- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions

Images

[1] => [[1]] => {{1}} => {{1}} => {{1}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 142 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[] => [] => {} => {} => {}

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.

Map

rows

Description

The set partition whose blocks are the rows of the tableau.

Map

complement

Description

The complement of a set partition obtained by replacing $i$ with $n+1-i$.

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.