Identifier
- St000823: Set partitions ⟶ ℤ
Values
=>
Cc0009;cc-rep
{{1,2}}=>2
{{1},{2}}=>1
{{1,2,3}}=>3
{{1,2},{3}}=>2
{{1,3},{2}}=>2
{{1},{2,3}}=>1
{{1},{2},{3}}=>1
{{1,2,3,4}}=>4
{{1,2,3},{4}}=>3
{{1,2,4},{3}}=>3
{{1,2},{3,4}}=>2
{{1,2},{3},{4}}=>2
{{1,3,4},{2}}=>3
{{1,3},{2,4}}=>2
{{1,3},{2},{4}}=>1
{{1,4},{2,3}}=>2
{{1},{2,3,4}}=>1
{{1},{2,3},{4}}=>1
{{1,4},{2},{3}}=>2
{{1},{2,4},{3}}=>1
{{1},{2},{3,4}}=>1
{{1},{2},{3},{4}}=>1
{{1,2,3,4,5}}=>5
{{1,2,3,4},{5}}=>4
{{1,2,3,5},{4}}=>4
{{1,2,3},{4,5}}=>3
{{1,2,3},{4},{5}}=>3
{{1,2,4,5},{3}}=>4
{{1,2,4},{3,5}}=>3
{{1,2,4},{3},{5}}=>2
{{1,2,5},{3,4}}=>3
{{1,2},{3,4,5}}=>2
{{1,2},{3,4},{5}}=>2
{{1,2,5},{3},{4}}=>3
{{1,2},{3,5},{4}}=>2
{{1,2},{3},{4,5}}=>2
{{1,2},{3},{4},{5}}=>2
{{1,3,4,5},{2}}=>4
{{1,3,4},{2,5}}=>3
{{1,3,4},{2},{5}}=>1
{{1,3,5},{2,4}}=>3
{{1,3},{2,4,5}}=>2
{{1,3},{2,4},{5}}=>2
{{1,3,5},{2},{4}}=>2
{{1,3},{2,5},{4}}=>1
{{1,3},{2},{4,5}}=>1
{{1,3},{2},{4},{5}}=>1
{{1,4,5},{2,3}}=>3
{{1,4},{2,3,5}}=>2
{{1,4},{2,3},{5}}=>1
{{1,5},{2,3,4}}=>2
{{1},{2,3,4,5}}=>1
{{1},{2,3,4},{5}}=>1
{{1,5},{2,3},{4}}=>2
{{1},{2,3,5},{4}}=>1
{{1},{2,3},{4,5}}=>1
{{1},{2,3},{4},{5}}=>1
{{1,4,5},{2},{3}}=>3
{{1,4},{2,5},{3}}=>2
{{1,4},{2},{3,5}}=>1
{{1,4},{2},{3},{5}}=>1
{{1,5},{2,4},{3}}=>2
{{1},{2,4,5},{3}}=>1
{{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}}=>1
{{1},{2},{3,4},{5}}=>1
{{1,5},{2},{3},{4}}=>2
{{1},{2,5},{3},{4}}=>1
{{1},{2},{3,5},{4}}=>1
{{1},{2},{3},{4,5}}=>1
{{1},{2},{3},{4},{5}}=>1
{{1,2,3,4,5,6}}=>6
{{1,2,3,4,5},{6}}=>5
{{1,2,3,4,6},{5}}=>5
{{1,2,3,4},{5,6}}=>4
{{1,2,3,4},{5},{6}}=>4
{{1,2,3,5,6},{4}}=>5
{{1,2,3,5},{4,6}}=>4
{{1,2,3,5},{4},{6}}=>3
{{1,2,3,6},{4,5}}=>4
{{1,2,3},{4,5,6}}=>3
{{1,2,3},{4,5},{6}}=>3
{{1,2,3,6},{4},{5}}=>4
{{1,2,3},{4,6},{5}}=>3
{{1,2,3},{4},{5,6}}=>3
{{1,2,3},{4},{5},{6}}=>3
{{1,2,4,5,6},{3}}=>5
{{1,2,4,5},{3,6}}=>4
{{1,2,4,5},{3},{6}}=>2
{{1,2,4,6},{3,5}}=>4
{{1,2,4},{3,5,6}}=>3
{{1,2,4},{3,5},{6}}=>3
{{1,2,4,6},{3},{5}}=>3
{{1,2,4},{3,6},{5}}=>2
{{1,2,4},{3},{5,6}}=>2
{{1,2,4},{3},{5},{6}}=>2
{{1,2,5,6},{3,4}}=>4
{{1,2,5},{3,4,6}}=>3
{{1,2,5},{3,4},{6}}=>2
{{1,2,6},{3,4,5}}=>3
{{1,2},{3,4,5,6}}=>2
{{1,2},{3,4,5},{6}}=>2
{{1,2,6},{3,4},{5}}=>3
{{1,2},{3,4,6},{5}}=>2
{{1,2},{3,4},{5,6}}=>2
{{1,2},{3,4},{5},{6}}=>2
{{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}}=>2
{{1,2},{3,5},{4},{6}}=>2
{{1,2,6},{3},{4,5}}=>3
{{1,2},{3,6},{4,5}}=>2
{{1,2},{3},{4,5,6}}=>2
{{1,2},{3},{4,5},{6}}=>2
{{1,2,6},{3},{4},{5}}=>3
{{1,2},{3,6},{4},{5}}=>2
{{1,2},{3},{4,6},{5}}=>2
{{1,2},{3},{4},{5,6}}=>2
{{1,2},{3},{4},{5},{6}}=>2
{{1,3,4,5,6},{2}}=>5
{{1,3,4,5},{2,6}}=>4
{{1,3,4,5},{2},{6}}=>1
{{1,3,4,6},{2,5}}=>4
{{1,3,4},{2,5,6}}=>3
{{1,3,4},{2,5},{6}}=>3
{{1,3,4,6},{2},{5}}=>2
{{1,3,4},{2,6},{5}}=>1
{{1,3,4},{2},{5,6}}=>1
{{1,3,4},{2},{5},{6}}=>1
{{1,3,5,6},{2,4}}=>4
{{1,3,5},{2,4,6}}=>3
{{1,3,5},{2,4},{6}}=>2
{{1,3,6},{2,4,5}}=>3
{{1,3},{2,4,5,6}}=>2
{{1,3},{2,4,5},{6}}=>2
{{1,3,6},{2,4},{5}}=>3
{{1,3},{2,4,6},{5}}=>2
{{1,3},{2,4},{5,6}}=>2
{{1,3},{2,4},{5},{6}}=>2
{{1,3,5,6},{2},{4}}=>3
{{1,3,5},{2,6},{4}}=>2
{{1,3,5},{2},{4,6}}=>1
{{1,3,5},{2},{4},{6}}=>1
{{1,3,6},{2,5},{4}}=>2
{{1,3},{2,5,6},{4}}=>1
{{1,3},{2,5},{4,6}}=>1
{{1,3},{2,5},{4},{6}}=>1
{{1,3,6},{2},{4,5}}=>2
{{1,3},{2,6},{4,5}}=>1
{{1,3},{2},{4,5,6}}=>1
{{1,3},{2},{4,5},{6}}=>1
{{1,3,6},{2},{4},{5}}=>2
{{1,3},{2,6},{4},{5}}=>1
{{1,3},{2},{4,6},{5}}=>1
{{1,3},{2},{4},{5,6}}=>1
{{1,3},{2},{4},{5},{6}}=>1
{{1,4,5,6},{2,3}}=>4
{{1,4,5},{2,3,6}}=>3
{{1,4,5},{2,3},{6}}=>1
{{1,4,6},{2,3,5}}=>3
{{1,4},{2,3,5,6}}=>2
{{1,4},{2,3,5},{6}}=>2
{{1,4,6},{2,3},{5}}=>2
{{1,4},{2,3,6},{5}}=>1
{{1,4},{2,3},{5,6}}=>1
{{1,4},{2,3},{5},{6}}=>1
{{1,5,6},{2,3,4}}=>3
{{1,5},{2,3,4,6}}=>2
{{1,5},{2,3,4},{6}}=>1
{{1,6},{2,3,4,5}}=>2
{{1},{2,3,4,5,6}}=>1
{{1},{2,3,4,5},{6}}=>1
{{1,6},{2,3,4},{5}}=>2
{{1},{2,3,4,6},{5}}=>1
{{1},{2,3,4},{5,6}}=>1
{{1},{2,3,4},{5},{6}}=>1
{{1,5,6},{2,3},{4}}=>3
{{1,5},{2,3,6},{4}}=>2
{{1,5},{2,3},{4,6}}=>1
{{1,5},{2,3},{4},{6}}=>1
{{1,6},{2,3,5},{4}}=>2
{{1},{2,3,5,6},{4}}=>1
{{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}}=>1
{{1},{2,3},{4,5},{6}}=>1
{{1,6},{2,3},{4},{5}}=>2
{{1},{2,3,6},{4},{5}}=>1
{{1},{2,3},{4,6},{5}}=>1
{{1},{2,3},{4},{5,6}}=>1
{{1},{2,3},{4},{5},{6}}=>1
{{1,4,5,6},{2},{3}}=>4
{{1,4,5},{2,6},{3}}=>3
{{1,4,5},{2},{3,6}}=>1
{{1,4,5},{2},{3},{6}}=>1
{{1,4,6},{2,5},{3}}=>3
{{1,4},{2,5,6},{3}}=>2
{{1,4},{2,5},{3,6}}=>2
{{1,4},{2,5},{3},{6}}=>1
{{1,4,6},{2},{3,5}}=>2
{{1,4},{2,6},{3,5}}=>1
{{1,4},{2},{3,5,6}}=>1
{{1,4},{2},{3,5},{6}}=>1
{{1,4,6},{2},{3},{5}}=>2
{{1,4},{2,6},{3},{5}}=>1
{{1,4},{2},{3,6},{5}}=>1
{{1,4},{2},{3},{5,6}}=>1
{{1,4},{2},{3},{5},{6}}=>1
{{1,5,6},{2,4},{3}}=>3
{{1,5},{2,4,6},{3}}=>2
{{1,5},{2,4},{3,6}}=>1
{{1,5},{2,4},{3},{6}}=>1
{{1,6},{2,4,5},{3}}=>2
{{1},{2,4,5,6},{3}}=>1
{{1},{2,4,5},{3,6}}=>1
{{1},{2,4,5},{3},{6}}=>1
{{1,6},{2,4},{3,5}}=>2
{{1},{2,4,6},{3,5}}=>1
{{1},{2,4},{3,5,6}}=>1
{{1},{2,4},{3,5},{6}}=>1
{{1,6},{2,4},{3},{5}}=>2
{{1},{2,4,6},{3},{5}}=>1
{{1},{2,4},{3,6},{5}}=>1
{{1},{2,4},{3},{5,6}}=>1
{{1},{2,4},{3},{5},{6}}=>1
{{1,5,6},{2},{3,4}}=>3
{{1,5},{2,6},{3,4}}=>2
{{1,5},{2},{3,4,6}}=>1
{{1,5},{2},{3,4},{6}}=>1
{{1,6},{2,5},{3,4}}=>2
{{1},{2,5,6},{3,4}}=>1
{{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}}=>1
{{1},{2},{3,4,5},{6}}=>1
{{1,6},{2},{3,4},{5}}=>2
{{1},{2,6},{3,4},{5}}=>1
{{1},{2},{3,4,6},{5}}=>1
{{1},{2},{3,4},{5,6}}=>1
{{1},{2},{3,4},{5},{6}}=>1
{{1,5,6},{2},{3},{4}}=>3
{{1,5},{2,6},{3},{4}}=>2
{{1,5},{2},{3,6},{4}}=>1
{{1,5},{2},{3},{4,6}}=>1
{{1,5},{2},{3},{4},{6}}=>1
{{1,6},{2,5},{3},{4}}=>2
{{1},{2,5,6},{3},{4}}=>1
{{1},{2,5},{3,6},{4}}=>1
{{1},{2,5},{3},{4,6}}=>1
{{1},{2,5},{3},{4},{6}}=>1
{{1,6},{2},{3,5},{4}}=>2
{{1},{2,6},{3,5},{4}}=>1
{{1},{2},{3,5,6},{4}}=>1
{{1},{2},{3,5},{4,6}}=>1
{{1},{2},{3,5},{4},{6}}=>1
{{1,6},{2},{3},{4,5}}=>2
{{1},{2,6},{3},{4,5}}=>1
{{1},{2},{3,6},{4,5}}=>1
{{1},{2},{3},{4,5,6}}=>1
{{1},{2},{3},{4,5},{6}}=>1
{{1,6},{2},{3},{4},{5}}=>2
{{1},{2,6},{3},{4},{5}}=>1
{{1},{2},{3,6},{4},{5}}=>1
{{1},{2},{3},{4,6},{5}}=>1
{{1},{2},{3},{4},{5,6}}=>1
{{1},{2},{3},{4},{5},{6}}=>1
{{1,2,3,4,5,6,7}}=>7
{{1,2,3,4,5,6},{7}}=>6
{{1,2,3,4,5,7},{6}}=>6
{{1,2,3,4,5},{6,7}}=>5
{{1,2,3,4,5},{6},{7}}=>5
{{1,2,3,4,6,7},{5}}=>6
{{1,2,3,4,6},{5,7}}=>5
{{1,2,3,4,6},{5},{7}}=>4
{{1,2,3,4,7},{5,6}}=>5
{{1,2,3,4},{5,6,7}}=>4
{{1,2,3,4},{5,6},{7}}=>4
{{1,2,3,4,7},{5},{6}}=>5
{{1,2,3,4},{5,7},{6}}=>4
{{1,2,3,4},{5},{6,7}}=>4
{{1,2,3,4},{5},{6},{7}}=>4
{{1,2,3,5,6,7},{4}}=>6
{{1,2,3,5,6},{4,7}}=>5
{{1,2,3,5,6},{4},{7}}=>3
{{1,2,3,5,7},{4,6}}=>5
{{1,2,3,5},{4,6,7}}=>4
{{1,2,3,5},{4,6},{7}}=>4
{{1,2,3,5,7},{4},{6}}=>4
{{1,2,3,5},{4,7},{6}}=>3
{{1,2,3,5},{4},{6,7}}=>3
{{1,2,3,5},{4},{6},{7}}=>3
{{1,2,3,6,7},{4,5}}=>5
{{1,2,3,6},{4,5,7}}=>4
{{1,2,3,6},{4,5},{7}}=>3
{{1,2,3,7},{4,5,6}}=>4
{{1,2,3},{4,5,6,7}}=>3
{{1,2,3},{4,5,6},{7}}=>3
{{1,2,3,7},{4,5},{6}}=>4
{{1,2,3},{4,5,7},{6}}=>3
{{1,2,3},{4,5},{6,7}}=>3
{{1,2,3},{4,5},{6},{7}}=>3
{{1,2,3,6,7},{4},{5}}=>5
{{1,2,3,6},{4,7},{5}}=>4
{{1,2,3,6},{4},{5,7}}=>3
{{1,2,3,6},{4},{5},{7}}=>3
{{1,2,3,7},{4,6},{5}}=>4
{{1,2,3},{4,6,7},{5}}=>3
{{1,2,3},{4,6},{5,7}}=>3
{{1,2,3},{4,6},{5},{7}}=>3
{{1,2,3,7},{4},{5,6}}=>4
{{1,2,3},{4,7},{5,6}}=>3
{{1,2,3},{4},{5,6,7}}=>3
{{1,2,3},{4},{5,6},{7}}=>3
{{1,2,3,7},{4},{5},{6}}=>4
{{1,2,3},{4,7},{5},{6}}=>3
{{1,2,3},{4},{5,7},{6}}=>3
{{1,2,3},{4},{5},{6,7}}=>3
{{1,2,3},{4},{5},{6},{7}}=>3
{{1,2,4,5,6,7},{3}}=>6
{{1,2,4,5,6},{3,7}}=>5
{{1,2,4,5,6},{3},{7}}=>2
{{1,2,4,5,7},{3,6}}=>5
{{1,2,4,5},{3,6,7}}=>4
{{1,2,4,5},{3,6},{7}}=>4
{{1,2,4,5,7},{3},{6}}=>3
{{1,2,4,5},{3,7},{6}}=>2
{{1,2,4,5},{3},{6,7}}=>2
{{1,2,4,5},{3},{6},{7}}=>2
{{1,2,4,6,7},{3,5}}=>5
{{1,2,4,6},{3,5,7}}=>4
{{1,2,4,6},{3,5},{7}}=>3
{{1,2,4,7},{3,5,6}}=>4
{{1,2,4},{3,5,6,7}}=>3
{{1,2,4},{3,5,6},{7}}=>3
{{1,2,4,7},{3,5},{6}}=>4
{{1,2,4},{3,5,7},{6}}=>3
{{1,2,4},{3,5},{6,7}}=>3
{{1,2,4},{3,5},{6},{7}}=>3
{{1,2,4,6,7},{3},{5}}=>4
{{1,2,4,6},{3,7},{5}}=>3
{{1,2,4,6},{3},{5,7}}=>2
{{1,2,4,6},{3},{5},{7}}=>2
{{1,2,4,7},{3,6},{5}}=>3
{{1,2,4},{3,6,7},{5}}=>2
{{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}}=>2
{{1,2,4},{3},{5,6},{7}}=>2
{{1,2,4,7},{3},{5},{6}}=>3
{{1,2,4},{3,7},{5},{6}}=>2
{{1,2,4},{3},{5,7},{6}}=>2
{{1,2,4},{3},{5},{6,7}}=>2
{{1,2,4},{3},{5},{6},{7}}=>2
{{1,2,5,6,7},{3,4}}=>5
{{1,2,5,6},{3,4,7}}=>4
{{1,2,5,6},{3,4},{7}}=>2
{{1,2,5,7},{3,4,6}}=>4
{{1,2,5},{3,4,6,7}}=>3
{{1,2,5},{3,4,6},{7}}=>3
{{1,2,5,7},{3,4},{6}}=>3
{{1,2,5},{3,4,7},{6}}=>2
{{1,2,5},{3,4},{6,7}}=>2
{{1,2,5},{3,4},{6},{7}}=>2
{{1,2,6,7},{3,4,5}}=>4
{{1,2,6},{3,4,5,7}}=>3
{{1,2,6},{3,4,5},{7}}=>2
{{1,2,7},{3,4,5,6}}=>3
{{1,2},{3,4,5,6,7}}=>2
{{1,2},{3,4,5,6},{7}}=>2
{{1,2,7},{3,4,5},{6}}=>3
{{1,2},{3,4,5,7},{6}}=>2
{{1,2},{3,4,5},{6,7}}=>2
{{1,2},{3,4,5},{6},{7}}=>2
{{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}}=>2
{{1,2},{3,4,6},{5},{7}}=>2
{{1,2,7},{3,4},{5,6}}=>3
{{1,2},{3,4,7},{5,6}}=>2
{{1,2},{3,4},{5,6,7}}=>2
{{1,2},{3,4},{5,6},{7}}=>2
{{1,2,7},{3,4},{5},{6}}=>3
{{1,2},{3,4,7},{5},{6}}=>2
{{1,2},{3,4},{5,7},{6}}=>2
{{1,2},{3,4},{5},{6,7}}=>2
{{1,2},{3,4},{5},{6},{7}}=>2
{{1,2,5,6,7},{3},{4}}=>5
{{1,2,5,6},{3,7},{4}}=>4
{{1,2,5,6},{3},{4,7}}=>2
{{1,2,5,6},{3},{4},{7}}=>2
{{1,2,5,7},{3,6},{4}}=>4
{{1,2,5},{3,6,7},{4}}=>3
{{1,2,5},{3,6},{4,7}}=>3
{{1,2,5},{3,6},{4},{7}}=>2
{{1,2,5,7},{3},{4,6}}=>3
{{1,2,5},{3,7},{4,6}}=>2
{{1,2,5},{3},{4,6,7}}=>2
{{1,2,5},{3},{4,6},{7}}=>2
{{1,2,5,7},{3},{4},{6}}=>3
{{1,2,5},{3,7},{4},{6}}=>2
{{1,2,5},{3},{4,7},{6}}=>2
{{1,2,5},{3},{4},{6,7}}=>2
{{1,2,5},{3},{4},{6},{7}}=>2
{{1,2,6,7},{3,5},{4}}=>4
{{1,2,6},{3,5,7},{4}}=>3
{{1,2,6},{3,5},{4,7}}=>2
{{1,2,6},{3,5},{4},{7}}=>2
{{1,2,7},{3,5,6},{4}}=>3
{{1,2},{3,5,6,7},{4}}=>2
{{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}}=>2
{{1,2},{3,5},{4,6},{7}}=>2
{{1,2,7},{3,5},{4},{6}}=>3
{{1,2},{3,5,7},{4},{6}}=>2
{{1,2},{3,5},{4,7},{6}}=>2
{{1,2},{3,5},{4},{6,7}}=>2
{{1,2},{3,5},{4},{6},{7}}=>2
{{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}}=>2
{{1,2},{3,6},{4,5},{7}}=>2
{{1,2,7},{3},{4,5,6}}=>3
{{1,2},{3,7},{4,5,6}}=>2
{{1,2},{3},{4,5,6,7}}=>2
{{1,2},{3},{4,5,6},{7}}=>2
{{1,2,7},{3},{4,5},{6}}=>3
{{1,2},{3,7},{4,5},{6}}=>2
{{1,2},{3},{4,5,7},{6}}=>2
{{1,2},{3},{4,5},{6,7}}=>2
{{1,2},{3},{4,5},{6},{7}}=>2
{{1,2,6,7},{3},{4},{5}}=>4
{{1,2,6},{3,7},{4},{5}}=>3
{{1,2,6},{3},{4,7},{5}}=>2
{{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},{4,7},{5}}=>2
{{1,2},{3,6},{4},{5,7}}=>2
{{1,2},{3,6},{4},{5},{7}}=>2
{{1,2,7},{3},{4,6},{5}}=>3
{{1,2},{3,7},{4,6},{5}}=>2
{{1,2},{3},{4,6,7},{5}}=>2
{{1,2},{3},{4,6},{5,7}}=>2
{{1,2},{3},{4,6},{5},{7}}=>2
{{1,2,7},{3},{4},{5,6}}=>3
{{1,2},{3,7},{4},{5,6}}=>2
{{1,2},{3},{4,7},{5,6}}=>2
{{1,2},{3},{4},{5,6,7}}=>2
{{1,2},{3},{4},{5,6},{7}}=>2
{{1,2,7},{3},{4},{5},{6}}=>3
{{1,2},{3,7},{4},{5},{6}}=>2
{{1,2},{3},{4,7},{5},{6}}=>2
{{1,2},{3},{4},{5,7},{6}}=>2
{{1,2},{3},{4},{5},{6,7}}=>2
{{1,2},{3},{4},{5},{6},{7}}=>2
{{1,3,4,5,6,7},{2}}=>6
{{1,3,4,5,6},{2,7}}=>5
{{1,3,4,5,6},{2},{7}}=>1
{{1,3,4,5,7},{2,6}}=>5
{{1,3,4,5},{2,6,7}}=>4
{{1,3,4,5},{2,6},{7}}=>4
{{1,3,4,5,7},{2},{6}}=>2
{{1,3,4,5},{2,7},{6}}=>1
{{1,3,4,5},{2},{6,7}}=>1
{{1,3,4,5},{2},{6},{7}}=>1
{{1,3,4,6,7},{2,5}}=>5
{{1,3,4,6},{2,5,7}}=>4
{{1,3,4,6},{2,5},{7}}=>3
{{1,3,4,7},{2,5,6}}=>4
{{1,3,4},{2,5,6,7}}=>3
{{1,3,4},{2,5,6},{7}}=>3
{{1,3,4,7},{2,5},{6}}=>4
{{1,3,4},{2,5,7},{6}}=>3
{{1,3,4},{2,5},{6,7}}=>3
{{1,3,4},{2,5},{6},{7}}=>3
{{1,3,4,6,7},{2},{5}}=>3
{{1,3,4,6},{2,7},{5}}=>2
{{1,3,4,6},{2},{5,7}}=>1
{{1,3,4,6},{2},{5},{7}}=>1
{{1,3,4,7},{2,6},{5}}=>2
{{1,3,4},{2,6,7},{5}}=>1
{{1,3,4},{2,6},{5,7}}=>1
{{1,3,4},{2,6},{5},{7}}=>1
{{1,3,4,7},{2},{5,6}}=>2
{{1,3,4},{2,7},{5,6}}=>1
{{1,3,4},{2},{5,6,7}}=>1
{{1,3,4},{2},{5,6},{7}}=>1
{{1,3,4,7},{2},{5},{6}}=>2
{{1,3,4},{2,7},{5},{6}}=>1
{{1,3,4},{2},{5,7},{6}}=>1
{{1,3,4},{2},{5},{6,7}}=>1
{{1,3,4},{2},{5},{6},{7}}=>1
{{1,3,5,6,7},{2,4}}=>5
{{1,3,5,6},{2,4,7}}=>4
{{1,3,5,6},{2,4},{7}}=>2
{{1,3,5,7},{2,4,6}}=>4
{{1,3,5},{2,4,6,7}}=>3
{{1,3,5},{2,4,6},{7}}=>3
{{1,3,5,7},{2,4},{6}}=>3
{{1,3,5},{2,4,7},{6}}=>2
{{1,3,5},{2,4},{6,7}}=>2
{{1,3,5},{2,4},{6},{7}}=>2
{{1,3,6,7},{2,4,5}}=>4
{{1,3,6},{2,4,5,7}}=>3
{{1,3,6},{2,4,5},{7}}=>2
{{1,3,7},{2,4,5,6}}=>3
{{1,3},{2,4,5,6,7}}=>2
{{1,3},{2,4,5,6},{7}}=>2
{{1,3,7},{2,4,5},{6}}=>3
{{1,3},{2,4,5,7},{6}}=>2
{{1,3},{2,4,5},{6,7}}=>2
{{1,3},{2,4,5},{6},{7}}=>2
{{1,3,6,7},{2,4},{5}}=>4
{{1,3,6},{2,4,7},{5}}=>3
{{1,3,6},{2,4},{5,7}}=>2
{{1,3,6},{2,4},{5},{7}}=>2
{{1,3,7},{2,4,6},{5}}=>3
{{1,3},{2,4,6,7},{5}}=>2
{{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}}=>2
{{1,3},{2,4},{5,6},{7}}=>2
{{1,3,7},{2,4},{5},{6}}=>3
{{1,3},{2,4,7},{5},{6}}=>2
{{1,3},{2,4},{5,7},{6}}=>2
{{1,3},{2,4},{5},{6,7}}=>2
{{1,3},{2,4},{5},{6},{7}}=>2
{{1,3,5,6,7},{2},{4}}=>4
{{1,3,5,6},{2,7},{4}}=>3
{{1,3,5,6},{2},{4,7}}=>1
{{1,3,5,6},{2},{4},{7}}=>1
{{1,3,5,7},{2,6},{4}}=>3
{{1,3,5},{2,6,7},{4}}=>2
{{1,3,5},{2,6},{4,7}}=>2
{{1,3,5},{2,6},{4},{7}}=>1
{{1,3,5,7},{2},{4,6}}=>2
{{1,3,5},{2,7},{4,6}}=>1
{{1,3,5},{2},{4,6,7}}=>1
{{1,3,5},{2},{4,6},{7}}=>1
{{1,3,5,7},{2},{4},{6}}=>2
{{1,3,5},{2,7},{4},{6}}=>1
{{1,3,5},{2},{4,7},{6}}=>1
{{1,3,5},{2},{4},{6,7}}=>1
{{1,3,5},{2},{4},{6},{7}}=>1
{{1,3,6,7},{2,5},{4}}=>3
{{1,3,6},{2,5,7},{4}}=>2
{{1,3,6},{2,5},{4,7}}=>1
{{1,3,6},{2,5},{4},{7}}=>1
{{1,3,7},{2,5,6},{4}}=>2
{{1,3},{2,5,6,7},{4}}=>1
{{1,3},{2,5,6},{4,7}}=>1
{{1,3},{2,5,6},{4},{7}}=>1
{{1,3,7},{2,5},{4,6}}=>2
{{1,3},{2,5,7},{4,6}}=>1
{{1,3},{2,5},{4,6,7}}=>1
{{1,3},{2,5},{4,6},{7}}=>1
{{1,3,7},{2,5},{4},{6}}=>2
{{1,3},{2,5,7},{4},{6}}=>1
{{1,3},{2,5},{4,7},{6}}=>1
{{1,3},{2,5},{4},{6,7}}=>1
{{1,3},{2,5},{4},{6},{7}}=>1
{{1,3,6,7},{2},{4,5}}=>3
{{1,3,6},{2,7},{4,5}}=>2
{{1,3,6},{2},{4,5,7}}=>1
{{1,3,6},{2},{4,5},{7}}=>1
{{1,3,7},{2,6},{4,5}}=>2
{{1,3},{2,6,7},{4,5}}=>1
{{1,3},{2,6},{4,5,7}}=>1
{{1,3},{2,6},{4,5},{7}}=>1
{{1,3,7},{2},{4,5,6}}=>2
{{1,3},{2,7},{4,5,6}}=>1
{{1,3},{2},{4,5,6,7}}=>1
{{1,3},{2},{4,5,6},{7}}=>1
{{1,3,7},{2},{4,5},{6}}=>2
{{1,3},{2,7},{4,5},{6}}=>1
{{1,3},{2},{4,5,7},{6}}=>1
{{1,3},{2},{4,5},{6,7}}=>1
{{1,3},{2},{4,5},{6},{7}}=>1
{{1,3,6,7},{2},{4},{5}}=>3
{{1,3,6},{2,7},{4},{5}}=>2
{{1,3,6},{2},{4,7},{5}}=>1
{{1,3,6},{2},{4},{5,7}}=>1
{{1,3,6},{2},{4},{5},{7}}=>1
{{1,3,7},{2,6},{4},{5}}=>2
{{1,3},{2,6,7},{4},{5}}=>1
{{1,3},{2,6},{4,7},{5}}=>1
{{1,3},{2,6},{4},{5,7}}=>1
{{1,3},{2,6},{4},{5},{7}}=>1
{{1,3,7},{2},{4,6},{5}}=>2
{{1,3},{2,7},{4,6},{5}}=>1
{{1,3},{2},{4,6,7},{5}}=>1
{{1,3},{2},{4,6},{5,7}}=>1
{{1,3},{2},{4,6},{5},{7}}=>1
{{1,3,7},{2},{4},{5,6}}=>2
{{1,3},{2,7},{4},{5,6}}=>1
{{1,3},{2},{4,7},{5,6}}=>1
{{1,3},{2},{4},{5,6,7}}=>1
{{1,3},{2},{4},{5,6},{7}}=>1
{{1,3,7},{2},{4},{5},{6}}=>2
{{1,3},{2,7},{4},{5},{6}}=>1
{{1,3},{2},{4,7},{5},{6}}=>1
{{1,3},{2},{4},{5,7},{6}}=>1
{{1,3},{2},{4},{5},{6,7}}=>1
{{1,3},{2},{4},{5},{6},{7}}=>1
{{1,4,5,6,7},{2,3}}=>5
{{1,4,5,6},{2,3,7}}=>4
{{1,4,5,6},{2,3},{7}}=>1
{{1,4,5,7},{2,3,6}}=>4
{{1,4,5},{2,3,6,7}}=>3
{{1,4,5},{2,3,6},{7}}=>3
{{1,4,5,7},{2,3},{6}}=>2
{{1,4,5},{2,3,7},{6}}=>1
{{1,4,5},{2,3},{6,7}}=>1
{{1,4,5},{2,3},{6},{7}}=>1
{{1,4,6,7},{2,3,5}}=>4
{{1,4,6},{2,3,5,7}}=>3
{{1,4,6},{2,3,5},{7}}=>2
{{1,4,7},{2,3,5,6}}=>3
{{1,4},{2,3,5,6,7}}=>2
{{1,4},{2,3,5,6},{7}}=>2
{{1,4,7},{2,3,5},{6}}=>3
{{1,4},{2,3,5,7},{6}}=>2
{{1,4},{2,3,5},{6,7}}=>2
{{1,4},{2,3,5},{6},{7}}=>2
{{1,4,6,7},{2,3},{5}}=>3
{{1,4,6},{2,3,7},{5}}=>2
{{1,4,6},{2,3},{5,7}}=>1
{{1,4,6},{2,3},{5},{7}}=>1
{{1,4,7},{2,3,6},{5}}=>2
{{1,4},{2,3,6,7},{5}}=>1
{{1,4},{2,3,6},{5,7}}=>1
{{1,4},{2,3,6},{5},{7}}=>1
{{1,4,7},{2,3},{5,6}}=>2
{{1,4},{2,3,7},{5,6}}=>1
{{1,4},{2,3},{5,6,7}}=>1
{{1,4},{2,3},{5,6},{7}}=>1
{{1,4,7},{2,3},{5},{6}}=>2
{{1,4},{2,3,7},{5},{6}}=>1
{{1,4},{2,3},{5,7},{6}}=>1
{{1,4},{2,3},{5},{6,7}}=>1
{{1,4},{2,3},{5},{6},{7}}=>1
{{1,5,6,7},{2,3,4}}=>4
{{1,5,6},{2,3,4,7}}=>3
{{1,5,6},{2,3,4},{7}}=>1
{{1,5,7},{2,3,4,6}}=>3
{{1,5},{2,3,4,6,7}}=>2
{{1,5},{2,3,4,6},{7}}=>2
{{1,5,7},{2,3,4},{6}}=>2
{{1,5},{2,3,4,7},{6}}=>1
{{1,5},{2,3,4},{6,7}}=>1
{{1,5},{2,3,4},{6},{7}}=>1
{{1,6,7},{2,3,4,5}}=>3
{{1,6},{2,3,4,5,7}}=>2
{{1,6},{2,3,4,5},{7}}=>1
{{1,7},{2,3,4,5,6}}=>2
{{1},{2,3,4,5,6,7}}=>1
{{1},{2,3,4,5,6},{7}}=>1
{{1,7},{2,3,4,5},{6}}=>2
{{1},{2,3,4,5,7},{6}}=>1
{{1},{2,3,4,5},{6,7}}=>1
{{1},{2,3,4,5},{6},{7}}=>1
{{1,6,7},{2,3,4},{5}}=>3
{{1,6},{2,3,4,7},{5}}=>2
{{1,6},{2,3,4},{5,7}}=>1
{{1,6},{2,3,4},{5},{7}}=>1
{{1,7},{2,3,4,6},{5}}=>2
{{1},{2,3,4,6,7},{5}}=>1
{{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}}=>1
{{1},{2,3,4},{5,6},{7}}=>1
{{1,7},{2,3,4},{5},{6}}=>2
{{1},{2,3,4,7},{5},{6}}=>1
{{1},{2,3,4},{5,7},{6}}=>1
{{1},{2,3,4},{5},{6,7}}=>1
{{1},{2,3,4},{5},{6},{7}}=>1
{{1,5,6,7},{2,3},{4}}=>4
{{1,5,6},{2,3,7},{4}}=>3
{{1,5,6},{2,3},{4,7}}=>1
{{1,5,6},{2,3},{4},{7}}=>1
{{1,5,7},{2,3,6},{4}}=>3
{{1,5},{2,3,6,7},{4}}=>2
{{1,5},{2,3,6},{4,7}}=>2
{{1,5},{2,3,6},{4},{7}}=>1
{{1,5,7},{2,3},{4,6}}=>2
{{1,5},{2,3,7},{4,6}}=>1
{{1,5},{2,3},{4,6,7}}=>1
{{1,5},{2,3},{4,6},{7}}=>1
{{1,5,7},{2,3},{4},{6}}=>2
{{1,5},{2,3,7},{4},{6}}=>1
{{1,5},{2,3},{4,7},{6}}=>1
{{1,5},{2,3},{4},{6,7}}=>1
{{1,5},{2,3},{4},{6},{7}}=>1
{{1,6,7},{2,3,5},{4}}=>3
{{1,6},{2,3,5,7},{4}}=>2
{{1,6},{2,3,5},{4,7}}=>1
{{1,6},{2,3,5},{4},{7}}=>1
{{1,7},{2,3,5,6},{4}}=>2
{{1},{2,3,5,6,7},{4}}=>1
{{1},{2,3,5,6},{4,7}}=>1
{{1},{2,3,5,6},{4},{7}}=>1
{{1,7},{2,3,5},{4,6}}=>2
{{1},{2,3,5,7},{4,6}}=>1
{{1},{2,3,5},{4,6,7}}=>1
{{1},{2,3,5},{4,6},{7}}=>1
{{1,7},{2,3,5},{4},{6}}=>2
{{1},{2,3,5,7},{4},{6}}=>1
{{1},{2,3,5},{4,7},{6}}=>1
{{1},{2,3,5},{4},{6,7}}=>1
{{1},{2,3,5},{4},{6},{7}}=>1
{{1,6,7},{2,3},{4,5}}=>3
{{1,6},{2,3,7},{4,5}}=>2
{{1,6},{2,3},{4,5,7}}=>1
{{1,6},{2,3},{4,5},{7}}=>1
{{1,7},{2,3,6},{4,5}}=>2
{{1},{2,3,6,7},{4,5}}=>1
{{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}}=>1
{{1},{2,3},{4,5,6},{7}}=>1
{{1,7},{2,3},{4,5},{6}}=>2
{{1},{2,3,7},{4,5},{6}}=>1
{{1},{2,3},{4,5,7},{6}}=>1
{{1},{2,3},{4,5},{6,7}}=>1
{{1},{2,3},{4,5},{6},{7}}=>1
{{1,6,7},{2,3},{4},{5}}=>3
{{1,6},{2,3,7},{4},{5}}=>2
{{1,6},{2,3},{4,7},{5}}=>1
{{1,6},{2,3},{4},{5,7}}=>1
{{1,6},{2,3},{4},{5},{7}}=>1
{{1,7},{2,3,6},{4},{5}}=>2
{{1},{2,3,6,7},{4},{5}}=>1
{{1},{2,3,6},{4,7},{5}}=>1
{{1},{2,3,6},{4},{5,7}}=>1
{{1},{2,3,6},{4},{5},{7}}=>1
{{1,7},{2,3},{4,6},{5}}=>2
{{1},{2,3,7},{4,6},{5}}=>1
{{1},{2,3},{4,6,7},{5}}=>1
{{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},{2,3,7},{4},{5,6}}=>1
{{1},{2,3},{4,7},{5,6}}=>1
{{1},{2,3},{4},{5,6,7}}=>1
{{1},{2,3},{4},{5,6},{7}}=>1
{{1,7},{2,3},{4},{5},{6}}=>2
{{1},{2,3,7},{4},{5},{6}}=>1
{{1},{2,3},{4,7},{5},{6}}=>1
{{1},{2,3},{4},{5,7},{6}}=>1
{{1},{2,3},{4},{5},{6,7}}=>1
{{1},{2,3},{4},{5},{6},{7}}=>1
{{1,4,5,6,7},{2},{3}}=>5
{{1,4,5,6},{2,7},{3}}=>4
{{1,4,5,6},{2},{3,7}}=>1
{{1,4,5,6},{2},{3},{7}}=>1
{{1,4,5,7},{2,6},{3}}=>4
{{1,4,5},{2,6,7},{3}}=>3
{{1,4,5},{2,6},{3,7}}=>3
{{1,4,5},{2,6},{3},{7}}=>1
{{1,4,5,7},{2},{3,6}}=>2
{{1,4,5},{2,7},{3,6}}=>1
{{1,4,5},{2},{3,6,7}}=>1
{{1,4,5},{2},{3,6},{7}}=>1
{{1,4,5,7},{2},{3},{6}}=>2
{{1,4,5},{2,7},{3},{6}}=>1
{{1,4,5},{2},{3,7},{6}}=>1
{{1,4,5},{2},{3},{6,7}}=>1
{{1,4,5},{2},{3},{6},{7}}=>1
{{1,4,6,7},{2,5},{3}}=>4
{{1,4,6},{2,5,7},{3}}=>3
{{1,4,6},{2,5},{3,7}}=>2
{{1,4,6},{2,5},{3},{7}}=>1
{{1,4,7},{2,5,6},{3}}=>3
{{1,4},{2,5,6,7},{3}}=>2
{{1,4},{2,5,6},{3,7}}=>2
{{1,4},{2,5,6},{3},{7}}=>1
{{1,4,7},{2,5},{3,6}}=>3
{{1,4},{2,5,7},{3,6}}=>2
{{1,4},{2,5},{3,6,7}}=>2
{{1,4},{2,5},{3,6},{7}}=>2
{{1,4,7},{2,5},{3},{6}}=>2
{{1,4},{2,5,7},{3},{6}}=>1
{{1,4},{2,5},{3,7},{6}}=>1
{{1,4},{2,5},{3},{6,7}}=>1
{{1,4},{2,5},{3},{6},{7}}=>1
{{1,4,6,7},{2},{3,5}}=>3
{{1,4,6},{2,7},{3,5}}=>2
{{1,4,6},{2},{3,5,7}}=>1
{{1,4,6},{2},{3,5},{7}}=>1
{{1,4,7},{2,6},{3,5}}=>2
{{1,4},{2,6,7},{3,5}}=>1
{{1,4},{2,6},{3,5,7}}=>1
{{1,4},{2,6},{3,5},{7}}=>1
{{1,4,7},{2},{3,5,6}}=>2
{{1,4},{2,7},{3,5,6}}=>1
{{1,4},{2},{3,5,6,7}}=>1
{{1,4},{2},{3,5,6},{7}}=>1
{{1,4,7},{2},{3,5},{6}}=>2
{{1,4},{2,7},{3,5},{6}}=>1
{{1,4},{2},{3,5,7},{6}}=>1
{{1,4},{2},{3,5},{6,7}}=>1
{{1,4},{2},{3,5},{6},{7}}=>1
{{1,4,6,7},{2},{3},{5}}=>3
{{1,4,6},{2,7},{3},{5}}=>2
{{1,4,6},{2},{3,7},{5}}=>1
{{1,4,6},{2},{3},{5,7}}=>1
{{1,4,6},{2},{3},{5},{7}}=>1
{{1,4,7},{2,6},{3},{5}}=>2
{{1,4},{2,6,7},{3},{5}}=>1
{{1,4},{2,6},{3,7},{5}}=>1
{{1,4},{2,6},{3},{5,7}}=>1
{{1,4},{2,6},{3},{5},{7}}=>1
{{1,4,7},{2},{3,6},{5}}=>2
{{1,4},{2,7},{3,6},{5}}=>1
{{1,4},{2},{3,6,7},{5}}=>1
{{1,4},{2},{3,6},{5,7}}=>1
{{1,4},{2},{3,6},{5},{7}}=>1
{{1,4,7},{2},{3},{5,6}}=>2
{{1,4},{2,7},{3},{5,6}}=>1
{{1,4},{2},{3,7},{5,6}}=>1
{{1,4},{2},{3},{5,6,7}}=>1
{{1,4},{2},{3},{5,6},{7}}=>1
{{1,4,7},{2},{3},{5},{6}}=>2
{{1,4},{2,7},{3},{5},{6}}=>1
{{1,4},{2},{3,7},{5},{6}}=>1
{{1,4},{2},{3},{5,7},{6}}=>1
{{1,4},{2},{3},{5},{6,7}}=>1
{{1,4},{2},{3},{5},{6},{7}}=>1
{{1,5,6,7},{2,4},{3}}=>4
{{1,5,6},{2,4,7},{3}}=>3
{{1,5,6},{2,4},{3,7}}=>1
{{1,5,6},{2,4},{3},{7}}=>1
{{1,5,7},{2,4,6},{3}}=>3
{{1,5},{2,4,6,7},{3}}=>2
{{1,5},{2,4,6},{3,7}}=>2
{{1,5},{2,4,6},{3},{7}}=>1
{{1,5,7},{2,4},{3,6}}=>2
{{1,5},{2,4,7},{3,6}}=>1
{{1,5},{2,4},{3,6,7}}=>1
{{1,5},{2,4},{3,6},{7}}=>1
{{1,5,7},{2,4},{3},{6}}=>2
{{1,5},{2,4,7},{3},{6}}=>1
{{1,5},{2,4},{3,7},{6}}=>1
{{1,5},{2,4},{3},{6,7}}=>1
{{1,5},{2,4},{3},{6},{7}}=>1
{{1,6,7},{2,4,5},{3}}=>3
{{1,6},{2,4,5,7},{3}}=>2
{{1,6},{2,4,5},{3,7}}=>1
{{1,6},{2,4,5},{3},{7}}=>1
{{1,7},{2,4,5,6},{3}}=>2
{{1},{2,4,5,6,7},{3}}=>1
{{1},{2,4,5,6},{3,7}}=>1
{{1},{2,4,5,6},{3},{7}}=>1
{{1,7},{2,4,5},{3,6}}=>2
{{1},{2,4,5,7},{3,6}}=>1
{{1},{2,4,5},{3,6,7}}=>1
{{1},{2,4,5},{3,6},{7}}=>1
{{1,7},{2,4,5},{3},{6}}=>2
{{1},{2,4,5,7},{3},{6}}=>1
{{1},{2,4,5},{3,7},{6}}=>1
{{1},{2,4,5},{3},{6,7}}=>1
{{1},{2,4,5},{3},{6},{7}}=>1
{{1,6,7},{2,4},{3,5}}=>3
{{1,6},{2,4,7},{3,5}}=>2
{{1,6},{2,4},{3,5,7}}=>1
{{1,6},{2,4},{3,5},{7}}=>1
{{1,7},{2,4,6},{3,5}}=>2
{{1},{2,4,6,7},{3,5}}=>1
{{1},{2,4,6},{3,5,7}}=>1
{{1},{2,4,6},{3,5},{7}}=>1
{{1,7},{2,4},{3,5,6}}=>2
{{1},{2,4,7},{3,5,6}}=>1
{{1},{2,4},{3,5,6,7}}=>1
{{1},{2,4},{3,5,6},{7}}=>1
{{1,7},{2,4},{3,5},{6}}=>2
{{1},{2,4,7},{3,5},{6}}=>1
{{1},{2,4},{3,5,7},{6}}=>1
{{1},{2,4},{3,5},{6,7}}=>1
{{1},{2,4},{3,5},{6},{7}}=>1
{{1,6,7},{2,4},{3},{5}}=>3
{{1,6},{2,4,7},{3},{5}}=>2
{{1,6},{2,4},{3,7},{5}}=>1
{{1,6},{2,4},{3},{5,7}}=>1
{{1,6},{2,4},{3},{5},{7}}=>1
{{1,7},{2,4,6},{3},{5}}=>2
{{1},{2,4,6,7},{3},{5}}=>1
{{1},{2,4,6},{3,7},{5}}=>1
{{1},{2,4,6},{3},{5,7}}=>1
{{1},{2,4,6},{3},{5},{7}}=>1
{{1,7},{2,4},{3,6},{5}}=>2
{{1},{2,4,7},{3,6},{5}}=>1
{{1},{2,4},{3,6,7},{5}}=>1
{{1},{2,4},{3,6},{5,7}}=>1
{{1},{2,4},{3,6},{5},{7}}=>1
{{1,7},{2,4},{3},{5,6}}=>2
{{1},{2,4,7},{3},{5,6}}=>1
{{1},{2,4},{3,7},{5,6}}=>1
{{1},{2,4},{3},{5,6,7}}=>1
{{1},{2,4},{3},{5,6},{7}}=>1
{{1,7},{2,4},{3},{5},{6}}=>2
{{1},{2,4,7},{3},{5},{6}}=>1
{{1},{2,4},{3,7},{5},{6}}=>1
{{1},{2,4},{3},{5,7},{6}}=>1
{{1},{2,4},{3},{5},{6,7}}=>1
{{1},{2,4},{3},{5},{6},{7}}=>1
{{1,5,6,7},{2},{3,4}}=>4
{{1,5,6},{2,7},{3,4}}=>3
{{1,5,6},{2},{3,4,7}}=>1
{{1,5,6},{2},{3,4},{7}}=>1
{{1,5,7},{2,6},{3,4}}=>3
{{1,5},{2,6,7},{3,4}}=>2
{{1,5},{2,6},{3,4,7}}=>2
{{1,5},{2,6},{3,4},{7}}=>1
{{1,5,7},{2},{3,4,6}}=>2
{{1,5},{2,7},{3,4,6}}=>1
{{1,5},{2},{3,4,6,7}}=>1
{{1,5},{2},{3,4,6},{7}}=>1
{{1,5,7},{2},{3,4},{6}}=>2
{{1,5},{2,7},{3,4},{6}}=>1
{{1,5},{2},{3,4,7},{6}}=>1
{{1,5},{2},{3,4},{6,7}}=>1
{{1,5},{2},{3,4},{6},{7}}=>1
{{1,6,7},{2,5},{3,4}}=>3
{{1,6},{2,5,7},{3,4}}=>2
{{1,6},{2,5},{3,4,7}}=>1
{{1,6},{2,5},{3,4},{7}}=>1
{{1,7},{2,5,6},{3,4}}=>2
{{1},{2,5,6,7},{3,4}}=>1
{{1},{2,5,6},{3,4,7}}=>1
{{1},{2,5,6},{3,4},{7}}=>1
{{1,7},{2,5},{3,4,6}}=>2
{{1},{2,5,7},{3,4,6}}=>1
{{1},{2,5},{3,4,6,7}}=>1
{{1},{2,5},{3,4,6},{7}}=>1
{{1,7},{2,5},{3,4},{6}}=>2
{{1},{2,5,7},{3,4},{6}}=>1
{{1},{2,5},{3,4,7},{6}}=>1
{{1},{2,5},{3,4},{6,7}}=>1
{{1},{2,5},{3,4},{6},{7}}=>1
{{1,6,7},{2},{3,4,5}}=>3
{{1,6},{2,7},{3,4,5}}=>2
{{1,6},{2},{3,4,5,7}}=>1
{{1,6},{2},{3,4,5},{7}}=>1
{{1,7},{2,6},{3,4,5}}=>2
{{1},{2,6,7},{3,4,5}}=>1
{{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}}=>1
{{1},{2},{3,4,5,6},{7}}=>1
{{1,7},{2},{3,4,5},{6}}=>2
{{1},{2,7},{3,4,5},{6}}=>1
{{1},{2},{3,4,5,7},{6}}=>1
{{1},{2},{3,4,5},{6,7}}=>1
{{1},{2},{3,4,5},{6},{7}}=>1
{{1,6,7},{2},{3,4},{5}}=>3
{{1,6},{2,7},{3,4},{5}}=>2
{{1,6},{2},{3,4,7},{5}}=>1
{{1,6},{2},{3,4},{5,7}}=>1
{{1,6},{2},{3,4},{5},{7}}=>1
{{1,7},{2,6},{3,4},{5}}=>2
{{1},{2,6,7},{3,4},{5}}=>1
{{1},{2,6},{3,4,7},{5}}=>1
{{1},{2,6},{3,4},{5,7}}=>1
{{1},{2,6},{3,4},{5},{7}}=>1
{{1,7},{2},{3,4,6},{5}}=>2
{{1},{2,7},{3,4,6},{5}}=>1
{{1},{2},{3,4,6,7},{5}}=>1
{{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},{2,7},{3,4},{5,6}}=>1
{{1},{2},{3,4,7},{5,6}}=>1
{{1},{2},{3,4},{5,6,7}}=>1
{{1},{2},{3,4},{5,6},{7}}=>1
{{1,7},{2},{3,4},{5},{6}}=>2
{{1},{2,7},{3,4},{5},{6}}=>1
{{1},{2},{3,4,7},{5},{6}}=>1
{{1},{2},{3,4},{5,7},{6}}=>1
{{1},{2},{3,4},{5},{6,7}}=>1
{{1},{2},{3,4},{5},{6},{7}}=>1
{{1,5,6,7},{2},{3},{4}}=>4
{{1,5,6},{2,7},{3},{4}}=>3
{{1,5,6},{2},{3,7},{4}}=>1
{{1,5,6},{2},{3},{4,7}}=>1
{{1,5,6},{2},{3},{4},{7}}=>1
{{1,5,7},{2,6},{3},{4}}=>3
{{1,5},{2,6,7},{3},{4}}=>2
{{1,5},{2,6},{3,7},{4}}=>2
{{1,5},{2,6},{3},{4,7}}=>1
{{1,5},{2,6},{3},{4},{7}}=>1
{{1,5,7},{2},{3,6},{4}}=>2
{{1,5},{2,7},{3,6},{4}}=>1
{{1,5},{2},{3,6,7},{4}}=>1
{{1,5},{2},{3,6},{4,7}}=>1
{{1,5},{2},{3,6},{4},{7}}=>1
{{1,5,7},{2},{3},{4,6}}=>2
{{1,5},{2,7},{3},{4,6}}=>1
{{1,5},{2},{3,7},{4,6}}=>1
{{1,5},{2},{3},{4,6,7}}=>1
{{1,5},{2},{3},{4,6},{7}}=>1
{{1,5,7},{2},{3},{4},{6}}=>2
{{1,5},{2,7},{3},{4},{6}}=>1
{{1,5},{2},{3,7},{4},{6}}=>1
{{1,5},{2},{3},{4,7},{6}}=>1
{{1,5},{2},{3},{4},{6,7}}=>1
{{1,5},{2},{3},{4},{6},{7}}=>1
{{1,6,7},{2,5},{3},{4}}=>3
{{1,6},{2,5,7},{3},{4}}=>2
{{1,6},{2,5},{3,7},{4}}=>1
{{1,6},{2,5},{3},{4,7}}=>1
{{1,6},{2,5},{3},{4},{7}}=>1
{{1,7},{2,5,6},{3},{4}}=>2
{{1},{2,5,6,7},{3},{4}}=>1
{{1},{2,5,6},{3,7},{4}}=>1
{{1},{2,5,6},{3},{4,7}}=>1
{{1},{2,5,6},{3},{4},{7}}=>1
{{1,7},{2,5},{3,6},{4}}=>2
{{1},{2,5,7},{3,6},{4}}=>1
{{1},{2,5},{3,6,7},{4}}=>1
{{1},{2,5},{3,6},{4,7}}=>1
{{1},{2,5},{3,6},{4},{7}}=>1
{{1,7},{2,5},{3},{4,6}}=>2
{{1},{2,5,7},{3},{4,6}}=>1
{{1},{2,5},{3,7},{4,6}}=>1
{{1},{2,5},{3},{4,6,7}}=>1
{{1},{2,5},{3},{4,6},{7}}=>1
{{1,7},{2,5},{3},{4},{6}}=>2
{{1},{2,5,7},{3},{4},{6}}=>1
{{1},{2,5},{3,7},{4},{6}}=>1
{{1},{2,5},{3},{4,7},{6}}=>1
{{1},{2,5},{3},{4},{6,7}}=>1
{{1},{2,5},{3},{4},{6},{7}}=>1
{{1,6,7},{2},{3,5},{4}}=>3
{{1,6},{2,7},{3,5},{4}}=>2
{{1,6},{2},{3,5,7},{4}}=>1
{{1,6},{2},{3,5},{4,7}}=>1
{{1,6},{2},{3,5},{4},{7}}=>1
{{1,7},{2,6},{3,5},{4}}=>2
{{1},{2,6,7},{3,5},{4}}=>1
{{1},{2,6},{3,5,7},{4}}=>1
{{1},{2,6},{3,5},{4,7}}=>1
{{1},{2,6},{3,5},{4},{7}}=>1
{{1,7},{2},{3,5,6},{4}}=>2
{{1},{2,7},{3,5,6},{4}}=>1
{{1},{2},{3,5,6,7},{4}}=>1
{{1},{2},{3,5,6},{4,7}}=>1
{{1},{2},{3,5,6},{4},{7}}=>1
{{1,7},{2},{3,5},{4,6}}=>2
{{1},{2,7},{3,5},{4,6}}=>1
{{1},{2},{3,5,7},{4,6}}=>1
{{1},{2},{3,5},{4,6,7}}=>1
{{1},{2},{3,5},{4,6},{7}}=>1
{{1,7},{2},{3,5},{4},{6}}=>2
{{1},{2,7},{3,5},{4},{6}}=>1
{{1},{2},{3,5,7},{4},{6}}=>1
{{1},{2},{3,5},{4,7},{6}}=>1
{{1},{2},{3,5},{4},{6,7}}=>1
{{1},{2},{3,5},{4},{6},{7}}=>1
{{1,6,7},{2},{3},{4,5}}=>3
{{1,6},{2,7},{3},{4,5}}=>2
{{1,6},{2},{3,7},{4,5}}=>1
{{1,6},{2},{3},{4,5,7}}=>1
{{1,6},{2},{3},{4,5},{7}}=>1
{{1,7},{2,6},{3},{4,5}}=>2
{{1},{2,6,7},{3},{4,5}}=>1
{{1},{2,6},{3,7},{4,5}}=>1
{{1},{2,6},{3},{4,5,7}}=>1
{{1},{2,6},{3},{4,5},{7}}=>1
{{1,7},{2},{3,6},{4,5}}=>2
{{1},{2,7},{3,6},{4,5}}=>1
{{1},{2},{3,6,7},{4,5}}=>1
{{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},{2,7},{3},{4,5,6}}=>1
{{1},{2},{3,7},{4,5,6}}=>1
{{1},{2},{3},{4,5,6,7}}=>1
{{1},{2},{3},{4,5,6},{7}}=>1
{{1,7},{2},{3},{4,5},{6}}=>2
{{1},{2,7},{3},{4,5},{6}}=>1
{{1},{2},{3,7},{4,5},{6}}=>1
{{1},{2},{3},{4,5,7},{6}}=>1
{{1},{2},{3},{4,5},{6,7}}=>1
{{1},{2},{3},{4,5},{6},{7}}=>1
{{1,6,7},{2},{3},{4},{5}}=>3
{{1,6},{2,7},{3},{4},{5}}=>2
{{1,6},{2},{3,7},{4},{5}}=>1
{{1,6},{2},{3},{4,7},{5}}=>1
{{1,6},{2},{3},{4},{5,7}}=>1
{{1,6},{2},{3},{4},{5},{7}}=>1
{{1,7},{2,6},{3},{4},{5}}=>2
{{1},{2,6,7},{3},{4},{5}}=>1
{{1},{2,6},{3,7},{4},{5}}=>1
{{1},{2,6},{3},{4,7},{5}}=>1
{{1},{2,6},{3},{4},{5,7}}=>1
{{1},{2,6},{3},{4},{5},{7}}=>1
{{1,7},{2},{3,6},{4},{5}}=>2
{{1},{2,7},{3,6},{4},{5}}=>1
{{1},{2},{3,6,7},{4},{5}}=>1
{{1},{2},{3,6},{4,7},{5}}=>1
{{1},{2},{3,6},{4},{5,7}}=>1
{{1},{2},{3,6},{4},{5},{7}}=>1
{{1,7},{2},{3},{4,6},{5}}=>2
{{1},{2,7},{3},{4,6},{5}}=>1
{{1},{2},{3,7},{4,6},{5}}=>1
{{1},{2},{3},{4,6,7},{5}}=>1
{{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},{2,7},{3},{4},{5,6}}=>1
{{1},{2},{3,7},{4},{5,6}}=>1
{{1},{2},{3},{4,7},{5,6}}=>1
{{1},{2},{3},{4},{5,6,7}}=>1
{{1},{2},{3},{4},{5,6},{7}}=>1
{{1,7},{2},{3},{4},{5},{6}}=>2
{{1},{2,7},{3},{4},{5},{6}}=>1
{{1},{2},{3,7},{4},{5},{6}}=>1
{{1},{2},{3},{4,7},{5},{6}}=>1
{{1},{2},{3},{4},{5,7},{6}}=>1
{{1},{2},{3},{4},{5},{6,7}}=>1
{{1},{2},{3},{4},{5},{6},{7}}=>1
{{1},{2},{3,4,5,6,7,8}}=>1
{{1},{2,4,5,6,7,8},{3}}=>1
{{1},{2,3,5,6,7,8},{4}}=>1
{{1},{2,3,4,6,7,8},{5}}=>1
{{1},{2,3,4,5,7,8},{6}}=>1
{{1},{2,3,4,5,6,7},{8}}=>1
{{1},{2,3,4,5,6,8},{7}}=>1
{{1},{2,3,4,5,6,7,8}}=>1
{{1,2},{3,4,5,6,7,8}}=>2
{{1,4,5,6,7,8},{2},{3}}=>6
{{1,3,5,6,7,8},{2},{4}}=>5
{{1,3,4,5,6,7,8},{2}}=>7
{{1,4,5,6,7,8},{2,3}}=>6
{{1,2,4,5,6,7,8},{3}}=>7
{{1,2,5,6,7,8},{3,4}}=>6
{{1,2,3,5,6,7,8},{4}}=>7
{{1,2,3,6,7,8},{4,5}}=>6
{{1,2,3,4,6,7,8},{5}}=>7
{{1,2,3,4,5,6},{7,8}}=>6
{{1,2,3,4,7,8},{5,6}}=>6
{{1,2,3,4,5,7,8},{6}}=>7
{{1,2,3,4,5,6,7},{8}}=>7
{{1,8},{2,3,4,5,6,7}}=>2
{{1,2,3,4,5,8},{6,7}}=>6
{{1,2,3,4,5,6,8},{7}}=>7
{{1,2,3,4,5,6,7,8}}=>8
{{1,3,5,6,7,8},{2,4}}=>6
{{1,3,4,6,7,8},{2,5}}=>6
{{1,2,4,6,7,8},{3,5}}=>6
{{1,3,4,5,7,8},{2,6}}=>6
{{1,2,4,5,7,8},{3,6}}=>6
{{1,2,3,5,7,8},{4,6}}=>6
{{1,3,4,5,6,8},{2,7}}=>6
{{1,2,4,5,6,8},{3,7}}=>6
{{1,2,3,5,6,8},{4,7}}=>6
{{1,2,3,4,6,8},{5,7}}=>6
{{1,3,4,5,6,7},{2,8}}=>6
{{1,2,4,5,6,7},{3,8}}=>6
{{1,2,3,5,6,7},{4,8}}=>6
{{1,2,3,4,6,7},{5,8}}=>6
{{1,2,3,4,5,7},{6,8}}=>6
{{1,3},{2,4,5,6,7,8}}=>2
{{1,4},{2,3,5,6,7,8}}=>2
{{1,5},{2,3,4,6,7,8}}=>2
{{1,6},{2,3,4,5,7,8}}=>2
{{1,7},{2,3,4,5,6,8}}=>2
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Description
The number of unsplittable factors of the set partition.
Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks
$$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$
if $k\leq \ell$ and
$$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$
otherwise.
Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks
$$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$
if $k\leq \ell$ and
$$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$
otherwise.
References
[1] Chen, W. Y. C., Li, T. X. S., Wang, D. G. L. A bijection between atomic partitions and unsplitable partitions MathSciNet:2770112
[2] Triangle read by rows: T(n,k) = number of noncommutative symmetric polynomials of degree n that have exactly k different variables appearing in each monomial and which generate the algebra of all noncommutative symmetric polynomials (n >= 1, 1 <= k <= n). OEIS:A055105
[2] Triangle read by rows: T(n,k) = number of noncommutative symmetric polynomials of degree n that have exactly k different variables appearing in each monomial and which generate the algebra of all noncommutative symmetric polynomials (n >= 1, 1 <= k <= n). OEIS:A055105
Code
def statistic(pi):
return len(split(pi))
def split_product(pi, sigma):
"""
An associative product.
sage: split_product([[1,2]],[[1,3],[2]])
{{1, 2, 3, 5}, {4}}
sage: split_product([[1,3],[2]],[[1,2]])
{{1, 3, 4, 5}, {2}}
"""
pi = sorted([sorted(B) for B in pi])
sigma = sorted([sorted(B) for B in sigma])
k = len(pi)
l = len(sigma)
m = SetPartition(pi).size()
if k <= l:
return SetPartition([B + [c+m for c in C] for B, C in zip(pi, sigma)]
+ [[c+m for c in C] for C in sigma[k:]])
else:
return SetPartition([B + [c+m for c in C] for B, C in zip(pi, sigma)]
+ pi[l:])
@cached_function
def unsplittable(n):
"""
sage: [len(unsplittable(n)) for n in range(1,8)]
[1, 1, 2, 6, 22, 92, 426]
sage: oeis(_)
0: A074664: Number of algebraically independent elements of degree n in the algebra of symmetric polynomials in noncommuting variables.
"""
return Set(SetPartitions(n)).difference(Set([split_product(pi, sigma) for k in range(1,n) for pi in SetPartitions(k) for sigma in SetPartitions(n-k)]))
def split(pi):
pi = SetPartition(pi)
n = pi.size()
if pi in unsplittable(n):
return [pi]
for l in range(1, n):
for sigma in unsplittable(l):
for kappa in SetPartitions(n-l):
if pi == split_product(sigma, kappa):
return [sigma] + split(kappa)
Created
May 30, 2017 at 09:43 by Martin Rubey
Updated
Nov 18, 2017 at 22:17 by Martin Rubey
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