{3},{4},{5},{6}}=>0
{{1,3,4,5,6},{2}}=>4
{{1,3,4,5},{2,6}}=>3
{{1,3,4,5},{2},{6}}=>3
{{1,3,4,6},{2,5}}=>3
{{1,3,4},{2,5,6}}=>2
{{1,3,4},{2,5},{6}}=>2
{{1,3,4,6},{2},{5}}=>4
{{1,3,4},{2,6},{5}}=>3
{{1,3,4},{2},{5,6}}=>2
{{1,3,4},{2},{5},{6}}=>2
{{1,3,5,6},{2,4}}=>3
{{1,3,5},{2,4,6}}=>2
{{1,3,5},{2,4},{6}}=>2
{{1,3,6},{2,4,5}}=>2
{{1,3},{2,4,5,6}}=>1
{{1,3},{2,4,5},{6}}=>1
{{1,3,6},{2,4},{5}}=>3
{{1,3},{2,4,6},{5}}=>2
{{1,3},{2,4},{5,6}}=>1
{{1,3},{2,4},{5},{6}}=>1
{{1,3,5,6},{2},{4}}=>5
{{1,3,5},{2,6},{4}}=>4
{{1,3,5},{2},{4,6}}=>3
{{1,3,5},{2},{4},{6}}=>3
{{1,3,6},{2,5},{4}}=>4
{{1,3},{2,5,6},{4}}=>3
{{1,3},{2,5},{4,6}}=>2
{{1,3},{2,5},{4},{6}}=>2
{{1,3,6},{2},{4,5}}=>3
{{1,3},{2,6},{4,5}}=>2
{{1,3},{2},{4,5,6}}=>1
{{1,3},{2},{4,5},{6}}=>1
{{1,3,6},{2},{4},{5}}=>4
{{1,3},{2,6},{4},{5}}=>3
{{1,3},{2},{4,6},{5}}=>2
{{1,3},{2},{4},{5,6}}=>1
{{1,3},{2},{4},{5},{6}}=>1
{{1,4,5,6},{2,3}}=>3
{{1,4,5},{2,3,6}}=>2
{{1,4,5},{2,3},{6}}=>2
{{1,4,6},{2,3,5}}=>2
{{1,4},{2,3,5,6}}=>1
{{1,4},{2,3,5},{6}}=>1
{{1,4,6},{2,3},{5}}=>3
{{1,4},{2,3,6},{5}}=>2
{{1,4},{2,3},{5,6}}=>1
{{1,4},{2,3},{5},{6}}=>1
{{1,5,6},{2,3,4}}=>2
{{1,5},{2,3,4,6}}=>1
{{1,5},{2,3,4},{6}}=>1
{{1,6},{2,3,4,5}}=>1
{{1},{2,3,4,5,6}}=>0
{{1},{2,3,4,5},{6}}=>0
{{1,6},{2,3,4},{5}}=>2
{{1},{2,3,4,6},{5}}=>1
{{1},{2,3,4},{5,6}}=>0
{{1},{2,3,4},{5},{6}}=>0
{{1,5,6},{2,3},{4}}=>4
{{1,5},{2,3,6},{4}}=>3
{{1,5},{2,3},{4,6}}=>2
{{1,5},{2,3},{4},{6}}=>2
{{1,6},{2,3,5},{4}}=>3
{{1},{2,3,5,6},{4}}=>2
{{1},{2,3,5},{4,6}}=>1
{{1},{2,3,5},{4},{6}}=>1
{{1,6},{2,3},{4,5}}=>2
{{1},{2,3,6},{4,5}}=>1
{{1},{2,3},{4,5,6}}=>0
{{1},{2,3},{4,5},{6}}=>0
{{1,6},{2,3},{4},{5}}=>3
{{1},{2,3,6},{4},{5}}=>2
{{1},{2,3},{4,6},{5}}=>1
{{1},{2,3},{4},{5,6}}=>0
{{1},{2,3},{4},{5},{6}}=>0
{{1,4,5,6},{2},{3}}=>6
{{1,4,5},{2,6},{3}}=>5
{{1,4,5},{2},{3,6}}=>4
{{1,4,5},{2},{3},{6}}=>4
{{1,4,6},{2,5},{3}}=>5
{{1,4},{2,5,6},{3}}=>4
{{1,4},{2,5},{3,6}}=>3
{{1,4},{2,5},{3},{6}}=>3
{{1,4,6},{2},{3,5}}=>4
{{1,4},{2,6},{3,5}}=>3
{{1,4},{2},{3,5,6}}=>2
{{1,4},{2},{3,5},{6}}=>2
{{1,4,6},{2},{3},{5}}=>5
{{1,4},{2,6},{3},{5}}=>4
{{1,4},{2},{3,6},{5}}=>3
{{1,4},{2},{3},{5,6}}=>2
{{1,4},{2},{3},{5},{6}}=>2
{{1,5,6},{2,4},{3}}=>5
{{1,5},{2,4,6},{3}}=>4
{{1,5},{2,4},{3,6}}=>3
{{1,5},{2,4},{3},{6}}=>3
{{1,6},{2,4,5},{3}}=>4
{{1},{2,4,5,6},{3}}=>3
{{1},{2,4,5},{3,6}}=>2
{{1},{2,4,5},{3},{6}}=>2
{{1,6},{2,4},{3,5}}=>3
{{1},{2,4,6},{3,5}}=>2
{{1},{2,4},{3,5,6}}=>1
{{1},{2,4},{3,5},{6}}=>1
{{1,6},{2,4},{3},{5}}=>4
{{1},{2,4,6},{3},{5}}=>3
{{1},{2,4},{3,6},{5}}=>2
{{1},{2,4},{3},{5,6}}=>1
{{1},{2,4},{3},{5},{6}}=>1
{{1,5,6},{2},{3,4}}=>4
{{1,5},{2,6},{3,4}}=>3
{{1,5},{2},{3,4,6}}=>2
{{1,5},{2},{3,4},{6}}=>2
{{1,6},{2,5},{3,4}}=>3
{{1},{2,5,6},{3,4}}=>2
{{1},{2,5},{3,4,6}}=>1
{{1},{2,5},{3,4},{6}}=>1
{{1,6},{2},{3,4,5}}=>2
{{1},{2,6},{3,4,5}}=>1
{{1},{2},{3,4,5,6}}=>0
{{1},{2},{3,4,5},{6}}=>0
{{1,6},{2},{3,4},{5}}=>3
{{1},{2,6},{3,4},{5}}=>2
{{1},{2},{3,4,6},{5}}=>1
{{1},{2},{3,4},{5,6}}=>0
{{1},{2},{3,4},{5},{6}}=>0
{{1,5,6},{2},{3},{4}}=>6
{{1,5},{2,6},{3},{4}}=>5
{{1,5},{2},{3,6},{4}}=>4
{{1,5},{2},{3},{4,6}}=>3
{{1,5},{2},{3},{4},{6}}=>3
{{1,6},{2,5},{3},{4}}=>5
{{1},{2,5,6},{3},{4}}=>4
{{1},{2,5},{3,6},{4}}=>3
{{1},{2,5},{3},{4,6}}=>2
{{1},{2,5},{3},{4},{6}}=>2
{{1,6},{2},{3,5},{4}}=>4
{{1},{2,6},{3,5},{4}}=>3
{{1},{2},{3,5,6},{4}}=>2
{{1},{2},{3,5},{4,6}}=>1
{{1},{2},{3,5},{4},{6}}=>1
{{1,6},{2},{3},{4,5}}=>3
{{1},{2,6},{3},{4,5}}=>2
{{1},{2},{3,6},{4,5}}=>1
{{1},{2},{3},{4,5,6}}=>0
{{1},{2},{3},{4,5},{6}}=>0
{{1,6},{2},{3},{4},{5}}=>4
{{1},{2,6},{3},{4},{5}}=>3
{{1},{2},{3,6},{4},{5}}=>2
{{1},{2},{3},{4,6},{5}}=>1
{{1},{2},{3},{4},{5,6}}=>0
{{1},{2},{3},{4},{5},{6}}=>0
{{1,2,3,4,5,6,7}}=>0
{{1,2,3,4,5,6},{7}}=>0
{{1,2,3,4,5,7},{6}}=>1
{{1,2,3,4,5},{6,7}}=>0
{{1,2,3,4,5},{6},{7}}=>0
{{1,2,3,4,6,7},{5}}=>2
{{1,2,3,4,6},{5,7}}=>1
{{1,2,3,4,6},{5},{7}}=>1
{{1,2,3,4,7},{5,6}}=>1
{{1,2,3,4},{5,6,7}}=>0
{{1,2,3,4},{5,6},{7}}=>0
{{1,2,3,4,7},{5},{6}}=>2
{{1,2,3,4},{5,7},{6}}=>1
{{1,2,3,4},{5},{6,7}}=>0
{{1,2,3,4},{5},{6},{7}}=>0
{{1,2,3,5,6,7},{4}}=>3
{{1,2,3,5,6},{4,7}}=>2
{{1,2,3,5,6},{4},{7}}=>2
{{1,2,3,5,7},{4,6}}=>2
{{1,2,3,5},{4,6,7}}=>1
{{1,2,3,5},{4,6},{7}}=>1
{{1,2,3,5,7},{4},{6}}=>3
{{1,2,3,5},{4,7},{6}}=>2
{{1,2,3,5},{4},{6,7}}=>1
{{1,2,3,5},{4},{6},{7}}=>1
{{1,2,3,6,7},{4,5}}=>2
{{1,2,3,6},{4,5,7}}=>1
{{1,2,3,6},{4,5},{7}}=>1
{{1,2,3,7},{4,5,6}}=>1
{{1,2,3},{4,5,6,7}}=>0
{{1,2,3},{4,5,6},{7}}=>0
{{1,2,3,7},{4,5},{6}}=>2
{{1,2,3},{4,5,7},{6}}=>1
{{1,2,3},{4,5},{6,7}}=>0
{{1,2,3},{4,5},{6},{7}}=>0
{{1,2,3,6,7},{4},{5}}=>4
{{1,2,3,6},{4,7},{5}}=>3
{{1,2,3,6},{4},{5,7}}=>2
{{1,2,3,6},{4},{5},{7}}=>2
{{1,2,3,7},{4,6},{5}}=>3
{{1,2,3},{4,6,7},{5}}=>2
{{1,2,3},{4,6},{5,7}}=>1
{{1,2,3},{4,6},{5},{7}}=>1
{{1,2,3,7},{4},{5,6}}=>2
{{1,2,3},{4,7},{5,6}}=>1
{{1,2,3},{4},{5,6,7}}=>0
{{1,2,3},{4},{5,6},{7}}=>0
{{1,2,3,7},{4},{5},{6}}=>3
{{1,2,3},{4,7},{5},{6}}=>2
{{1,2,3},{4},{5,7},{6}}=>1
{{1,2,3},{4},{5},{6,7}}=>0
{{1,2,3},{4},{5},{6},{7}}=>0
{{1,2,4,5,6,7},{3}}=>4
{{1,2,4,5,6},{3,7}}=>3
{{1,2,4,5,6},{3},{7}}=>3
{{1,2,4,5,7},{3,6}}=>3
{{1,2,4,5},{3,6,7}}=>2
{{1,2,4,5},{3,6},{7}}=>2
{{1,2,4,5,7},{3},{6}}=>4
{{1,2,4,5},{3,7},{6}}=>3
{{1,2,4,5},{3},{6,7}}=>2
{{1,2,4,5},{3},{6},{7}}=>2
{{1,2,4,6,7},{3,5}}=>3
{{1,2,4,6},{3,5,7}}=>2
{{1,2,4,6},{3,5},{7}}=>2
{{1,2,4,7},{3,5,6}}=>2
{{1,2,4},{3,5,6,7}}=>1
{{1,2,4},{3,5,6},{7}}=>1
{{1,2,4,7},{3,5},{6}}=>3
{{1,2,4},{3,5,7},{6}}=>2
{{1,2,4},{3,5},{6,7}}=>1
{{1,2,4},{3,5},{6},{7}}=>1
{{1,2,4,6,7},{3},{5}}=>5
{{1,2,4,6},{3,7},{5}}=>4
{{1,2,4,6},{3},{5,7}}=>3
{{1,2,4,6},{3},{5},{7}}=>3
{{1,2,4,7},{3,6},{5}}=>4
{{1,2,4},{3,6,7},{5}}=>3
{{1,2,4},{3,6},{5,7}}=>2
{{1,2,4},{3,6},{5},{7}}=>2
{{1,2,4,7},{3},{5,6}}=>3
{{1,2,4},{3,7},{5,6}}=>2
{{1,2,4},{3},{5,6,7}}=>1
{{1,2,4},{3},{5,6},{7}}=>1
{{1,2,4,7},{3},{5},{6}}=>4
{{1,2,4},{3,7},{5},{6}}=>3
{{1,2,4},{3},{5,7},{6}}=>2
{{1,2,4},{3},{5},{6,7}}=>1
{{1,2,4},{3},{5},{6},{7}}=>1
{{1,2,5,6,7},{3,4}}=>3
{{1,2,5,6},{3,4,7}}=>2
{{1,2,5,6},{3,4},{7}}=>2
{{1,2,5,7},{3,4,6}}=>2
{{1,2,5},{3,4,6,7}}=>1
{{1,2,5},{3,4,6},{7}}=>1
{{1,2,5,7},{3,4},{6}}=>3
{{1,2,5},{3,4,7},{6}}=>2
{{1,2,5},{3,4},{6,7}}=>1
{{1,2,5},{3,4},{6},{7}}=>1
{{1,2,6,7},{3,4,5}}=>2
{{1,2,6},{3,4,5,7}}=>1
{{1,2,6},{3,4,5},{7}}=>1
{{1,2,7},{3,4,5,6}}=>1
{{1,2},{3,4,5,6,7}}=>0
{{1,2},{3,4,5,6},{7}}=>0
{{1,2,7},{3,4,5},{6}}=>2
{{1,2},{3,4,5,7},{6}}=>1
{{1,2},{3,4,5},{6,7}}=>0
{{1,2},{3,4,5},{6},{7}}=>0
{{1,2,6,7},{3,4},{5}}=>4
{{1,2,6},{3,4,7},{5}}=>3
{{1,2,6},{3,4},{5,7}}=>2
{{1,2,6},{3,4},{5},{7}}=>2
{{1,2,7},{3,4,6},{5}}=>3
{{1,2},{3,4,6,7},{5}}=>2
{{1,2},{3,4,6},{5,7}}=>1
{{1,2},{3,4,6},{5},{7}}=>1
{{1,2,7},{3,4},{5,6}}=>2
{{1,2},{3,4,7},{5,6}}=>1
{{1,2},{3,4},{5,6,7}}=>0
{{1,2},{3,4},{5,6},{7}}=>0
{{1,2,7},{3,4},{5},{6}}=>3
{{1,2},{3,4,7},{5},{6}}=>2
{{1,2},{3,4},{5,7},{6}}=>1
{{1,2},{3,4},{5},{6,7}}=>0
{{1,2},{3,4},{5},{6},{7}}=>0
{{1,2,5,6,7},{3},{4}}=>6
{{1,2,5,6},{3,7},{4}}=>5
{{1,2,5,6},{3},{4,7}}=>4
{{1,2,5,6},{3},{4},{7}}=>4
{{1,2,5,7},{3,6},{4}}=>5
{{1,2,5},{3,6,7},{4}}=>4
{{1,2,5},{3,6},{4,7}}=>3
{{1,2,5},{3,6},{4},{7}}=>3
{{1,2,5,7},{3},{4,6}}=>4
{{1,2,5},{3,7},{4,6}}=>3
{{1,2,5},{3},{4,6,7}}=>2
{{1,2,5},{3},{4,6},{7}}=>2
{{1,2,5,7},{3},{4},{6}}=>5
{{1,2,5},{3,7},{4},{6}}=>4
{{1,2,5},{3},{4,7},{6}}=>3
{{1,2,5},{3},{4},{6,7}}=>2
{{1,2,5},{3},{4},{6},{7}}=>2
{{1,2,6,7},{3,5},{4}}=>5
{{1,2,6},{3,5,7},{4}}=>4
{{1,2,6},{3,5},{4,7}}=>3
{{1,2,6},{3,5},{4},{7}}=>3
{{1,2,7},{3,5,6},{4}}=>4
{{1,2},{3,5,6,7},{4}}=>3
{{1,2},{3,5,6},{4,7}}=>2
{{1,2},{3,5,6},{4},{7}}=>2
{{1,2,7},{3,5},{4,6}}=>3
{{1,2},{3,5,7},{4,6}}=>2
{{1,2},{3,5},{4,6,7}}=>1
{{1,2},{3,5},{4,6},{7}}=>1
{{1,2,7},{3,5},{4},{6}}=>4
{{1,2},{3,5,7},{4},{6}}=>3
{{1,2},{3,5},{4,7},{6}}=>2
{{1,2},{3,5},{4},{6,7}}=>1
{{1,2},{3,5},{4},{6},{7}}=>1
{{1,2,6,7},{3},{4,5}}=>4
{{1,2,6},{3,7},{4,5}}=>3
{{1,2,6},{3},{4,5,7}}=>2
{{1,2,6},{3},{4,5},{7}}=>2
{{1,2,7},{3,6},{4,5}}=>3
{{1,2},{3,6,7},{4,5}}=>2
{{1,2},{3,6},{4,5,7}}=>1
{{1,2},{3,6},{4,5},{7}}=>1
{{1,2,7},{3},{4,5,6}}=>2
{{1,2},{3,7},{4,5,6}}=>1
{{1,2},{3},{4,5,6,7}}=>0
{{1,2},{3},{4,5,6},{7}}=>0
{{1,2,7},{3},{4,5},{6}}=>3
{{1,2},{3,7},{4,5},{6}}=>2
{{1,2},{3},{4,5,7},{6}}=>1
{{1,2},{3},{4,5},{6,7}}=>0
{{1,2},{3},{4,5},{6},{7}}=>0
{{1,2,6,7},{3},{4},{5}}=>6
{{1,2,6},{3,7},{4},{5}}=>5
{{1,2,6},{3},{4,7},{5}}=>4
{{1,2,6},{3},{4},{5,7}}=>3
{{1,2,6},{3},{4},{5},{7}}=>3
{{1,2,7},{3,6},{4},{5}}=>5
{{1,2},{3,6,7},{4},{5}}=>4
{{1,2},{3,6},{4,7},{5}}=>3
{{1,2},{3,6},{4},{5,7}}=>2
{{1,2},{3,6},{4},{5},{7}}=>2
{{1,2,7},{3},{4,6},{5}}=>4
{{1,2},{3,7},{4,6},{5}}=>3
{{1,2},{3},{4,6,7},{5}}=>2
{{1,2},{3},{4,6},{5,7}}=>1
{{1,2},{3},{4,6},{5},{7}}=>1
{{1,2,7},{3},{4},{5,6}}=>3
{{1,2},{3,7},{4},{5,6}}=>2
{{1,2},{3},{4,7},{5,6}}=>1
{{1,2},{3},{4},{5,6,7}}=>0
{{1,2},{3},{4},{5,6},{7}}=>0
{{1,2,7},{3},{4},{5},{6}}=>4
{{1,2},{3,7},{4},{5},{6}}=>3
{{1,2},{3},{4,7},{5},{6}}=>2
{{1,2},{3},{4},{5,7},{6}}=>1
{{1,2},{3},{4},{5},{6,7}}=>0
{{1,2},{3},{4},{5},{6},{7}}=>0
{{1,3,4,5,6,7},{2}}=>5
{{1,3,4,5,6},{2,7}}=>4
{{1,3,4,5,6},{2},{7}}=>4
{{1,3,4,5,7},{2,6}}=>4
{{1,3,4,5},{2,6,7}}=>3
{{1,3,4,5},{2,6},{7}}=>3
{{1,3,4,5,7},{2},{6}}=>5
{{1,3,4,5},{2,7},{6}}=>4
{{1,3,4,5},{2},{6,7}}=>3
{{1,3,4,5},{2},{6},{7}}=>3
{{1,3,4,6,7},{2,5}}=>4
{{1,3,4,6},{2,5,7}}=>3
{{1,3,4,6},{2,5},{7}}=>3
{{1,3,4,7},{2,5,6}}=>3
{{1,3,4},{2,5,6,7}}=>2
{{1,3,4},{2,5,6},{7}}=>2
{{1,3,4,7},{2,5},{6}}=>4
{{1,3,4},{2,5,7},{6}}=>3
{{1,3,4},{2,5},{6,7}}=>2
{{1,3,4},{2,5},{6},{7}}=>2
{{1,3,4,6,7},{2},{5}}=>6
{{1,3,4,6},{2,7},{5}}=>5
{{1,3,4,6},{2},{5,7}}=>4
{{1,3,4,6},{2},{5},{7}}=>4
{{1,3,4,7},{2,6},{5}}=>5
{{1,3,4},{2,6,7},{5}}=>4
{{1,3,4},{2,6},{5,7}}=>3
{{1,3,4},{2,6},{5},{7}}=>3
{{1,3,4,7},{2},{5,6}}=>4
{{1,3,4},{2,7},{5,6}}=>3
{{1,3,4},{2},{5,6,7}}=>2
{{1,3,4},{2},{5,6},{7}}=>2
{{1,3,4,7},{2},{5},{6}}=>5
{{1,3,4},{2,7},{5},{6}}=>4
{{1,3,4},{2},{5,7},{6}}=>3
{{1,3,4},{2},{5},{6,7}}=>2
{{1,3,4},{2},{5},{6},{7}}=>2
{{1,3,5,6,7},{2,4}}=>4
{{1,3,5,6},{2,4,7}}=>3
{{1,3,5,6},{2,4},{7}}=>3
{{1,3,5,7},{2,4,6}}=>3
{{1,3,5},{2,4,6,7}}=>2
{{1,3,5},{2,4,6},{7}}=>2
{{1,3,5,7},{2,4},{6}}=>4
{{1,3,5},{2,4,7},{6}}=>3
{{1,3,5},{2,4},{6,7}}=>2
{{1,3,5},{2,4},{6},{7}}=>2
{{1,3,6,7},{2,4,5}}=>3
{{1,3,6},{2,4,5,7}}=>2
{{1,3,6},{2,4,5},{7}}=>2
{{1,3,7},{2,4,5,6}}=>2
{{1,3},{2,4,5,6,7}}=>1
{{1,3},{2,4,5,6},{7}}=>1
{{1,3,7},{2,4,5},{6}}=>3
{{1,3},{2,4,5,7},{6}}=>2
{{1,3},{2,4,5},{6,7}}=>1
{{1,3},{2,4,5},{6},{7}}=>1
{{1,3,6,7},{2,4},{5}}=>5
{{1,3,6},{2,4,7},{5}}=>4
{{1,3,6},{2,4},{5,7}}=>3
{{1,3,6},{2,4},{5},{7}}=>3
{{1,3,7},{2,4,6},{5}}=>4
{{1,3},{2,4,6,7},{5}}=>3
{{1,3},{2,4,6},{5,7}}=>2
{{1,3},{2,4,6},{5},{7}}=>2
{{1,3,7},{2,4},{5,6}}=>3
{{1,3},{2,4,7},{5,6}}=>2
{{1,3},{2,4},{5,6,7}}=>1
{{1,3},{2,4},{5,6},{7}}=>1
{{1,3,7},{2,4},{5},{6}}=>4
{{1,3},{2,4,7},{5},{6}}=>3
{{1,3},{2,4},{5,7},{6}}=>2
{{1,3},{2,4},{5},{6,7}}=>1
{{1,3},{2,4},{5},{6},{7}}=>1
{{1,3,5,6,7},{2},{4}}=>7
{{1,3,5,6},{2,7},{4}}=>6
{{1,3,5,6},{2},{4,7}}=>5
{{1,3,5,6},{2},{4},{7}}=>5
{{1,3,5,7},{2,6},{4}}=>6
{{1,3,5},{2,6,7},{4}}=>5
{{1,3,5},{2,6},{4,7}}=>4
{{1,3,5},{2,6},{4},{7}}=>4
{{1,3,5,7},{2},{4,6}}=>5
{{1,3,5},{2,7},{4,6}}=>4
{{1,3,5},{2},{4,6,7}}=>3
{{1,3,5},{2},{4,6},{7}}=>3
{{1,3,5,7},{2},{4},{6}}=>6
{{1,3,5},{2,7},{4},{6}}=>5
{{1,3,5},{2},{4,7},{6}}=>4
{{1,3,5},{2},{4},{6,7}}=>3
{{1,3,5},{2},{4},{6},{7}}=>3
{{1,3,6,7},{2,5},{4}}=>6
{{1,3,6},{2,5,7},{4}}=>5
{{1,3,6},{2,5},{4,7}}=>4
{{1,3,6},{2,5},{4},{7}}=>4
{{1,3,7},{2,5,6},{4}}=>5
{{1,3},{2,5,6,7},{4}}=>4
{{1,3},{2,5,6},{4,7}}=>3
{{1,3},{2,5,6},{4},{7}}=>3
{{1,3,7},{2,5},{4,6}}=>4
{{1,3},{2,5,7},{4,6}}=>3
{{1,3},{2,5},{4,6,7}}=>2
{{1,3},{2,5},{4,6},{7}}=>2
{{1,3,7},{2,5},{4},{6}}=>5
{{1,3},{2,5,7},{4},{6}}=>4
{{1,3},{2,5},{4,7},{6}}=>3
{{1,3},{2,5},{4},{6,7}}=>2
{{1,3},{2,5},{4},{6},{7}}=>2
{{1,3,6,7},{2},{4,5}}=>5
{{1,3,6},{2,7},{4,5}}=>4
{{1,3,6},{2},{4,5,7}}=>3
{{1,3,6},{2},{4,5},{7}}=>3
{{1,3,7},{2,6},{4,5}}=>4
{{1,3},{2,6,7},{4,5}}=>3
{{1,3},{2,6},{4,5,7}}=>2
{{1,3},{2,6},{4,5},{7}}=>2
{{1,3,7},{2},{4,5,6}}=>3
{{1,3},{2,7},{4,5,6}}=>2
{{1,3},{2},{4,5,6,7}}=>1
{{1,3},{2},{4,5,6},{7}}=>1
{{1,3,7},{2},{4,5},{6}}=>4
{{1,3},{2,7},{4,5},{6}}=>3
{{1,3},{2},{4,5,7},{6}}=>2
{{1,3},{2},{4,5},{6,7}}=>1
{{1,3},{2},{4,5},{6},{7}}=>1
{{1,3,6,7},{2},{4},{5}}=>7
{{1,3,6},{2,7},{4},{5}}=>6
{{1,3,6},{2},{4,7},{5}}=>5
{{1,3,6},{2},{4},{5,7}}=>4
{{1,3,6},{2},{4},{5},{7}}=>4
{{1,3,7},{2,6},{4},{5}}=>6
{{1,3},{2,6,7},{4},{5}}=>5
{{1,3},{2,6},{4,7},{5}}=>4
{{1,3},{2,6},{4},{5,7}}=>3
{{1,3},{2,6},{4},{5},{7}}=>3
{{1,3,7},{2},{4,6},{5}}=>5
{{1,3},{2,7},{4,6},{5}}=>4
{{1,3},{2},{4,6,7},{5}}=>3
{{1,3},{2},{4,6},{5,7}}=>2
{{1,3},{2},{4,6},{5},{7}}=>2
{{1,3,7},{2},{4},{5,6}}=>4
{{1,3},{2,7},{4},{5,6}}=>3
{{1,3},{2},{4,7},{5,6}}=>2
{{1,3},{2},{4},{5,6,7}}=>1
{{1,3},{2},{4},{5,6},{7}}=>1
{{1,3,7},{2},{4},{5},{6}}=>5
{{1,3},{2,7},{4},{5},{6}}=>4
{{1,3},{2},{4,7},{5},{6}}=>3
{{1,3},{2},{4},{5,7},{6}}=>2
{{1,3},{2},{4},{5},{6,7}}=>1
{{1,3},{2},{4},{5},{6},{7}}=>1
{{1,4,5,6,7},{2,3}}=>4
{{1,4,5,6},{2,3,7}}=>3
{{1,4,5,6},{2,3},{7}}=>3
{{1,4,5,7},{2,3,6}}=>3
{{1,4,5},{2,3,6,7}}=>2
{{1,4,5},{2,3,6},{7}}=>2
{{1,4,5,7},{2,3},{6}}=>4
{{1,4,5},{2,3,7},{6}}=>3
{{1,4,5},{2,3},{6,7}}=>2
{{1,4,5},{2,3},{6},{7}}=>2
{{1,4,6,7},{2,3,5}}=>3
{{1,4,6},{2,3,5,7}}=>2
{{1,4,6},{2,3,5},{7}}=>2
{{1,4,7},{2,3,5,6}}=>2
{{1,4},{2,3,5,6,7}}=>1
{{1,4},{2,3,5,6},{7}}=>1
{{1,4,7},{2,3,5},{6}}=>3
{{1,4},{2,3,5,7},{6}}=>2
{{1,4},{2,3,5},{6,7}}=>1
{{1,4},{2,3,5},{6},{7}}=>1
{{1,4,6,7},{2,3},{5}}=>5
{{1,4,6},{2,3,7},{5}}=>4
{{1,4,6},{2,3},{5,7}}=>3
{{1,4,6},{2,3},{5},{7}}=>3
{{1,4,7},{2,3,6},{5}}=>4
{{1,4},{2,3,6,7},{5}}=>3
{{1,4},{2,3,6},{5,7}}=>2
{{1,4},{2,3,6},{5},{7}}=>2
{{1,4,7},{2,3},{5,6}}=>3
{{1,4},{2,3,7},{5,6}}=>2
{{1,4},{2,3},{5,6,7}}=>1
{{1,4},{2,3},{5,6},{7}}=>1
{{1,4,7},{2,3},{5},{6}}=>4
{{1,4},{2,3,7},{5},{6}}=>3
{{1,4},{2,3},{5,7},{6}}=>2
{{1,4},{2,3},{5},{6,7}}=>1
{{1,4},{2,3},{5},{6},{7}}=>1
{{1,5,6,7},{2,3,4}}=>3
{{1,5,6},{2,3,4,7}}=>2
{{1,5,6},{2,3,4},{7}}=>2
{{1,5,7},{2,3,4,6}}=>2
{{1,5},{2,3,4,6,7}}=>1
{{1,5},{2,3,4,6},{7}}=>1
{{1,5,7},{2,3,4},{6}}=>3
{{1,5},{2,3,4,7},{6}}=>2
{{1,5},{2,3,4},{6,7}}=>1
{{1,5},{2,3,4},{6},{7}}=>1
{{1,6,7},{2,3,4,5}}=>2
{{1,6},{2,3,4,5,7}}=>1
{{1,6},{2,3,4,5},{7}}=>1
{{1,7},{2,3,4,5,6}}=>1
{{1},{2,3,4,5,6,7}}=>0
{{1},{2,3,4,5,6},{7}}=>0
{{1,7},{2,3,4,5},{6}}=>2
{{1},{2,3,4,5,7},{6}}=>1
{{1},{2,3,4,5},{6,7}}=>0
{{1},{2,3,4,5},{6},{7}}=>0
{{1,6,7},{2,3,4},{5}}=>4
{{1,6},{2,3,4,7},{5}}=>3
{{1,6},{2,3,4},{5,7}}=>2
{{1,6},{2,3,4},{5},{7}}=>2
{{1,7},{2,3,4,6},{5}}=>3
{{1},{2,3,4,6,7},{5}}=>2
{{1},{2,3,4,6},{5,7}}=>1
{{1},{2,3,4,6},{5},{7}}=>1
{{1,7},{2,3,4},{5,6}}=>2
{{1},{2,3,4,7},{5,6}}=>1
{{1},{2,3,4},{5,6,7}}=>0
{{1},{2,3,4},{5,6},{7}}=>0
{{1,7},{2,3,4},{5},{6}}=>3
{{1},{2,3,4,7},{5},{6}}=>2
{{1},{2,3,4},{5,7},{6}}=>1
{{1},{2,3,4},{5},{6,7}}=>0
{{1},{2,3,4},{5},{6},{7}}=>0
{{1,5,6,7},{2,3},{4}}=>6
{{1,5,6},{2,3,7},{4}}=>5
{{1,5,6},{2,3},{4,7}}=>4
{{1,5,6},{2,3},{4},{7}}=>4
{{1,5,7},{2,3,6},{4}}=>5
{{1,5},{2,3,6,7},{4}}=>4
{{1,5},{2,3,6},{4,7}}=>3
{{1,5},{2,3,6},{4},{7}}=>3
{{1,5,7},{2,3},{4,6}}=>4
{{1,5},{2,3,7},{4,6}}=>3
{{1,5},{2,3},{4,6,7}}=>2
{{1,5},{2,3},{4,6},{7}}=>2
{{1,5,7},{2,3},{4},{6}}=>5
{{1,5},{2,3,7},{4},{6}}=>4
{{1,5},{2,3},{4,7},{6}}=>3
{{1,5},{2,3},{4},{6,7}}=>2
{{1,5},{2,3},{4},{6},{7}}=>2
{{1,6,7},{2,3,5},{4}}=>5
{{1,6},{2,3,5,7},{4}}=>4
{{1,6},{2,3,5},{4,7}}=>3
{{1,6},{2,3,5},{4},{7}}=>3
{{1,7},{2,3,5,6},{4}}=>4
{{1},{2,3,5,6,7},{4}}=>3
{{1},{2,3,5,6},{4,7}}=>2
{{1},{2,3,5,6},{4},{7}}=>2
{{1,7},{2,3,5},{4,6}}=>3
{{1},{2,3,5,7},{4,6}}=>2
{{1},{2,3,5},{4,6,7}}=>1
{{1},{2,3,5},{4,6},{7}}=>1
{{1,7},{2,3,5},{4},{6}}=>4
{{1},{2,3,5,7},{4},{6}}=>3
{{1},{2,3,5},{4,7},{6}}=>2
{{1},{2,3,5},{4},{6,7}}=>1
{{1},{2,3,5},{4},{6},{7}}=>1
{{1,6,7},{2,3},{4,5}}=>4
{{1,6},{2,3,7},{4,5}}=>3
{{1,6},{2,3},{4,5,7}}=>2
{{1,6},{2,3},{4,5},{7}}=>2
{{1,7},{2,3,6},{4,5}}=>3
{{1},{2,3,6,7},{4,5}}=>2
{{1},{2,3,6},{4,5,7}}=>1
{{1},{2,3,6},{4,5},{7}}=>1
{{1,7},{2,3},{4,5,6}}=>2
{{1},{2,3,7},{4,5,6}}=>1
{{1},{2,3},{4,5,6,7}}=>0
{{1},{2,3},{4,5,6},{7}}=>0
{{1,7},{2,3},{4,5},{6}}=>3
{{1},{2,3,7},{4,5},{6}}=>2
{{1},{2,3},{4,5,7},{6}}=>1
{{1},{2,3},{4,5},{6,7}}=>0
{{1},{2,3},{4,5},{6},{7}}=>0
{{1,6,7},{2,3},{4},{5}}=>6
{{1,6},{2,3,7},{4},{5}}=>5
{{1,6},{2,3},{4,7},{5}}=>4
{{1,6},{2,3},{4},{5,7}}=>3
{{1,6},{2,3},{4},{5},{7}}=>3
{{1,7},{2,3,6},{4},{5}}=>5
{{1},{2,3,6,7},{4},{5}}=>4
{{1},{2,3,6},{4,7},{5}}=>3
{{1},{2,3,6},{4},{5,7}}=>2
{{1},{2,3,6},{4},{5},{7}}=>2
{{1,7},{2,3},{4,6},{5}}=>4
{{1},{2,3,7},{4,6},{5}}=>3
{{1},{2,3},{4,6,7},{5}}=>2
{{1},{2,3},{4,6},{5,7}}=>1
{{1},{2,3},{4,6},{5},{7}}=>1
{{1,7},{2,3},{4},{5,6}}=>3
{{1},{2,3,7},{4},{5,6}}=>2
{{1},{2,3},{4,7},{5,6}}=>1
{{1},{2,3},{4},{5,6,7}}=>0
{{1},{2,3},{4},{5,6},{7}}=>0
{{1,7},{2,3},{4},{5},{6}}=>4
{{1},{2,3,7},{4},{5},{6}}=>3
{{1},{2,3},{4,7},{5},{6}}=>2
{{1},{2,3},{4},{5,7},{6}}=>1
{{1},{2,3},{4},{5},{6,7}}=>0
{{1},{2,3},{4},{5},{6},{7}}=>0
{{1,4,5,6,7},{2},{3}}=>8
{{1,4,5,6},{2,7},{3}}=>7
{{1,4,5,6},{2},{3,7}}=>6
{{1,4,5,6},{2},{3},{7}}=>6
{{1,4,5,7},{2,6},{3}}=>7
{{1,4,5},{2,6,7},{3}}=>6
{{1,4,5},{2,6},{3,7}}=>5
{{1,4,5},{2,6},{3},{7}}=>5
{{1,4,5,7},{2},{3,6}}=>6
{{1,4,5},{2,7},{3,6}}=>5
{{1,4,5},{2},{3,6,7}}=>4
{{1,4,5},{2},{3,6},{7}}=>4
{{1,4,5,7},{2},{3},{6}}=>7
{{1,4,5},{2,7},{3},{6}}=>6
{{1,4,5},{2},{3,7},{6}}=>5
{{1,4,5},{2},{3},{6,7}}=>4
{{1,4,5},{2},{3},{6},{7}}=>4
{{1,4,6,7},{2,5},{3}}=>7
{{1,4,6},{2,5,7},{3}}=>6
{{1,4,6},{2,5},{3,7}}=>5
{{1,4,6},{2,5},{3},{7}}=>5
{{1,4,7},{2,5,6},{3}}=>6
{{1,4},{2,5,6,7},{3}}=>5
{{1,4},{2,5,6},{3,7}}=>4
{{1,4},{2,5,6},{3},{7}}=>4
{{1,4,7},{2,5},{3,6}}=>5
{{1,4},{2,5,7},{3,6}}=>4
{{1,4},{2,5},{3,6,7}}=>3
{{1,4},{2,5},{3,6},{7}}=>3
{{1,4,7},{2,5},{3},{6}}=>6
{{1,4},{2,5,7},{3},{6}}=>5
{{1,4},{2,5},{3,7},{6}}=>4
{{1,4},{2,5},{3},{6,7}}=>3
{{1,4},{2,5},{3},{6},{7}}=>3
{{1,4,6,7},{2},{3,5}}=>6
{{1,4,6},{2,7},{3,5}}=>5
{{1,4,6},{2},{3,5,7}}=>4
{{1,4,6},{2},{3,5},{7}}=>4
{{1,4,7},{2,6},{3,5}}=>5
{{1,4},{2,6,7},{3,5}}=>4
{{1,4},{2,6},{3,5,7}}=>3
{{1,4},{2,6},{3,5},{7}}=>3
{{1,4,7},{2},{3,5,6}}=>4
{{1,4},{2,7},{3,5,6}}=>3
{{1,4},{2},{3,5,6,7}}=>2
{{1,4},{2},{3,5,6},{7}}=>2
{{1,4,7},{2},{3,5},{6}}=>5
{{1,4},{2,7},{3,5},{6}}=>4
{{1,4},{2},{3,5,7},{6}}=>3
{{1,4},{2},{3,5},{6,7}}=>2
{{1,4},{2},{3,5},{6},{7}}=>2
{{1,4,6,7},{2},{3},{5}}=>8
{{1,4,6},{2,7},{3},{5}}=>7
{{1,4,6},{2},{3,7},{5}}=>6
{{1,4,6},{2},{3},{5,7}}=>5
{{1,4,6},{2},{3},{5},{7}}=>5
{{1,4,7},{2,6},{3},{5}}=>7
{{1,4},{2,6,7},{3},{5}}=>6
{{1,4},{2,6},{3,7},{5}}=>5
{{1,4},{2,6},{3},{5,7}}=>4
{{1,4},{2,6},{3},{5},{7}}=>4
{{1,4,7},{2},{3,6},{5}}=>6
{{1,4},{2,7},{3,6},{5}}=>5
{{1,4},{2},{3,6,7},{5}}=>4
{{1,4},{2},{3,6},{5,7}}=>3
{{1,4},{2},{3,6},{5},{7}}=>3
{{1,4,7},{2},{3},{5,6}}=>5
{{1,4},{2,7},{3},{5,6}}=>4
{{1,4},{2},{3,7},{5,6}}=>3
{{1,4},{2},{3},{5,6,7}}=>2
{{1,4},{2},{3},{5,6},{7}}=>2
{{1,4,7},{2},{3},{5},{6}}=>6
{{1,4},{2,7},{3},{5},{6}}=>5
{{1,4},{2},{3,7},{5},{6}}=>4
{{1,4},{2},{3},{5,7},{6}}=>3
{{1,4},{2},{3},{5},{6,7}}=>2
{{1,4},{2},{3},{5},{6},{7}}=>2
{{1,5,6,7},{2,4},{3}}=>7
{{1,5,6},{2,4,7},{3}}=>6
{{1,5,6},{2,4},{3,7}}=>5
{{1,5,6},{2,4},{3},{7}}=>5
{{1,5,7},{2,4,6},{3}}=>6
{{1,5},{2,4,6,7},{3}}=>5
{{1,5},{2,4,6},{3,7}}=>4
{{1,5},{2,4,6},{3},{7}}=>4
{{1,5,7},{2,4},{3,6}}=>5
{{1,5},{2,4,7},{3,6}}=>4
{{1,5},{2,4},{3,6,7}}=>3
{{1,5},{2,4},{3,6},{7}}=>3
{{1,5,7},{2,4},{3},{6}}=>6
{{1,5},{2,4,7},{3},{6}}=>5
{{1,5},{2,4},{3,7},{6}}=>4
{{1,5},{2,4},{3},{6,7}}=>3
{{1,5},{2,4},{3},{6},{7}}=>3
{{1,6,7},{2,4,5},{3}}=>6
{{1,6},{2,4,5,7},{3}}=>5
{{1,6},{2,4,5},{3,7}}=>4
{{1,6},{2,4,5},{3},{7}}=>4
{{1,7},{2,4,5,6},{3}}=>5
{{1},{2,4,5,6,7},{3}}=>4
{{1},{2,4,5,6},{3,7}}=>3
{{1},{2,4,5,6},{3},{7}}=>3
{{1,7},{2,4,5},{3,6}}=>4
{{1},{2,4,5,7},{3,6}}=>3
{{1},{2,4,5},{3,6,7}}=>2
{{1},{2,4,5},{3,6},{7}}=>2
{{1,7},{2,4,5},{3},{6}}=>5
{{1},{2,4,5,7},{3},{6}}=>4
{{1},{2,4,5},{3,7},{6}}=>3
{{1},{2,4,5},{3},{6,7}}=>2
{{1},{2,4,5},{3},{6},{7}}=>2
{{1,6,7},{2,4},{3,5}}=>5
{{1,6},{2,4,7},{3,5}}=>4
{{1,6},{2,4},{3,5,7}}=>3
{{1,6},{2,4},{3,5},{7}}=>3
{{1,7},{2,4,6},{3,5}}=>4
{{1},{2,4,6,7},{3,5}}=>3
{{1},{2,4,6},{3,5,7}}=>2
{{1},{2,4,6},{3,5},{7}}=>2
{{1,7},{2,4},{3,5,6}}=>3
{{1},{2,4,7},{3,5,6}}=>2
{{1},{2,4},{3,5,6,7}}=>1
{{1},{2,4},{3,5,6},{7}}=>1
{{1,7},{2,4},{3,5},{6}}=>4
{{1},{2,4,7},{3,5},{6}}=>3
{{1},{2,4},{3,5,7},{6}}=>2
{{1},{2,4},{3,5},{6,7}}=>1
{{1},{2,4},{3,5},{6},{7}}=>1
{{1,6,7},{2,4},{3},{5}}=>7
{{1,6},{2,4,7},{3},{5}}=>6
{{1,6},{2,4},{3,7},{5}}=>5
{{1,6},{2,4},{3},{5,7}}=>4
{{1,6},{2,4},{3},{5},{7}}=>4
{{1,7},{2,4,6},{3},{5}}=>6
{{1},{2,4,6,7},{3},{5}}=>5
{{1},{2,4,6},{3,7},{5}}=>4
{{1},{2,4,6},{3},{5,7}}=>3
{{1},{2,4,6},{3},{5},{7}}=>3
{{1,7},{2,4},{3,6},{5}}=>5
{{1},{2,4,7},{3,6},{5}}=>4
{{1},{2,4},{3,6,7},{5}}=>3
{{1},{2,4},{3,6},{5,7}}=>2
{{1},{2,4},{3,6},{5},{7}}=>2
{{1,7},{2,4},{3},{5,6}}=>4
{{1},{2,4,7},{3},{5,6}}=>3
{{1},{2,4},{3,7},{5,6}}=>2
{{1},{2,4},{3},{5,6,7}}=>1
{{1},{2,4},{3},{5,6},{7}}=>1
{{1,7},{2,4},{3},{5},{6}}=>5
{{1},{2,4,7},{3},{5},{6}}=>4
{{1},{2,4},{3,7},{5},{6}}=>3
{{1},{2,4},{3},{5,7},{6}}=>2
{{1},{2,4},{3},{5},{6,7}}=>1
{{1},{2,4},{3},{5},{6},{7}}=>1
{{1,5,6,7},{2},{3,4}}=>6
{{1,5,6},{2,7},{3,4}}=>5
{{1,5,6},{2},{3,4,7}}=>4
{{1,5,6},{2},{3,4},{7}}=>4
{{1,5,7},{2,6},{3,4}}=>5
{{1,5},{2,6,7},{3,4}}=>4
{{1,5},{2,6},{3,4,7}}=>3
{{1,5},{2,6},{3,4},{7}}=>3
{{1,5,7},{2},{3,4,6}}=>4
{{1,5},{2,7},{3,4,6}}=>3
{{1,5},{2},{3,4,6,7}}=>2
{{1,5},{2},{3,4,6},{7}}=>2
{{1,5,7},{2},{3,4},{6}}=>5
{{1,5},{2,7},{3,4},{6}}=>4
{{1,5},{2},{3,4,7},{6}}=>3
{{1,5},{2},{3,4},{6,7}}=>2
{{1,5},{2},{3,4},{6},{7}}=>2
{{1,6,7},{2,5},{3,4}}=>5
{{1,6},{2,5,7},{3,4}}=>4
{{1,6},{2,5},{3,4,7}}=>3
{{1,6},{2,5},{3,4},{7}}=>3
{{1,7},{2,5,6},{3,4}}=>4
{{1},{2,5,6,7},{3,4}}=>3
{{1},{2,5,6},{3,4,7}}=>2
{{1},{2,5,6},{3,4},{7}}=>2
{{1,7},{2,5},{3,4,6}}=>3
{{1},{2,5,7},{3,4,6}}=>2
{{1},{2,5},{3,4,6,7}}=>1
{{1},{2,5},{3,4,6},{7}}=>1
{{1,7},{2,5},{3,4},{6}}=>4
{{1},{2,5,7},{3,4},{6}}=>3
{{1},{2,5},{3,4,7},{6}}=>2
{{1},{2,5},{3,4},{6,7}}=>1
{{1},{2,5},{3,4},{6},{7}}=>1
{{1,6,7},{2},{3,4,5}}=>4
{{1,6},{2,7},{3,4,5}}=>3
{{1,6},{2},{3,4,5,7}}=>2
{{1,6},{2},{3,4,5},{7}}=>2
{{1,7},{2,6},{3,4,5}}=>3
{{1},{2,6,7},{3,4,5}}=>2
{{1},{2,6},{3,4,5,7}}=>1
{{1},{2,6},{3,4,5},{7}}=>1
{{1,7},{2},{3,4,5,6}}=>2
{{1},{2,7},{3,4,5,6}}=>1
{{1},{2},{3,4,5,6,7}}=>0
{{1},{2},{3,4,5,6},{7}}=>0
{{1,7},{2},{3,4,5},{6}}=>3
{{1},{2,7},{3,4,5},{6}}=>2
{{1},{2},{3,4,5,7},{6}}=>1
{{1},{2},{3,4,5},{6,7}}=>0
{{1},{2},{3,4,5},{6},{7}}=>0
{{1,6,7},{2},{3,4},{5}}=>6
{{1,6},{2,7},{3,4},{5}}=>5
{{1,6},{2},{3,4,7},{5}}=>4
{{1,6},{2},{3,4},{5,7}}=>3
{{1,6},{2},{3,4},{5},{7}}=>3
{{1,7},{2,6},{3,4},{5}}=>5
{{1},{2,6,7},{3,4},{5}}=>4
{{1},{2,6},{3,4,7},{5}}=>3
{{1},{2,6},{3,4},{5,7}}=>2
{{1},{2,6},{3,4},{5},{7}}=>2
{{1,7},{2},{3,4,6},{5}}=>4
{{1},{2,7},{3,4,6},{5}}=>3
{{1},{2},{3,4,6,7},{5}}=>2
{{1},{2},{3,4,6},{5,7}}=>1
{{1},{2},{3,4,6},{5},{7}}=>1
{{1,7},{2},{3,4},{5,6}}=>3
{{1},{2,7},{3,4},{5,6}}=>2
{{1},{2},{3,4,7},{5,6}}=>1
{{1},{2},{3,4},{5,6,7}}=>0
{{1},{2},{3,4},{5,6},{7}}=>0
{{1,7},{2},{3,4},{5},{6}}=>4
{{1},{2,7},{3,4},{5},{6}}=>3
{{1},{2},{3,4,7},{5},{6}}=>2
{{1},{2},{3,4},{5,7},{6}}=>1
{{1},{2},{3,4},{5},{6,7}}=>0
{{1},{2},{3,4},{5},{6},{7}}=>0
{{1,5,6,7},{2},{3},{4}}=>9
{{1,5,6},{2,7},{3},{4}}=>8
{{1,5,6},{2},{3,7},{4}}=>7
{{1,5,6},{2},{3},{4,7}}=>6
{{1,5,6},{2},{3},{4},{7}}=>6
{{1,5,7},{2,6},{3},{4}}=>8
{{1,5},{2,6,7},{3},{4}}=>7
{{1,5},{2,6},{3,7},{4}}=>6
{{1,5},{2,6},{3},{4,7}}=>5
{{1,5},{2,6},{3},{4},{7}}=>5
{{1,5,7},{2},{3,6},{4}}=>7
{{1,5},{2,7},{3,6},{4}}=>6
{{1,5},{2},{3,6,7},{4}}=>5
{{1,5},{2},{3,6},{4,7}}=>4
{{1,5},{2},{3,6},{4},{7}}=>4
{{1,5,7},{2},{3},{4,6}}=>6
{{1,5},{2,7},{3},{4,6}}=>5
{{1,5},{2},{3,7},{4,6}}=>4
{{1,5},{2},{3},{4,6,7}}=>3
{{1,5},{2},{3},{4,6},{7}}=>3
{{1,5,7},{2},{3},{4},{6}}=>7
{{1,5},{2,7},{3},{4},{6}}=>6
{{1,5},{2},{3,7},{4},{6}}=>5
{{1,5},{2},{3},{4,7},{6}}=>4
{{1,5},{2},{3},{4},{6,7}}=>3
{{1,5},{2},{3},{4},{6},{7}}=>3
{{1,6,7},{2,5},{3},{4}}=>8
{{1,6},{2,5,7},{3},{4}}=>7
{{1,6},{2,5},{3,7},{4}}=>6
{{1,6},{2,5},{3},{4,7}}=>5
{{1,6},{2,5},{3},{4},{7}}=>5
{{1,7},{2,5,6},{3},{4}}=>7
{{1},{2,5,6,7},{3},{4}}=>6
{{1},{2,5,6},{3,7},{4}}=>5
{{1},{2,5,6},{3},{4,7}}=>4
{{1},{2,5,6},{3},{4},{7}}=>4
{{1,7},{2,5},{3,6},{4}}=>6
{{1},{2,5,7},{3,6},{4}}=>5
{{1},{2,5},{3,6,7},{4}}=>4
{{1},{2,5},{3,6},{4,7}}=>3
{{1},{2,5},{3,6},{4},{7}}=>3
{{1,7},{2,5},{3},{4,6}}=>5
{{1},{2,5,7},{3},{4,6}}=>4
{{1},{2,5},{3,7},{4,6}}=>3
{{1},{2,5},{3},{4,6,7}}=>2
{{1},{2,5},{3},{4,6},{7}}=>2
{{1,7},{2,5},{3},{4},{6}}=>6
{{1},{2,5,7},{3},{4},{6}}=>5
{{1},{2,5},{3,7},{4},{6}}=>4
{{1},{2,5},{3},{4,7},{6}}=>3
{{1},{2,5},{3},{4},{6,7}}=>2
{{1},{2,5},{3},{4},{6},{7}}=>2
{{1,6,7},{2},{3,5},{4}}=>7
{{1,6},{2,7},{3,5},{4}}=>6
{{1,6},{2},{3,5,7},{4}}=>5
{{1,6},{2},{3,5},{4,7}}=>4
{{1,6},{2},{3,5},{4},{7}}=>4
{{1,7},{2,6},{3,5},{4}}=>6
{{1},{2,6,7},{3,5},{4}}=>5
{{1},{2,6},{3,5,7},{4}}=>4
{{1},{2,6},{3,5},{4,7}}=>3
{{1},{2,6},{3,5},{4},{7}}=>3
{{1,7},{2},{3,5,6},{4}}=>5
{{1},{2,7},{3,5,6},{4}}=>4
{{1},{2},{3,5,6,7},{4}}=>3
{{1},{2},{3,5,6},{4,7}}=>2
{{1},{2},{3,5,6},{4},{7}}=>2
{{1,7},{2},{3,5},{4,6}}=>4
{{1},{2,7},{3,5},{4,6}}=>3
{{1},{2},{3,5,7},{4,6}}=>2
{{1},{2},{3,5},{4,6,7}}=>1
{{1},{2},{3,5},{4,6},{7}}=>1
{{1,7},{2},{3,5},{4},{6}}=>5
{{1},{2,7},{3,5},{4},{6}}=>4
{{1},{2},{3,5,7},{4},{6}}=>3
{{1},{2},{3,5},{4,7},{6}}=>2
{{1},{2},{3,5},{4},{6,7}}=>1
{{1},{2},{3,5},{4},{6},{7}}=>1
{{1,6,7},{2},{3},{4,5}}=>6
{{1,6},{2,7},{3},{4,5}}=>5
{{1,6},{2},{3,7},{4,5}}=>4
{{1,6},{2},{3},{4,5,7}}=>3
{{1,6},{2},{3},{4,5},{7}}=>3
{{1,7},{2,6},{3},{4,5}}=>5
{{1},{2,6,7},{3},{4,5}}=>4
{{1},{2,6},{3,7},{4,5}}=>3
{{1},{2,6},{3},{4,5,7}}=>2
{{1},{2,6},{3},{4,5},{7}}=>2
{{1,7},{2},{3,6},{4,5}}=>4
{{1},{2,7},{3,6},{4,5}}=>3
{{1},{2},{3,6,7},{4,5}}=>2
{{1},{2},{3,6},{4,5,7}}=>1
{{1},{2},{3,6},{4,5},{7}}=>1
{{1,7},{2},{3},{4,5,6}}=>3
{{1},{2,7},{3},{4,5,6}}=>2
{{1},{2},{3,7},{4,5,6}}=>1
{{1},{2},{3},{4,5,6,7}}=>0
{{1},{2},{3},{4,5,6},{7}}=>0
{{1,7},{2},{3},{4,5},{6}}=>4
{{1},{2,7},{3},{4,5},{6}}=>3
{{1},{2},{3,7},{4,5},{6}}=>2
{{1},{2},{3},{4,5,7},{6}}=>1
{{1},{2},{3},{4,5},{6,7}}=>0
{{1},{2},{3},{4,5},{6},{7}}=>0
{{1,6,7},{2},{3},{4},{5}}=>8
{{1,6},{2,7},{3},{4},{5}}=>7
{{1,6},{2},{3,7},{4},{5}}=>6
{{1,6},{2},{3},{4,7},{5}}=>5
{{1,6},{2},{3},{4},{5,7}}=>4
{{1,6},{2},{3},{4},{5},{7}}=>4
{{1,7},{2,6},{3},{4},{5}}=>7
{{1},{2,6,7},{3},{4},{5}}=>6
{{1},{2,6},{3,7},{4},{5}}=>5
{{1},{2,6},{3},{4,7},{5}}=>4
{{1},{2,6},{3},{4},{5,7}}=>3
{{1},{2,6},{3},{4},{5},{7}}=>3
{{1,7},{2},{3,6},{4},{5}}=>6
{{1},{2,7},{3,6},{4},{5}}=>5
{{1},{2},{3,6,7},{4},{5}}=>4
{{1},{2},{3,6},{4,7},{5}}=>3
{{1},{2},{3,6},{4},{5,7}}=>2
{{1},{2},{3,6},{4},{5},{7}}=>2
{{1,7},{2},{3},{4,6},{5}}=>5
{{1},{2,7},{3},{4,6},{5}}=>4
{{1},{2},{3,7},{4,6},{5}}=>3
{{1},{2},{3},{4,6,7},{5}}=>2
{{1},{2},{3},{4,6},{5,7}}=>1
{{1},{2},{3},{4,6},{5},{7}}=>1
{{1,7},{2},{3},{4},{5,6}}=>4
{{1},{2,7},{3},{4},{5,6}}=>3
{{1},{2},{3,7},{4},{5,6}}=>2
{{1},{2},{3},{4,7},{5,6}}=>1
{{1},{2},{3},{4},{5,6,7}}=>0
{{1},{2},{3},{4},{5,6},{7}}=>0
{{1,7},{2},{3},{4},{5},{6}}=>5
{{1},{2,7},{3},{4},{5},{6}}=>4
{{1},{2},{3,7},{4},{5},{6}}=>3
{{1},{2},{3},{4,7},{5},{6}}=>2
{{1},{2},{3},{4},{5,7},{6}}=>1
{{1},{2},{3},{4},{5},{6,7}}=>0
{{1},{2},{3},{4},{5},{6},{7}}=>0
{{1},{2},{3,4,5,6,7,8}}=>0
{{1},{2,4,5,6,7,8},{3}}=>5
{{1},{2,3,5,6,7,8},{4}}=>4
{{1},{2,3,4,6,7,8},{5}}=>3
{{1},{2,3,4,5,7,8},{6}}=>2
{{1},{2,3,4,5,6,7},{8}}=>0
{{1},{2,3,4,5,6,8},{7}}=>1
{{1},{2,3,4,5,6,7,8}}=>0
{{1,2},{3,4,5,6,7,8}}=>0
{{1,4,5,6,7,8},{2},{3}}=>10
{{1,3,5,6,7,8},{2},{4}}=>9
{{1,3,4,5,6,7,8},{2}}=>6
{{1,4,5,6,7,8},{2,3}}=>5
{{1,2,4,5,6,7,8},{3}}=>5
{{1,2,5,6,7,8},{3,4}}=>4
{{1,2,3,5,6,7,8},{4}}=>4
{{1,2,3,6,7,8},{4,5}}=>3
{{1,2,3,4,6,7,8},{5}}=>3
{{1,2,3,4,5,6},{7,8}}=>0
{{1,2,3,4,7,8},{5,6}}=>2
{{1,2,3,4,5,7,8},{6}}=>2
{{1,2,3,4,5,6,7},{8}}=>0
{{1,8},{2,3,4,5,6,7}}=>1
{{1,2,3,4,5,8},{6,7}}=>1
{{1,2,3,4,5,6,8},{7}}=>1
{{1,2,3,4,5,6,7,8}}=>0
{{1,3,5,6,7,8},{2,4}}=>5
{{1,3,4,6,7,8},{2,5}}=>5
{{1,2,4,6,7,8},{3,5}}=>4
{{1,3,4,5,7,8},{2,6}}=>5
{{1,2,4,5,7,8},{3,6}}=>4
{{1,2,3,5,7,8},{4,6}}=>3
{{1,3,4,5,6,8},{2,7}}=>5
{{1,2,4,5,6,8},{3,7}}=>4
{{1,2,3,5,6,8},{4,7}}=>3
{{1,2,3,4,6,8},{5,7}}=>2
{{1,3,4,5,6,7},{2,8}}=>5
{{1,2,4,5,6,7},{3,8}}=>4
{{1,2,3,5,6,7},{4,8}}=>3
{{1,2,3,4,6,7},{5,8}}=>2
{{1,2,3,4,5,7},{6,8}}=>1
{{1,3},{2,4,5,6,7,8}}=>1
{{1,4},{2,3,5,6,7,8}}=>1
{{1,5},{2,3,4,6,7,8}}=>1
{{1,6},{2,3,4,5,7,8}}=>1
{{1,7},{2,3,4,5,6,8}}=>1
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called ros in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called ros in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
References
[1] Steingrimsson, E. Statistics on ordered partitions of sets arXiv:math/0605670
[2] Ishikawa, M., Kasraoui, A., Zeng, J. Euler-Mahonian statistics on ordered partitions and SteingrÃmsson's conjecture—a survey MathSciNet:2467717
[3] Rhoades, B. Ordered set partition statistics and the Delta Conjecture arXiv:1605.04007
[4] Sagan, B. E. A maj statistic for set partitions MathSciNet:1087650
[2] Ishikawa, M., Kasraoui, A., Zeng, J. Euler-Mahonian statistics on ordered partitions and SteingrÃmsson's conjecture—a survey MathSciNet:2467717
[3] Rhoades, B. Ordered set partition statistics and the Delta Conjecture arXiv:1605.04007
[4] Sagan, B. E. A maj statistic for set partitions MathSciNet:1087650
Code
def statistic(S):
S = sorted( sorted(B) for B in S )
n = sum( len(B) for B in S )
ros = 0
for k in range(len(S)):
j = S[k][0]
for l in range(k):
for i in S[l]:
if i > j:
ros += 1
return ros
def statistic_alt(pi):
return len(pattern_occurrences(pi, [[1,3],[2]], [1,2], [], [], []))
def pattern_occurrences(pi, P, First, Last, Arcs, Consecutives):
"""We assume that pi is a SetPartition of {1,2,...,n} and P is a
SetPartition of {1,2,...,k}.
"""
occurrences = []
pi = SetPartition(pi)
P = SetPartition(P)
openers = [min(B) for B in pi]
closers = [max(B) for B in pi]
pi_sorted = sorted([sorted(b) for b in pi])
edges = [(a,b) for B in pi_sorted for (a,b) in zip(B, B[1:])]
for s in Subsets(pi.base_set(), P.size()):
s = sorted(s)
pi_r = pi.restriction(s)
if pi_r.standardization() != P:
continue
X = pi_r.base_set()
if any(X[i-1] not in openers for i in First):
continue
if any(X[i-1] not in closers for i in Last):
continue
if any((X[i-1], X[j-1]) not in edges for (i,j) in Arcs):
continue
if any(abs(X[i-1]-X[j-1]) != 1 for (i,j) in Consecutives):
continue
occurrences += [s]
return occurrences
Created
May 16, 2016 at 20:36 by Christian Stump
Updated
Aug 11, 2016 at 09:42 by Martin Rubey
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!