- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions

Images

[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}} => {{1,3},{2}}

[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}} => {{1,4},{2},{3}}

[1,2,1] => [1,0,1,1,0,0,1,0] => {{1},{2,3},{4}} => {{1,3},{2},{4}}

[1,3] => [1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => {{1,3,4},{2}}

[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}} => {{1,2,4},{3}}

[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}} => {{1,5},{2},{3},{4}}

[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5}} => {{1,4},{2},{3},{5}}

[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => {{1,4,5},{2},{3}}

[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4},{5}} => {{1,3},{2},{4},{5}}

[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5}} => {{1,3,5},{2},{4}}

[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => {{1},{2,3,4},{5}} => {{1,3,4},{2},{5}}

[1,4] => [1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => {{1,3,4,5},{2}}

[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}} => {{1,2,5},{3},{4}}

[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5}} => {{1,2,4},{3},{5}}

[2,3] => [1,1,0,0,1,1,1,0,0,0] => {{1,2},{3,4,5}} => {{1,2,4,5},{3}}

[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}} => {{1,2,3,5},{4}}

[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}} => {{1,6},{2},{3},{4},{5}}

[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3},{4,5},{6}} => {{1,5},{2},{3},{4},{6}}

[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4,5,6}} => {{1,5,6},{2},{3},{4}}

[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => {{1},{2},{3,4},{5},{6}} => {{1,4},{2},{3},{5},{6}}

[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => {{1},{2},{3,4},{5,6}} => {{1,4,6},{2},{3},{5}}

[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => {{1},{2},{3,4,5},{6}} => {{1,4,5},{2},{3},{6}}

[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3,4,5,6}} => {{1,4,5,6},{2},{3}}

[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6}} => {{1,3},{2},{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}} => {{1,3,6},{2},{4},{5}}

[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2,3},{4,5},{6}} => {{1,3,5},{2},{4},{6}}

[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => {{1},{2,3},{4,5,6}} => {{1,3,5,6},{2},{4}}

[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2,3,4},{5},{6}} => {{1,3,4},{2},{5},{6}}

[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2,3,4},{5,6}} => {{1,3,4,6},{2},{5}}

[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => {{1},{2,3,4,5},{6}} => {{1,3,4,5},{2},{6}}

[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2,3,4,5,6}} => {{1,3,4,5,6},{2}}

[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}} => {{1,2,6},{3},{4},{5}}

[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => {{1,2},{3},{4,5},{6}} => {{1,2,5},{3},{4},{6}}

[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => {{1,2},{3},{4,5,6}} => {{1,2,5,6},{3},{4}}

[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => {{1,2},{3,4},{5},{6}} => {{1,2,4},{3},{5},{6}}

[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => {{1,2},{3,4},{5,6}} => {{1,2,4,6},{3},{5}}

[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => {{1,2},{3,4,5},{6}} => {{1,2,4,5},{3},{6}}

[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => {{1,2},{3,4,5,6}} => {{1,2,4,5,6},{3}}

[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}} => {{1,2,3,6},{4},{5}}

[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => {{1,2,3},{4,5},{6}} => {{1,2,3,5},{4},{6}}

[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => {{1,2,3},{4,5,6}} => {{1,2,3,5,6},{4}}

[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}} => {{1,2,3,4,6},{5}}

[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}} => {{1,7},{2},{3},{4},{5},{6}}

[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}} => {{1,6},{2},{3},{4},{5},{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}} => {{1,6,7},{2},{3},{4},{5}}

[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}} => {{1,5},{2},{3},{4},{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}} => {{1,5,7},{2},{3},{4},{6}}

[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}} => {{1,5,6},{2},{3},{4},{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}} => {{1,5,6,7},{2},{3},{4}}

[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}} => {{1,4},{2},{3},{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}} => {{1,4,7},{2},{3},{5},{6}}

[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}} => {{1,4,6},{2},{3},{5},{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}} => {{1,4,6,7},{2},{3},{5}}

[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}} => {{1,4,5},{2},{3},{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}} => {{1,4,5,7},{2},{3},{6}}

[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}} => {{1,4,5,6},{2},{3},{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}} => {{1,4,5,6,7},{2},{3}}

[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}} => {{1,3},{2},{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}} => {{1,3,7},{2},{4},{5},{6}}

[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}} => {{1,3,6},{2},{4},{5},{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}} => {{1,3,6,7},{2},{4},{5}}

[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}} => {{1,3,5},{2},{4},{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}} => {{1,3,5,7},{2},{4},{6}}

[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}} => {{1,3,5,6},{2},{4},{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}} => {{1,3,5,6,7},{2},{4}}

[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}} => {{1,3,4},{2},{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}} => {{1,3,4,7},{2},{5},{6}}

[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}} => {{1,3,4,6},{2},{5},{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}} => {{1,3,4,6,7},{2},{5}}

[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}} => {{1,3,4,5},{2},{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}} => {{1,3,4,5,7},{2},{6}}

[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => {{1},{2,3,4,5,6},{7}} => {{1,3,4,5,6},{2},{7}}

[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => {{1},{2,3,4,5,6,7}} => {{1,3,4,5,6,7},{2}}

[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}} => {{1,2,7},{3},{4},{5},{6}}

[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}} => {{1,2,6},{3},{4},{5},{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}} => {{1,2,6,7},{3},{4},{5}}

[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}} => {{1,2,5},{3},{4},{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}} => {{1,2,5,7},{3},{4},{6}}

>>> Load all 288 entries. <<<

[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}} => {{1,2,5,6},{3},{4},{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}} => {{1,2,5,6,7},{3},{4}}

[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}} => {{1,2,4},{3},{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}} => {{1,2,4,7},{3},{5},{6}}

[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}} => {{1,2,4,6},{3},{5},{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}} => {{1,2,4,6,7},{3},{5}}

[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}} => {{1,2,4,5},{3},{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}} => {{1,2,4,5,7},{3},{6}}

[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => {{1,2},{3,4,5,6},{7}} => {{1,2,4,5,6},{3},{7}}

[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => {{1,2},{3,4,5,6,7}} => {{1,2,4,5,6,7},{3}}

[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}} => {{1,2,3,7},{4},{5},{6}}

[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}} => {{1,2,3,6},{4},{5},{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}} => {{1,2,3,6,7},{4},{5}}

[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}} => {{1,2,3,5},{4},{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}} => {{1,2,3,5,7},{4},{6}}

[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => {{1,2,3},{4,5,6},{7}} => {{1,2,3,5,6},{4},{7}}

[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => {{1,2,3},{4,5,6,7}} => {{1,2,3,5,6,7},{4}}

[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}} => {{1,2,3,4,7},{5},{6}}

[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => {{1,2,3,4},{5,6},{7}} => {{1,2,3,4,6},{5},{7}}

[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => {{1,2,3,4},{5,6,7}} => {{1,2,3,4,6,7},{5}}

[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}} => {{1,2,3,4,5,7},{6}}

[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}} => {{1,8},{2},{3},{4},{5},{6},{7}}

[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}} => {{1,7,8},{2},{3},{4},{5},{6}}

[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}} => {{1,6},{2},{3},{4},{5},{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}} => {{1,6,8},{2},{3},{4},{5},{7}}

[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}} => {{1,6,7},{2},{3},{4},{5},{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}} => {{1,6,7,8},{2},{3},{4},{5}}

[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}} => {{1,5,8},{2},{3},{4},{6},{7}}

[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}} => {{1,5,6,7,8},{2},{3},{4}}

[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}} => {{1,4,8},{2},{3},{5},{6},{7}}

[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}} => {{1,4,6},{2},{3},{5},{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}} => {{1,4,6,7,8},{2},{3},{5}}

[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}} => {{1,4,5},{2},{3},{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}} => {{1,4,5,7,8},{2},{3},{6}}

[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}} => {{1,4,5,6,8},{2},{3},{7}}

[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}} => {{1,4,5,6,7},{2},{3},{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}} => {{1,4,5,6,7,8},{2},{3}}

[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}} => {{1,3,8},{2},{4},{5},{6},{7}}

[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}} => {{1,3,7},{2},{4},{5},{6},{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}} => {{1,3,6,7,8},{2},{4},{5}}

[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}} => {{1,3,5,7},{2},{4},{6},{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}} => {{1,3,5,7,8},{2},{4},{6}}

[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}} => {{1,3,5,6,8},{2},{4},{7}}

[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}} => {{1,3,5,6,7,8},{2},{4}}

[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}} => {{1,3,4},{2},{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}} => {{1,3,4,7,8},{2},{5},{6}}

[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}} => {{1,3,4,6,8},{2},{5},{7}}

[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}} => {{1,3,4,6,7},{2},{5},{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}} => {{1,3,4,5,6},{2},{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}} => {{1,3,4,5,6,8},{2},{7}}

[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}} => {{1,3,4,5,6,7},{2},{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}} => {{1,3,4,5,6,7,8},{2}}

[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}} => {{1,2,8},{3},{4},{5},{6},{7}}

[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}} => {{1,2,7},{3},{4},{5},{6},{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}} => {{1,2,7,8},{3},{4},{5},{6}}

[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}} => {{1,2,6},{3},{4},{5},{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}} => {{1,2,6,7,8},{3},{4},{5}}

[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}} => {{1,2,5},{3},{4},{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}} => {{1,2,5,7,8},{3},{4},{6}}

[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}} => {{1,2,5,6,8},{3},{4},{7}}

[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}} => {{1,2,5,6,7,8},{3},{4}}

[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}} => {{1,2,4},{3},{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}} => {{1,2,4,7,8},{3},{5},{6}}

[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}} => {{1,2,4,6},{3},{5},{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}} => {{1,2,4,6,8},{3},{5},{7}}

[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}} => {{1,2,4,6,7,8},{3},{5}}

[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}} => {{1,2,4,5},{3},{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}} => {{1,2,4,5,6,8},{3},{7}}

[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}} => {{1,2,4,5,6,7},{3},{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}} => {{1,2,4,5,6,7,8},{3}}

[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}} => {{1,2,3,8},{4},{5},{6},{7}}

[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}} => {{1,2,3,7,8},{4},{5},{6}}

[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}} => {{1,2,3,6,8},{4},{5},{7}}

[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}} => {{1,2,3,5},{4},{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}} => {{1,2,3,5,7},{4},{6},{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}} => {{1,2,3,5,6},{4},{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}} => {{1,2,3,5,6,8},{4},{7}}

[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}} => {{1,2,3,5,6,7},{4},{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}} => {{1,2,3,5,6,7,8},{4}}

[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}}

[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0] => {{1,2,3,4},{5},{6,7},{8}} => {{1,2,3,4,7},{5},{6},{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}} => {{1,2,3,4,6},{5},{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}} => {{1,2,3,4,6,8},{5},{7}}

[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}} => {{1,2,3,4,6,7},{5},{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}} => {{1,2,3,4,6,7,8},{5}}

[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}}

[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => {{1,2,3,4,5},{6,7},{8}} => {{1,2,3,4,5,7},{6},{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}} => {{1,2,3,4,5,7,8},{6}}

[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}} => {{1,2,3,4,5,6,8},{7}}

[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}} => {{1,9},{2},{3},{4},{5},{6},{7},{8}}

[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}} => {{1,8,9},{2},{3},{4},{5},{6},{7}}

[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}} => {{1,7,9},{2},{3},{4},{5},{6},{8}}

[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}} => {{1,5},{2},{3},{4},{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}} => {{1,4},{2},{3},{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}} => {{1,4,5,6,7,8,9},{2},{3}}

[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}} => {{1,3,9},{2},{4},{5},{6},{7},{8}}

[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}} => {{1,3,5},{2},{4},{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}} => {{1,3,4},{2},{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}} => {{1,3,4,5,6,7,9},{2},{8}}

[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}} => {{1,3,4,5,6,7,8},{2},{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}} => {{1,3,4,5,6,7,8,9},{2}}

[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}} => {{1,2,9},{3},{4},{5},{6},{7},{8}}

[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}} => {{1,2,8},{3},{4},{5},{6},{7},{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}} => {{1,2,5},{3},{4},{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}} => {{1,2,4},{3},{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}} => {{1,2,4,6},{3},{5},{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}} => {{1,2,4,6,8},{3},{5},{7},{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}} => {{1,2,4,5,6,7,8},{3},{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}} => {{1,2,4,5,6,7,8,9},{3}}

[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}} => {{1,2,3,5},{4},{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}} => {{1,2,3,5,7,9},{4},{6},{8}}

[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}} => {{1,2,3,5,6},{4},{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}} => {{1,2,3,5,6,8,9},{4},{7}}

[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}} => {{1,2,3,5,6,7,8,9},{4}}

[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}} => {{1,2,3,4,6},{5},{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}} => {{1,2,3,4,6,8},{5},{7},{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}} => {{1,2,3,4,6,7},{5},{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}} => {{1,2,3,4,6,7,9},{5},{8}}

[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}} => {{1,2,3,4,6,7,8},{5},{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}} => {{1,2,3,4,6,7,8,9},{5}}

[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}} => {{1,2,3,4,5,7},{6},{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}} => {{1,2,3,4,5,7,8},{6},{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}} => {{1,2,3,4,5,7,8,9},{6}}

[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}} => {{1,2,3,4,5,6,8},{7},{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}} => {{1,2,3,4,5,6,8,9},{7}}

[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}} => {{1,2,3,4,5,6,7,9},{8}}

[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}} => {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}

[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}} => {{1,7},{2},{3},{4},{5},{6},{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}} => {{1,3,4,5,6,7,8,9},{2},{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}} => {{1,3,4,5,6,7,8,9,10},{2}}

[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}} => {{1,2,10},{3},{4},{5},{6},{7},{8},{9}}

[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}} => {{1,2,4},{3},{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}} => {{1,2,4,6},{3},{5},{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}} => {{1,2,4,6,8},{3},{5},{7},{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}} => {{1,2,4,6,8,10},{3},{5},{7},{9}}

[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}} => {{1,2,4,5,6,7,8,9,10},{3}}

[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}} => {{1,2,3,5},{4},{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}} => {{1,2,3,5,7,9},{4},{6},{8},{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}} => {{1,2,3,5,6,8,10},{4},{7},{9}}

[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}} => {{1,2,3,5,6,7,8,9,10},{4}}

[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}} => {{1,2,3,4,6,8,10},{5},{7},{9}}

[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}} => {{1,2,3,4,6,7,9,10},{5},{8}}

[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}} => {{1,2,3,4,6,7,8},{5},{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}} => {{1,2,3,4,6,7,8,10},{5},{9}}

[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}} => {{1,2,3,4,6,7,8,9,10},{5}}

[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}} => {{1,2,3,4,5,7,9},{6},{8},{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}} => {{1,2,3,4,5,7,8,9},{6},{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}} => {{1,2,3,4,5,7,8,9,10},{6}}

[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}} => {{1,2,3,4,5,6,8,9},{7},{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}} => {{1,2,3,4,5,6,8,9,10},{7}}

[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}} => {{1,2,3,4,5,6,7,9},{8},{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}} => {{1,2,3,4,5,6,7,9,10},{8}}

[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}} => {{1,2,3,4,5,6,7,8,10},{9}}

[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}} => {{1,3,4,5,6,7,8,9,10,11},{2}}

[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}} => {{1,2,4,6,8,10,12},{3},{5},{7},{9},{11}}

[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}} => {{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}

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

intertwining number to dual major index

Description

A bijection sending the intertwining number of a set partition to its dual major index.
More precisely, St000490The intertwining number of a set partition.$(P) = $St000493The los statistic of a set partition.$(\phi(P))$ for all set partitions $P$.