Identifier
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Images
=>
Cc0005;cc-rep-1Cc0009;cc-rep-2Cc0009;cc-rep-3
[1]=>[1,0]=>{{1}}=>{{1}}
[1,1]=>[1,0,1,0]=>{{1},{2}}=>{{1},{2}}
[2]=>[1,1,0,0]=>{{1,2}}=>{{1,2}}
[1,1,1]=>[1,0,1,0,1,0]=>{{1},{2},{3}}=>{{1},{2},{3}}
[1,2]=>[1,0,1,1,0,0]=>{{1},{2,3}}=>{{1,2},{3}}
[2,1]=>[1,1,0,0,1,0]=>{{1,2},{3}}=>{{1},{2,3}}
[3]=>[1,1,1,0,0,0]=>{{1,2,3}}=>{{1,2,3}}
[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4}}=>{{1},{2},{3},{4}}
[1,1,2]=>[1,0,1,0,1,1,0,0]=>{{1},{2},{3,4}}=>{{1,2},{3},{4}}
[1,2,1]=>[1,0,1,1,0,0,1,0]=>{{1},{2,3},{4}}=>{{1},{2,3},{4}}
[1,3]=>[1,0,1,1,1,0,0,0]=>{{1},{2,3,4}}=>{{1,2,3},{4}}
[2,1,1]=>[1,1,0,0,1,0,1,0]=>{{1,2},{3},{4}}=>{{1},{2},{3,4}}
[2,2]=>[1,1,0,0,1,1,0,0]=>{{1,2},{3,4}}=>{{1,2},{3,4}}
[3,1]=>[1,1,1,0,0,0,1,0]=>{{1,2,3},{4}}=>{{1},{2,3,4}}
[4]=>[1,1,1,1,0,0,0,0]=>{{1,2,3,4}}=>{{1,2,3,4}}
[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5}}=>{{1},{2},{3},{4},{5}}
[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4,5}}=>{{1,2},{3},{4},{5}}
[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3,4},{5}}=>{{1},{2,3},{4},{5}}
[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3,4,5}}=>{{1,2,3},{4},{5}}
[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>{{1},{2,3},{4},{5}}=>{{1},{2},{3,4},{5}}
[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>{{1},{2,3},{4,5}}=>{{1,2},{3,4},{5}}
[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>{{1},{2,3,4},{5}}=>{{1},{2,3,4},{5}}
[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>{{1},{2,3,4,5}}=>{{1,2,3,4},{5}}
[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5}}=>{{1},{2},{3},{4,5}}
[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>{{1,2},{3},{4,5}}=>{{1,2},{3},{4,5}}
[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>{{1,2},{3,4},{5}}=>{{1},{2,3},{4,5}}
[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3,4,5}}=>{{1,2,3},{4,5}}
[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>{{1,2,3},{4},{5}}=>{{1},{2},{3,4,5}}
[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>{{1,2,3},{4,5}}=>{{1,2},{3,4,5}}
[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>{{1,2,3,4},{5}}=>{{1},{2,3,4,5}}
[5]=>[1,1,1,1,1,0,0,0,0,0]=>{{1,2,3,4,5}}=>{{1,2,3,4,5}}
[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6}}=>{{1},{2},{3},{4},{5},{6}}
[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5,6}}=>{{1,2},{3},{4},{5},{6}}
[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4,5},{6}}=>{{1},{2,3},{4},{5},{6}}
[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4,5,6}}=>{{1,2,3},{4},{5},{6}}
[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3,4},{5},{6}}=>{{1},{2},{3,4},{5},{6}}
[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3,4},{5,6}}=>{{1,2},{3,4},{5},{6}}
[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3,4,5},{6}}=>{{1},{2,3,4},{5},{6}}
[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3,4,5,6}}=>{{1,2,3,4},{5},{6}}
[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2,3},{4},{5},{6}}=>{{1},{2},{3},{4,5},{6}}
[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5,6}}=>{{1,2},{3},{4,5},{6}}
[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2,3},{4,5},{6}}=>{{1},{2,3},{4,5},{6}}
[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2,3},{4,5,6}}=>{{1,2,3},{4,5},{6}}
[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>{{1},{2,3,4},{5},{6}}=>{{1},{2},{3,4,5},{6}}
[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>{{1},{2,3,4},{5,6}}=>{{1,2},{3,4,5},{6}}
[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>{{1},{2,3,4,5},{6}}=>{{1},{2,3,4,5},{6}}
[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2,3,4,5,6}}=>{{1,2,3,4,5},{6}}
[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6}}=>{{1},{2},{3},{4},{5,6}}
[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5,6}}=>{{1,2},{3},{4},{5,6}}
[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4,5},{6}}=>{{1},{2,3},{4},{5,6}}
[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>{{1,2},{3},{4,5,6}}=>{{1,2,3},{4},{5,6}}
[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>{{1,2},{3,4},{5},{6}}=>{{1},{2},{3,4},{5,6}}
[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2},{3,4},{5,6}}=>{{1,2},{3,4},{5,6}}
[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>{{1,2},{3,4,5},{6}}=>{{1},{2,3,4},{5,6}}
[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4,5,6}}=>{{1,2,3,4},{5,6}}
[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6}}=>{{1},{2},{3},{4,5,6}}
[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>{{1,2,3},{4},{5,6}}=>{{1,2},{3},{4,5,6}}
[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>{{1,2,3},{4,5},{6}}=>{{1},{2,3},{4,5,6}}
[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>{{1,2,3},{4,5,6}}=>{{1,2,3},{4,5,6}}
[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>{{1,2,3,4},{5},{6}}=>{{1},{2},{3,4,5,6}}
[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>{{1,2,3,4},{5,6}}=>{{1,2},{3,4,5,6}}
[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>{{1,2,3,4,5},{6}}=>{{1},{2,3,4,5,6}}
[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2,3,4,5,6}}=>{{1,2,3,4,5,6}}
[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7}}=>{{1},{2},{3},{4},{5},{6},{7}}
[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6,7}}=>{{1,2},{3},{4},{5},{6},{7}}
[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4},{5,6},{7}}=>{{1},{2,3},{4},{5},{6},{7}}
[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5,6,7}}=>{{1,2,3},{4},{5},{6},{7}}
[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3},{4,5},{6},{7}}=>{{1},{2},{3,4},{5},{6},{7}}
[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4,5},{6,7}}=>{{1,2},{3,4},{5},{6},{7}}
[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3},{4,5,6},{7}}=>{{1},{2,3,4},{5},{6},{7}}
[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4,5,6,7}}=>{{1,2,3,4},{5},{6},{7}}
[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2},{3,4},{5},{6},{7}}=>{{1},{2},{3},{4,5},{6},{7}}
[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>{{1},{2},{3,4},{5},{6,7}}=>{{1,2},{3},{4,5},{6},{7}}
[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2},{3,4},{5,6},{7}}=>{{1},{2,3},{4,5},{6},{7}}
[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2},{3,4},{5,6,7}}=>{{1,2,3},{4,5},{6},{7}}
[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>{{1},{2},{3,4,5},{6},{7}}=>{{1},{2},{3,4,5},{6},{7}}
[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>{{1},{2},{3,4,5},{6,7}}=>{{1,2},{3,4,5},{6},{7}}
[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>{{1},{2},{3,4,5,6},{7}}=>{{1},{2,3,4,5},{6},{7}}
[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7}}=>{{1,2,3,4,5},{6},{7}}
[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1},{2,3},{4},{5},{6},{7}}=>{{1},{2},{3},{4},{5,6},{7}}
[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5},{6,7}}=>{{1,2},{3},{4},{5,6},{7}}
[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>{{1},{2,3},{4},{5,6},{7}}=>{{1},{2,3},{4},{5,6},{7}}
[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>{{1},{2,3},{4},{5,6,7}}=>{{1,2,3},{4},{5,6},{7}}
[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>{{1},{2,3},{4,5},{6},{7}}=>{{1},{2},{3,4},{5,6},{7}}
[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2,3},{4,5},{6,7}}=>{{1,2},{3,4},{5,6},{7}}
[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>{{1},{2,3},{4,5,6},{7}}=>{{1},{2,3,4},{5,6},{7}}
[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1},{2,3},{4,5,6,7}}=>{{1,2,3,4},{5,6},{7}}
[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1},{2,3,4},{5},{6},{7}}=>{{1},{2},{3},{4,5,6},{7}}
[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>{{1},{2,3,4},{5},{6,7}}=>{{1,2},{3},{4,5,6},{7}}
[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>{{1},{2,3,4},{5,6},{7}}=>{{1},{2,3},{4,5,6},{7}}
[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2,3,4},{5,6,7}}=>{{1,2,3},{4,5,6},{7}}
[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>{{1},{2,3,4,5},{6},{7}}=>{{1},{2},{3,4,5,6},{7}}
[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1},{2,3,4,5},{6,7}}=>{{1,2},{3,4,5,6},{7}}
[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6},{7}}=>{{1},{2,3,4,5,6},{7}}
[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7}}=>{{1,2,3,4,5,6},{7}}
[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7}}=>{{1},{2},{3},{4},{5},{6,7}}
[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6,7}}=>{{1,2},{3},{4},{5},{6,7}}
[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4},{5,6},{7}}=>{{1},{2,3},{4},{5},{6,7}}
[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>{{1,2},{3},{4},{5,6,7}}=>{{1,2,3},{4},{5},{6,7}}
[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>{{1,2},{3},{4,5},{6},{7}}=>{{1},{2},{3,4},{5},{6,7}}
[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>{{1,2},{3},{4,5},{6,7}}=>{{1,2},{3,4},{5},{6,7}}
[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>{{1,2},{3},{4,5,6},{7}}=>{{1},{2,3,4},{5},{6,7}}
[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>{{1,2},{3},{4,5,6,7}}=>{{1,2,3,4},{5},{6,7}}
[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7}}=>{{1},{2},{3},{4,5},{6,7}}
[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>{{1,2},{3,4},{5},{6,7}}=>{{1,2},{3},{4,5},{6,7}}
[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2},{3,4},{5,6},{7}}=>{{1},{2,3},{4,5},{6,7}}
[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3,4},{5,6,7}}=>{{1,2,3},{4,5},{6,7}}
[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>{{1,2},{3,4,5},{6},{7}}=>{{1},{2},{3,4,5},{6,7}}
[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>{{1,2},{3,4,5},{6,7}}=>{{1,2},{3,4,5},{6,7}}
[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2},{3,4,5,6},{7}}=>{{1},{2,3,4,5},{6,7}}
[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2},{3,4,5,6,7}}=>{{1,2,3,4,5},{6,7}}
[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7}}=>{{1},{2},{3},{4},{5,6,7}}
[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>{{1,2,3},{4},{5},{6,7}}=>{{1,2},{3},{4},{5,6,7}}
[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>{{1,2,3},{4},{5,6},{7}}=>{{1},{2,3},{4},{5,6,7}}
[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>{{1,2,3},{4},{5,6,7}}=>{{1,2,3},{4},{5,6,7}}
[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>{{1,2,3},{4,5},{6},{7}}=>{{1},{2},{3,4},{5,6,7}}
[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3},{4,5},{6,7}}=>{{1,2},{3,4},{5,6,7}}
[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3},{4,5,6},{7}}=>{{1},{2,3,4},{5,6,7}}
[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3},{4,5,6,7}}=>{{1,2,3,4},{5,6,7}}
[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7}}=>{{1},{2},{3},{4,5,6,7}}
[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>{{1,2,3,4},{5},{6,7}}=>{{1,2},{3},{4,5,6,7}}
[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4},{5,6},{7}}=>{{1},{2,3},{4,5,6,7}}
[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4},{5,6,7}}=>{{1,2,3},{4,5,6,7}}
[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7}}=>{{1},{2},{3,4,5,6,7}}
[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5},{6,7}}=>{{1,2},{3,4,5,6,7}}
[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6},{7}}=>{{1},{2,3,4,5,6,7}}
[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7}}=>{{1,2,3,4,5,6,7}}
[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5},{6},{7},{8}}
[1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7,8}}=>{{1,2},{3},{4},{5},{6},{7},{8}}
[1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4},{5},{6,7},{8}}=>{{1},{2,3},{4},{5},{6},{7},{8}}
[1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5},{6,7,8}}=>{{1,2,3},{4},{5},{6},{7},{8}}
[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3},{4},{5,6},{7},{8}}=>{{1},{2},{3,4},{5},{6},{7},{8}}
[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4},{5,6},{7,8}}=>{{1,2},{3,4},{5},{6},{7},{8}}
[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3},{4},{5,6,7},{8}}=>{{1},{2,3,4},{5},{6},{7},{8}}
[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4},{5,6,7,8}}=>{{1,2,3,4},{5},{6},{7},{8}}
[1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2},{3},{4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5},{6},{7},{8}}
[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>{{1},{2},{3},{4,5},{6},{7,8}}=>{{1,2},{3},{4,5},{6},{7},{8}}
[1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2},{3},{4,5},{6,7},{8}}=>{{1},{2,3},{4,5},{6},{7},{8}}
[1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2},{3},{4,5},{6,7,8}}=>{{1,2,3},{4,5},{6},{7},{8}}
[1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>{{1},{2},{3},{4,5,6},{7},{8}}=>{{1},{2},{3,4,5},{6},{7},{8}}
[1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>{{1},{2},{3},{4,5,6,7},{8}}=>{{1},{2,3,4,5},{6},{7},{8}}
[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3},{4,5,6,7,8}}=>{{1,2,3,4,5},{6},{7},{8}}
[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3,4},{5},{6},{7,8}}=>{{1,2},{3},{4},{5,6},{7},{8}}
[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>{{1},{2},{3,4},{5,6},{7},{8}}=>{{1},{2},{3,4},{5,6},{7},{8}}
[1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3,4},{5,6},{7,8}}=>{{1,2},{3,4},{5,6},{7},{8}}
[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3,4},{5,6,7,8}}=>{{1,2,3,4},{5,6},{7},{8}}
[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1},{2},{3,4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5,6},{7},{8}}
[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2},{3,4,5},{6,7,8}}=>{{1,2,3},{4,5,6},{7},{8}}
[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1},{2},{3,4,5,6},{7,8}}=>{{1,2},{3,4,5,6},{7},{8}}
[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>{{1},{2},{3,4,5,6,7},{8}}=>{{1},{2,3,4,5,6},{7},{8}}
[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7,8}}=>{{1,2,3,4,5,6},{7},{8}}
[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2,3},{4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5},{6,7},{8}}
[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5},{6},{7,8}}=>{{1,2},{3},{4},{5},{6,7},{8}}
[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2,3},{4},{5},{6,7},{8}}=>{{1},{2,3},{4},{5},{6,7},{8}}
[1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2,3},{4},{5},{6,7,8}}=>{{1,2,3},{4},{5},{6,7},{8}}
[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2,3},{4},{5,6,7,8}}=>{{1,2,3,4},{5},{6,7},{8}}
[1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2,3},{4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5},{6,7},{8}}
[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2,3},{4,5},{6,7},{8}}=>{{1},{2,3},{4,5},{6,7},{8}}
[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2,3},{4,5},{6,7,8}}=>{{1,2,3},{4,5},{6,7},{8}}
[1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>{{1},{2,3},{4,5,6},{7},{8}}=>{{1},{2},{3,4,5},{6,7},{8}}
[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>{{1},{2,3},{4,5,6},{7,8}}=>{{1,2},{3,4,5},{6,7},{8}}
[1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1},{2,3},{4,5,6,7},{8}}=>{{1},{2,3,4,5},{6,7},{8}}
[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2,3},{4,5,6,7,8}}=>{{1,2,3,4,5},{6,7},{8}}
[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>{{1},{2,3,4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5,6,7},{8}}
[1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>{{1},{2,3,4},{5},{6,7},{8}}=>{{1},{2,3},{4},{5,6,7},{8}}
[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>{{1},{2,3,4},{5},{6,7,8}}=>{{1,2,3},{4},{5,6,7},{8}}
[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>{{1},{2,3,4},{5,6},{7,8}}=>{{1,2},{3,4},{5,6,7},{8}}
[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>{{1},{2,3,4},{5,6,7},{8}}=>{{1},{2,3,4},{5,6,7},{8}}
[1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>{{1},{2,3,4},{5,6,7,8}}=>{{1,2,3,4},{5,6,7},{8}}
[1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>{{1},{2,3,4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5,6,7},{8}}
[1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>{{1},{2,3,4,5},{6},{7,8}}=>{{1,2},{3},{4,5,6,7},{8}}
[1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>{{1},{2,3,4,5},{6,7},{8}}=>{{1},{2,3},{4,5,6,7},{8}}
[1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>{{1},{2,3,4,5},{6,7,8}}=>{{1,2,3},{4,5,6,7},{8}}
[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>{{1},{2,3,4,5,6},{7},{8}}=>{{1},{2},{3,4,5,6,7},{8}}
[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>{{1},{2,3,4,5,6},{7,8}}=>{{1,2},{3,4,5,6,7},{8}}
[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6,7},{8}}=>{{1},{2,3,4,5,6,7},{8}}
[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8}}=>{{1,2,3,4,5,6,7},{8}}
[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5},{6},{7,8}}
[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6},{7,8}}=>{{1,2},{3},{4},{5},{6},{7,8}}
[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4},{5},{6,7},{8}}=>{{1},{2,3},{4},{5},{6},{7,8}}
[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1,2},{3},{4},{5},{6,7,8}}=>{{1,2,3},{4},{5},{6},{7,8}}
[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>{{1,2},{3},{4},{5,6},{7},{8}}=>{{1},{2},{3,4},{5},{6},{7,8}}
[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1,2},{3},{4},{5,6,7,8}}=>{{1,2,3,4},{5},{6},{7,8}}
[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3},{4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5},{6},{7,8}}
[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3},{4,5},{6,7,8}}=>{{1,2,3},{4,5},{6},{7,8}}
[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>{{1,2},{3},{4,5,6},{7,8}}=>{{1,2},{3,4,5},{6},{7,8}}
[2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2},{3},{4,5,6,7},{8}}=>{{1},{2,3,4,5},{6},{7,8}}
[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2},{3},{4,5,6,7,8}}=>{{1,2,3,4,5},{6},{7,8}}
[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5,6},{7,8}}
[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>{{1,2},{3,4},{5},{6,7,8}}=>{{1,2,3},{4},{5,6},{7,8}}
[2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>{{1,2},{3,4},{5,6},{7},{8}}=>{{1},{2},{3,4},{5,6},{7,8}}
[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2},{3,4},{5,6},{7,8}}=>{{1,2},{3,4},{5,6},{7,8}}
[2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>{{1,2},{3,4},{5,6,7},{8}}=>{{1},{2,3,4},{5,6},{7,8}}
[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4},{5,6,7,8}}=>{{1,2,3,4},{5,6},{7,8}}
[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1,2},{3,4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5,6},{7,8}}
[2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>{{1,2},{3,4,5},{6},{7,8}}=>{{1,2},{3},{4,5,6},{7,8}}
[2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1,2},{3,4,5},{6,7,8}}=>{{1,2,3},{4,5,6},{7,8}}
[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1,2},{3,4,5,6},{7,8}}=>{{1,2},{3,4,5,6},{7,8}}
[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>{{1,2},{3,4,5,6,7},{8}}=>{{1},{2,3,4,5,6},{7,8}}
[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2},{3,4,5,6,7,8}}=>{{1,2,3,4,5,6},{7,8}}
[3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5},{6,7,8}}
[3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2,3},{4},{5},{6},{7,8}}=>{{1,2},{3},{4},{5},{6,7,8}}
[3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>{{1,2,3},{4},{5},{6,7,8}}=>{{1,2,3},{4},{5},{6,7,8}}
[3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>{{1,2,3},{4},{5,6},{7,8}}=>{{1,2},{3,4},{5},{6,7,8}}
[3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>{{1,2,3},{4},{5,6,7,8}}=>{{1,2,3,4},{5},{6,7,8}}
[3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2,3},{4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5},{6,7,8}}
[3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2,3},{4,5},{6,7},{8}}=>{{1},{2,3},{4,5},{6,7,8}}
[3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>{{1,2,3},{4,5,6},{7},{8}}=>{{1},{2},{3,4,5},{6,7,8}}
[3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>{{1,2,3},{4,5,6},{7,8}}=>{{1,2},{3,4,5},{6,7,8}}
[3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2,3},{4,5,6,7},{8}}=>{{1},{2,3,4,5},{6,7,8}}
[3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2,3},{4,5,6,7,8}}=>{{1,2,3,4,5},{6,7,8}}
[4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7},{8}}=>{{1},{2},{3},{4},{5,6,7,8}}
[4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>{{1,2,3,4},{5},{6},{7,8}}=>{{1,2},{3},{4},{5,6,7,8}}
[4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>{{1,2,3,4},{5},{6,7},{8}}=>{{1},{2,3},{4},{5,6,7,8}}
[4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>{{1,2,3,4},{5},{6,7,8}}=>{{1,2,3},{4},{5,6,7,8}}
[4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>{{1,2,3,4},{5,6},{7},{8}}=>{{1},{2},{3,4},{5,6,7,8}}
[4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3,4},{5,6},{7,8}}=>{{1,2},{3,4},{5,6,7,8}}
[4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3,4},{5,6,7},{8}}=>{{1},{2,3,4},{5,6,7,8}}
[4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4},{5,6,7,8}}=>{{1,2,3,4},{5,6,7,8}}
[5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7},{8}}=>{{1},{2},{3},{4,5,6,7,8}}
[5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>{{1,2,3,4,5},{6},{7,8}}=>{{1,2},{3},{4,5,6,7,8}}
[5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5},{6,7},{8}}=>{{1},{2,3},{4,5,6,7,8}}
[5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4,5},{6,7,8}}=>{{1,2,3},{4,5,6,7,8}}
[6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5,6},{7},{8}}=>{{1},{2},{3,4,5,6,7,8}}
[6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5,6},{7,8}}=>{{1,2},{3,4,5,6,7,8}}
[7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7},{8}}=>{{1},{2,3,4,5,6,7,8}}
[8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7,8}}=>{{1,2,3,4,5,6,7,8}}
[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
[1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8,9}}=>{{1,2},{3},{4},{5},{6},{7},{8},{9}}
[1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7,8},{9}}=>{{1},{2,3},{4},{5},{6},{7},{8},{9}}
[1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5},{6},{7,8,9}}=>{{1,2,3},{4},{5},{6},{7},{8},{9}}
[1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6,7},{8,9}}=>{{1,2},{3,4},{5},{6},{7},{8},{9}}
[1,1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>{{1},{2},{3},{4},{5},{6,7,8},{9}}=>{{1},{2,3,4},{5},{6},{7},{8},{9}}
[1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4},{5},{6,7,8,9}}=>{{1,2,3,4},{5},{6},{7},{8},{9}}
[1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5,6},{7,8,9}}=>{{1,2,3},{4,5},{6},{7},{8},{9}}
[1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3},{4},{5,6,7,8,9}}=>{{1,2,3,4,5},{6},{7},{8},{9}}
[1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4,5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5,6},{7},{8},{9}}
[1,1,1,2,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4,5},{6,7},{8,9}}=>{{1,2},{3,4},{5,6},{7},{8},{9}}
[1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4,5},{6,7,8,9}}=>{{1,2,3,4},{5,6},{7},{8},{9}}
[1,1,1,3,3]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2},{3},{4,5,6},{7,8,9}}=>{{1,2,3},{4,5,6},{7},{8},{9}}
[1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2},{3},{4,5,6,7,8,9}}=>{{1,2,3,4,5,6},{7},{8},{9}}
[1,1,2,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3,4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6,7},{8},{9}}
[1,1,2,2,3]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2},{3,4},{5,6},{7,8,9}}=>{{1,2,3},{4,5},{6,7},{8},{9}}
[1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3,4},{5,6,7,8,9}}=>{{1,2,3,4,5},{6,7},{8},{9}}
[1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7,8,9}}=>{{1,2,3,4,5,6,7},{8},{9}}
[1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2,3},{4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6},{7,8},{9}}
[1,2,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2,3},{4},{5},{6},{7},{8,9}}=>{{1,2},{3},{4},{5},{6},{7,8},{9}}
[1,2,2,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1},{2,3},{4,5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5,6},{7,8},{9}}
[1,2,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2,3},{4,5},{6,7},{8,9}}=>{{1,2},{3,4},{5,6},{7,8},{9}}
[1,2,2,4]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1},{2,3},{4,5},{6,7,8,9}}=>{{1,2,3,4},{5,6},{7,8},{9}}
[1,2,3,3]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2,3},{4,5,6},{7,8,9}}=>{{1,2,3},{4,5,6},{7,8},{9}}
[1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2,3},{4,5,6,7,8,9}}=>{{1,2,3,4,5,6},{7,8},{9}}
[1,3,1,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2,3,4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6,7,8},{9}}
[1,3,5]=>[1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2,3,4},{5,6,7,8,9}}=>{{1,2,3,4,5},{6,7,8},{9}}
[1,4,4]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1},{2,3,4,5},{6,7,8,9}}=>{{1,2,3,4},{5,6,7,8},{9}}
[1,6,2]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>{{1},{2,3,4,5,6,7},{8,9}}=>{{1,2},{3,4,5,6,7,8},{9}}
[1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6,7,8},{9}}=>{{1},{2,3,4,5,6,7,8},{9}}
[1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8,9}}=>{{1,2,3,4,5,6,7,8},{9}}
[2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6},{7},{8,9}}
[2,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6},{7},{8,9}}=>{{1,2},{3},{4},{5},{6},{7},{8,9}}
[2,1,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1,2},{3},{4},{5},{6},{7,8},{9}}=>{{1},{2,3},{4},{5},{6},{7},{8,9}}
[2,1,2,1,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4,5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5,6},{7},{8,9}}
[2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6,7},{8,9}}
[2,2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5,6},{7},{8},{9}}=>{{1},{2},{3},{4,5},{6,7},{8,9}}
[2,2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2},{3,4},{5,6},{7,8},{9}}=>{{1},{2,3},{4,5},{6,7},{8,9}}
[2,2,2,3]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1,2},{3,4},{5,6},{7,8,9}}=>{{1,2,3},{4,5},{6,7},{8,9}}
[2,2,5]=>[1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2},{3,4},{5,6,7,8,9}}=>{{1,2,3,4,5},{6,7},{8,9}}
[2,3,4]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4,5},{6,7,8,9}}=>{{1,2,3,4},{5,6,7},{8,9}}
[2,6,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>{{1,2},{3,4,5,6,7,8},{9}}=>{{1},{2,3,4,5,6,7},{8,9}}
[2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1,2},{3,4,5,6,7,8,9}}=>{{1,2,3,4,5,6,7},{8,9}}
[3,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6},{7,8,9}}
[3,2,1,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4,5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5,6},{7,8,9}}
[3,2,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3},{4,5},{6,7},{8,9}}=>{{1,2},{3,4},{5,6},{7,8,9}}
[3,3,1,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>{{1,2,3},{4,5,6},{7},{8},{9}}=>{{1},{2},{3},{4,5,6},{7,8,9}}
[3,3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1,2,3},{4,5,6},{7,8,9}}=>{{1,2,3},{4,5,6},{7,8,9}}
[3,4,2]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1,2,3},{4,5,6,7},{8,9}}=>{{1,2},{3,4,5,6},{7,8,9}}
[3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2,3},{4,5,6,7,8,9}}=>{{1,2,3,4,5,6},{7,8,9}}
[4,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5},{6,7,8,9}}
[4,2,1,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]=>{{1,2,3,4},{5,6},{7},{8},{9}}=>{{1},{2},{3},{4,5},{6,7,8,9}}
[4,2,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2,3,4},{5,6},{7,8},{9}}=>{{1},{2,3},{4,5},{6,7,8,9}}
[4,3,1,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]=>{{1,2,3,4},{5,6,7},{8},{9}}=>{{1},{2},{3,4,5},{6,7,8,9}}
[4,3,2]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]=>{{1,2,3,4},{5,6,7},{8,9}}=>{{1,2},{3,4,5},{6,7,8,9}}
[4,4,1]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2,3,4},{5,6,7,8},{9}}=>{{1},{2,3,4,5},{6,7,8,9}}
[4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2,3,4},{5,6,7,8,9}}=>{{1,2,3,4,5},{6,7,8,9}}
[5,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7},{8},{9}}=>{{1},{2},{3},{4},{5,6,7,8,9}}
[5,2,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]=>{{1,2,3,4,5},{6,7},{8},{9}}=>{{1},{2},{3,4},{5,6,7,8,9}}
[5,3,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3,4,5},{6,7,8},{9}}=>{{1},{2,3,4},{5,6,7,8,9}}
[5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4,5},{6,7,8,9}}=>{{1,2,3,4},{5,6,7,8,9}}
[6,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4,5,6},{7},{8},{9}}=>{{1},{2},{3},{4,5,6,7,8,9}}
[6,2,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5,6},{7,8},{9}}=>{{1},{2,3},{4,5,6,7,8,9}}
[6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4,5,6},{7,8,9}}=>{{1,2,3},{4,5,6,7,8,9}}
[7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5,6,7},{8},{9}}=>{{1},{2},{3,4,5,6,7,8,9}}
[7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5,6,7},{8,9}}=>{{1,2},{3,4,5,6,7,8,9}}
[8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7,8},{9}}=>{{1},{2,3,4,5,6,7,8,9}}
[9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7,8,9}}=>{{1,2,3,4,5,6,7,8,9}}
[1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}=>{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>{{1},{2},{3},{4},{5},{6},{7},{8,9},{10}}=>{{1},{2,3},{4},{5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8,9,10}}=>{{1,2,3},{4},{5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7,8},{9,10}}=>{{1,2},{3,4},{5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3},{4},{5},{6},{7,8,9,10}}=>{{1,2,3,4},{5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>{{1},{2},{3},{4},{5},{6,7},{8},{9},{10}}=>{{1},{2},{3},{4,5},{6},{7},{8},{9},{10}}
[1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2},{3},{4},{5},{6,7,8,9,10}}=>{{1,2,3,4,5},{6},{7},{8},{9},{10}}
[1,1,1,1,2,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3},{4},{5,6},{7,8},{9,10}}=>{{1,2},{3,4},{5,6},{7},{8},{9},{10}}
[1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2},{3},{4},{5,6,7,8,9,10}}=>{{1,2,3,4,5,6},{7},{8},{9},{10}}
[1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1},{2},{3},{4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7},{8},{9},{10}}
[1,1,2,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1},{2},{3,4},{5,6},{7,8},{9,10}}=>{{1,2},{3,4},{5,6},{7,8},{9},{10}}
[1,1,4,4]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1},{2},{3,4,5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6,7,8},{9},{10}}
[1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1},{2},{3,4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7,8},{9},{10}}
[1,2,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1},{2,3},{4,5},{6,7},{8,9},{10}}=>{{1},{2,3},{4,5},{6,7},{8,9},{10}}
[1,2,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>{{1},{2,3},{4,5},{6,7},{8,9,10}}=>{{1,2,3},{4,5},{6,7},{8,9},{10}}
[1,2,3,4]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>{{1},{2,3},{4,5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6,7},{8,9},{10}}
[1,2,7]=>[1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1},{2,3},{4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7},{8,9},{10}}
[1,3,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1},{2,3,4},{5,6,7},{8,9,10}}=>{{1,2,3},{4,5,6},{7,8,9},{10}}
[1,3,6]=>[1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1},{2,3,4},{5,6,7,8,9,10}}=>{{1,2,3,4,5,6},{7,8,9},{10}}
[1,4,5]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1},{2,3,4,5},{6,7,8,9,10}}=>{{1,2,3,4,5},{6,7,8,9},{10}}
[1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>{{1},{2,3,4,5,6,7,8,9},{10}}=>{{1},{2,3,4,5,6,7,8,9},{10}}
[1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7,8,9},{10}}
[2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
[2,1,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1,2},{3},{4},{5},{6},{7},{8},{9,10}}=>{{1,2},{3},{4},{5},{6},{7},{8},{9,10}}
[2,2,1,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6},{7,8},{9,10}}
[2,2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>{{1,2},{3,4},{5,6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5,6},{7,8},{9,10}}
[2,2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>{{1,2},{3,4},{5,6},{7,8},{9},{10}}=>{{1},{2},{3,4},{5,6},{7,8},{9,10}}
[2,2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2},{3,4},{5,6},{7,8},{9,10}}=>{{1,2},{3,4},{5,6},{7,8},{9,10}}
[2,2,2,4]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4},{5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6},{7,8},{9,10}}
[2,2,3,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1,2},{3,4},{5,6,7},{8,9,10}}=>{{1,2,3},{4,5,6},{7,8},{9,10}}
[2,4,4]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2},{3,4,5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6,7,8},{9,10}}
[2,6,2]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>{{1,2},{3,4,5,6,7,8},{9,10}}=>{{1,2},{3,4,5,6,7,8},{9,10}}
[2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>{{1,2},{3,4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7,8},{9,10}}
[3,1,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4},{5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6},{7},{8,9,10}}
[3,2,1,1,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3},{4,5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6,7},{8,9,10}}
[3,2,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2,3},{4,5},{6,7},{8,9},{10}}=>{{1},{2,3},{4,5},{6,7},{8,9,10}}
[3,3,2,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3},{4,5,6},{7,8},{9,10}}=>{{1,2},{3,4},{5,6,7},{8,9,10}}
[3,3,4]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3},{4,5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6,7},{8,9,10}}
[3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>{{1,2,3},{4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7},{8,9,10}}
[4,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4},{5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6},{7,8,9,10}}
[4,2,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2,3,4},{5,6},{7,8},{9,10}}=>{{1,2},{3,4},{5,6},{7,8,9,10}}
[4,2,4]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4},{5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6},{7,8,9,10}}
[4,3,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4},{5,6,7},{8,9,10}}=>{{1,2,3},{4,5,6},{7,8,9,10}}
[4,4,1,1]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>{{1,2,3,4},{5,6,7,8},{9},{10}}=>{{1},{2},{3,4,5,6},{7,8,9,10}}
[4,4,2]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>{{1,2,3,4},{5,6,7,8},{9,10}}=>{{1,2},{3,4,5,6},{7,8,9,10}}
[4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>{{1,2,3,4},{5,6,7,8,9,10}}=>{{1,2,3,4,5,6},{7,8,9,10}}
[5,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4,5},{6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5},{6,7,8,9,10}}
[5,2,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5},{6,7},{8,9},{10}}=>{{1},{2,3},{4,5},{6,7,8,9,10}}
[5,4,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]=>{{1,2,3,4,5},{6,7,8,9},{10}}=>{{1},{2,3,4,5},{6,7,8,9,10}}
[5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>{{1,2,3,4,5},{6,7,8,9,10}}=>{{1,2,3,4,5},{6,7,8,9,10}}
[6,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]=>{{1,2,3,4,5,6},{7},{8},{9},{10}}=>{{1},{2},{3},{4},{5,6,7,8,9,10}}
[6,3,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]=>{{1,2,3,4,5,6},{7,8,9},{10}}=>{{1},{2,3,4},{5,6,7,8,9,10}}
[6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>{{1,2,3,4,5,6},{7,8,9,10}}=>{{1,2,3,4},{5,6,7,8,9,10}}
[7,1,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]=>{{1,2,3,4,5,6,7},{8},{9},{10}}=>{{1},{2},{3},{4,5,6,7,8,9,10}}
[7,2,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]=>{{1,2,3,4,5,6,7},{8,9},{10}}=>{{1},{2,3},{4,5,6,7,8,9,10}}
[7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>{{1,2,3,4,5,6,7},{8,9,10}}=>{{1,2,3},{4,5,6,7,8,9,10}}
[8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>{{1,2,3,4,5,6,7,8},{9},{10}}=>{{1},{2},{3,4,5,6,7,8,9,10}}
[8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>{{1,2,3,4,5,6,7,8},{9,10}}=>{{1,2},{3,4,5,6,7,8,9,10}}
[9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7,8,9},{10}}=>{{1},{2,3,4,5,6,7,8,9,10}}
[10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>{{1,2,3,4,5,6,7,8,9,10}}=>{{1,2,3,4,5,6,7,8,9,10}}
[1,10]=>[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>{{1},{2,3,4,5,6,7,8,9,10,11}}=>{{1,2,3,4,5,6,7,8,9,10},{11}}
[10,1]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]=>{{1,2,3,4,5,6,7,8,9,10},{11}}=>{{1},{2,3,4,5,6,7,8,9,10,11}}
[2,2,2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>{{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}=>{{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
[1,1,1,1,1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}=>{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.
Map
inverse Wachs-White
Description
The inverse of a transformation of set partitions due to Wachs and White.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection $\gamma^{-1}$.
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with $(i,j)$ to the cell in the $i$-th column from the right and the $j$-th row from the top.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection $\gamma^{-1}$.
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with $(i,j)$ to the cell in the $i$-th column from the right and the $j$-th row from the top.
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