Identifier
Values
[(1,2)] => {{1,2}} => {{1,2}} => [2] => 2
[(1,2),(3,4)] => {{1,2},{3,4}} => {{1,2},{3,4}} => [2,2] => 2
[(1,3),(2,4)] => {{1,3},{2,4}} => {{1,3,4},{2}} => [3,1] => 1
[(1,4),(2,3)] => {{1,4},{2,3}} => {{1,4},{2,3}} => [2,2] => 2
[(1,2),(3,4),(5,6)] => {{1,2},{3,4},{5,6}} => {{1,2},{3,4},{5,6}} => [2,2,2] => 2
[(1,3),(2,4),(5,6)] => {{1,3},{2,4},{5,6}} => {{1,2},{3,5,6},{4}} => [2,3,1] => 1
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => {{1,2},{3,6},{4,5}} => [2,2,2] => 2
[(1,5),(2,3),(4,6)] => {{1,5},{2,3},{4,6}} => {{1,3,6},{2},{4,5}} => [3,1,2] => 2
[(1,6),(2,3),(4,5)] => {{1,6},{2,3},{4,5}} => {{1,6},{2,3},{4,5}} => [2,2,2] => 2
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => {{1,6},{2,4,5},{3}} => [2,3,1] => 1
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => {{1,4,6},{2,5},{3}} => [3,2,1] => 1
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => {{1,4,5,6},{2},{3}} => [4,1,1] => 1
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => {{1,3,5},{2,6},{4}} => [3,2,1] => 1
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => {{1,3,4},{2},{5,6}} => [3,1,2] => 2
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => {{1,4},{2,3},{5,6}} => [2,2,2] => 2
[(1,3),(2,6),(4,5)] => {{1,3},{2,6},{4,5}} => {{1,5},{2,3,6},{4}} => [2,3,1] => 1
[(1,4),(2,6),(3,5)] => {{1,4},{2,6},{3,5}} => {{1,5,6},{2,4},{3}} => [3,2,1] => 1
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => {{1,5,6},{2},{3,4}} => [3,1,2] => 2
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => {{1,6},{2,5},{3,4}} => [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}} => [2,2,2,2] => 2
[(1,3),(2,4),(5,6),(7,8)] => {{1,3},{2,4},{5,6},{7,8}} => {{1,2},{3,4},{5,7,8},{6}} => [2,2,3,1] => 1
[(1,4),(2,3),(5,6),(7,8)] => {{1,4},{2,3},{5,6},{7,8}} => {{1,2},{3,4},{5,8},{6,7}} => [2,2,2,2] => 2
[(1,5),(2,3),(4,6),(7,8)] => {{1,5},{2,3},{4,6},{7,8}} => {{1,2},{3,5,8},{4},{6,7}} => [2,3,1,2] => 2
[(1,6),(2,3),(4,5),(7,8)] => {{1,6},{2,3},{4,5},{7,8}} => {{1,2},{3,8},{4,5},{6,7}} => [2,2,2,2] => 2
[(1,7),(2,3),(4,5),(6,8)] => {{1,7},{2,3},{4,5},{6,8}} => {{1,3,8},{2},{4,5},{6,7}} => [3,1,2,2] => 2
[(1,8),(2,3),(4,5),(6,7)] => {{1,8},{2,3},{4,5},{6,7}} => {{1,8},{2,3},{4,5},{6,7}} => [2,2,2,2] => 2
[(1,8),(2,4),(3,5),(6,7)] => {{1,8},{2,4},{3,5},{6,7}} => {{1,8},{2,3},{4,6,7},{5}} => [2,2,3,1] => 1
[(1,7),(2,4),(3,5),(6,8)] => {{1,7},{2,4},{3,5},{6,8}} => {{1,3,8},{2},{4,6,7},{5}} => [3,1,3,1] => 1
[(1,6),(2,4),(3,5),(7,8)] => {{1,6},{2,4},{3,5},{7,8}} => {{1,2},{3,8},{4,6,7},{5}} => [2,2,3,1] => 1
[(1,5),(2,4),(3,6),(7,8)] => {{1,5},{2,4},{3,6},{7,8}} => {{1,2},{3,6,8},{4,7},{5}} => [2,3,2,1] => 1
[(1,3),(2,5),(4,6),(7,8)] => {{1,3},{2,5},{4,6},{7,8}} => {{1,2},{3,5,7},{4,8},{6}} => [2,3,2,1] => 1
[(1,2),(3,5),(4,6),(7,8)] => {{1,2},{3,5},{4,6},{7,8}} => {{1,2},{3,5,6},{4},{7,8}} => [2,3,1,2] => 2
[(1,2),(3,6),(4,5),(7,8)] => {{1,2},{3,6},{4,5},{7,8}} => {{1,2},{3,6},{4,5},{7,8}} => [2,2,2,2] => 2
[(1,3),(2,6),(4,5),(7,8)] => {{1,3},{2,6},{4,5},{7,8}} => {{1,2},{3,7},{4,5,8},{6}} => [2,2,3,1] => 1
[(1,4),(2,6),(3,5),(7,8)] => {{1,4},{2,6},{3,5},{7,8}} => {{1,2},{3,7,8},{4,6},{5}} => [2,3,2,1] => 1
[(1,5),(2,6),(3,4),(7,8)] => {{1,5},{2,6},{3,4},{7,8}} => {{1,2},{3,7,8},{4},{5,6}} => [2,3,1,2] => 2
[(1,6),(2,5),(3,4),(7,8)] => {{1,6},{2,5},{3,4},{7,8}} => {{1,2},{3,8},{4,7},{5,6}} => [2,2,2,2] => 2
[(1,7),(2,5),(3,4),(6,8)] => {{1,7},{2,5},{3,4},{6,8}} => {{1,3,8},{2},{4,7},{5,6}} => [3,1,2,2] => 2
[(1,8),(2,5),(3,4),(6,7)] => {{1,8},{2,5},{3,4},{6,7}} => {{1,8},{2,3},{4,7},{5,6}} => [2,2,2,2] => 2
[(1,8),(2,6),(3,4),(5,7)] => {{1,8},{2,6},{3,4},{5,7}} => {{1,8},{2,4,7},{3},{5,6}} => [2,3,1,2] => 2
[(1,7),(2,6),(3,4),(5,8)] => {{1,7},{2,6},{3,4},{5,8}} => {{1,4,8},{2,7},{3},{5,6}} => [3,2,1,2] => 2
[(1,5),(2,7),(3,4),(6,8)] => {{1,5},{2,7},{3,4},{6,8}} => {{1,3,7},{2,8},{4},{5,6}} => [3,2,1,2] => 2
[(1,4),(2,7),(3,5),(6,8)] => {{1,4},{2,7},{3,5},{6,8}} => {{1,3,7},{2,8},{4,6},{5}} => [3,2,2,1] => 1
[(1,3),(2,7),(4,5),(6,8)] => {{1,3},{2,7},{4,5},{6,8}} => {{1,3,7},{2},{4,5,8},{6}} => [3,1,3,1] => 1
[(1,2),(3,7),(4,5),(6,8)] => {{1,2},{3,7},{4,5},{6,8}} => {{1,3,6},{2},{4,5},{7,8}} => [3,1,2,2] => 2
[(1,2),(3,8),(4,5),(6,7)] => {{1,2},{3,8},{4,5},{6,7}} => {{1,6},{2,3},{4,5},{7,8}} => [2,2,2,2] => 2
[(1,3),(2,8),(4,5),(6,7)] => {{1,3},{2,8},{4,5},{6,7}} => {{1,7},{2,3},{4,5,8},{6}} => [2,2,3,1] => 1
[(1,4),(2,8),(3,5),(6,7)] => {{1,4},{2,8},{3,5},{6,7}} => {{1,7},{2,3,8},{4,6},{5}} => [2,3,2,1] => 1
[(1,5),(2,8),(3,4),(6,7)] => {{1,5},{2,8},{3,4},{6,7}} => {{1,7},{2,3,8},{4},{5,6}} => [2,3,1,2] => 2
[(1,6),(2,8),(3,4),(5,7)] => {{1,6},{2,8},{3,4},{5,7}} => {{1,7,8},{2,4},{3},{5,6}} => [3,2,1,2] => 2
[(1,7),(2,8),(3,4),(5,6)] => {{1,7},{2,8},{3,4},{5,6}} => {{1,7,8},{2},{3,4},{5,6}} => [3,1,2,2] => 2
[(1,8),(2,7),(3,4),(5,6)] => {{1,8},{2,7},{3,4},{5,6}} => {{1,8},{2,7},{3,4},{5,6}} => [2,2,2,2] => 2
[(1,8),(2,7),(3,5),(4,6)] => {{1,8},{2,7},{3,5},{4,6}} => {{1,8},{2,7},{3,5,6},{4}} => [2,2,3,1] => 1
[(1,7),(2,8),(3,5),(4,6)] => {{1,7},{2,8},{3,5},{4,6}} => {{1,7,8},{2},{3,5,6},{4}} => [3,1,3,1] => 1
[(1,6),(2,8),(3,5),(4,7)] => {{1,6},{2,8},{3,5},{4,7}} => {{1,7,8},{2,5},{3,6},{4}} => [3,2,2,1] => 1
[(1,5),(2,8),(3,6),(4,7)] => {{1,5},{2,8},{3,6},{4,7}} => {{1,7,8},{2,5,6},{3},{4}} => [3,3,1,1] => 1
[(1,3),(2,8),(4,6),(5,7)] => {{1,3},{2,8},{4,6},{5,7}} => {{1,7},{2,4,5},{3,8},{6}} => [2,3,2,1] => 1
[(1,2),(3,8),(4,6),(5,7)] => {{1,2},{3,8},{4,6},{5,7}} => {{1,6},{2,4,5},{3},{7,8}} => [2,3,1,2] => 2
[(1,2),(3,7),(4,6),(5,8)] => {{1,2},{3,7},{4,6},{5,8}} => {{1,4,6},{2,5},{3},{7,8}} => [3,2,1,2] => 2
[(1,3),(2,7),(4,6),(5,8)] => {{1,3},{2,7},{4,6},{5,8}} => {{1,4,7},{2,5},{3,8},{6}} => [3,2,2,1] => 1
[(1,4),(2,7),(3,6),(5,8)] => {{1,4},{2,7},{3,6},{5,8}} => {{1,4,7},{2,6,8},{3},{5}} => [3,3,1,1] => 1
[(1,6),(2,7),(3,5),(4,8)] => {{1,6},{2,7},{3,5},{4,8}} => {{1,5,7,8},{2},{3,6},{4}} => [4,1,2,1] => 1
[(1,7),(2,6),(3,5),(4,8)] => {{1,7},{2,6},{3,5},{4,8}} => {{1,5,8},{2,7},{3,6},{4}} => [3,2,2,1] => 1
[(1,8),(2,6),(3,5),(4,7)] => {{1,8},{2,6},{3,5},{4,7}} => {{1,8},{2,5,7},{3,6},{4}} => [2,3,2,1] => 1
[(1,7),(2,5),(3,6),(4,8)] => {{1,7},{2,5},{3,6},{4,8}} => {{1,5,8},{2,6,7},{3},{4}} => [3,3,1,1] => 1
[(1,5),(2,6),(3,7),(4,8)] => {{1,5},{2,6},{3,7},{4,8}} => {{1,5,6,7,8},{2},{3},{4}} => [5,1,1,1] => 1
[(1,3),(2,6),(4,7),(5,8)] => {{1,3},{2,6},{4,7},{5,8}} => {{1,4,5,7},{2},{3,8},{6}} => [4,1,2,1] => 1
[(1,2),(3,5),(4,7),(6,8)] => {{1,2},{3,5},{4,7},{6,8}} => {{1,3,5},{2,6},{4},{7,8}} => [3,2,1,2] => 2
[(1,3),(2,5),(4,7),(6,8)] => {{1,3},{2,5},{4,7},{6,8}} => {{1,3,5},{2,7},{4,8},{6}} => [3,2,2,1] => 1
[(1,4),(2,5),(3,7),(6,8)] => {{1,4},{2,5},{3,7},{6,8}} => {{1,3,6},{2,7,8},{4},{5}} => [3,3,1,1] => 1
[(1,5),(2,4),(3,7),(6,8)] => {{1,5},{2,4},{3,7},{6,8}} => {{1,3,6},{2,8},{4,7},{5}} => [3,2,2,1] => 1
[(1,6),(2,4),(3,7),(5,8)] => {{1,6},{2,4},{3,7},{5,8}} => {{1,4,6,8},{2},{3,7},{5}} => [4,1,2,1] => 1
[(1,7),(2,4),(3,6),(5,8)] => {{1,7},{2,4},{3,6},{5,8}} => {{1,4,8},{2,6},{3,7},{5}} => [3,2,2,1] => 1
[(1,8),(2,4),(3,6),(5,7)] => {{1,8},{2,4},{3,6},{5,7}} => {{1,8},{2,4,6},{3,7},{5}} => [2,3,2,1] => 1
[(1,8),(2,3),(4,6),(5,7)] => {{1,8},{2,3},{4,6},{5,7}} => {{1,8},{2,4,5},{3},{6,7}} => [2,3,1,2] => 2
[(1,7),(2,3),(4,6),(5,8)] => {{1,7},{2,3},{4,6},{5,8}} => {{1,4,8},{2,5},{3},{6,7}} => [3,2,1,2] => 2
[(1,5),(2,3),(4,7),(6,8)] => {{1,5},{2,3},{4,7},{6,8}} => {{1,3,5},{2,8},{4},{6,7}} => [3,2,1,2] => 2
[(1,4),(2,3),(5,7),(6,8)] => {{1,4},{2,3},{5,7},{6,8}} => {{1,3,4},{2},{5,8},{6,7}} => [3,1,2,2] => 2
[(1,3),(2,4),(5,7),(6,8)] => {{1,3},{2,4},{5,7},{6,8}} => {{1,3,4},{2},{5,7,8},{6}} => [3,1,3,1] => 1
[(1,2),(3,4),(5,7),(6,8)] => {{1,2},{3,4},{5,7},{6,8}} => {{1,3,4},{2},{5,6},{7,8}} => [3,1,2,2] => 2
[(1,2),(3,4),(5,8),(6,7)] => {{1,2},{3,4},{5,8},{6,7}} => {{1,4},{2,3},{5,6},{7,8}} => [2,2,2,2] => 2
[(1,3),(2,4),(5,8),(6,7)] => {{1,3},{2,4},{5,8},{6,7}} => {{1,4},{2,3},{5,7,8},{6}} => [2,2,3,1] => 1
[(1,4),(2,3),(5,8),(6,7)] => {{1,4},{2,3},{5,8},{6,7}} => {{1,4},{2,3},{5,8},{6,7}} => [2,2,2,2] => 2
[(1,5),(2,3),(4,8),(6,7)] => {{1,5},{2,3},{4,8},{6,7}} => {{1,5},{2,3,8},{4},{6,7}} => [2,3,1,2] => 2
[(1,6),(2,3),(4,8),(5,7)] => {{1,6},{2,3},{4,8},{5,7}} => {{1,5,8},{2,4},{3},{6,7}} => [3,2,1,2] => 2
[(1,7),(2,3),(4,8),(5,6)] => {{1,7},{2,3},{4,8},{5,6}} => {{1,5,8},{2},{3,4},{6,7}} => [3,1,2,2] => 2
[(1,8),(2,3),(4,7),(5,6)] => {{1,8},{2,3},{4,7},{5,6}} => {{1,8},{2,5},{3,4},{6,7}} => [2,2,2,2] => 2
[(1,8),(2,4),(3,7),(5,6)] => {{1,8},{2,4},{3,7},{5,6}} => {{1,8},{2,6},{3,4,7},{5}} => [2,2,3,1] => 1
[(1,7),(2,4),(3,8),(5,6)] => {{1,7},{2,4},{3,8},{5,6}} => {{1,6,8},{2},{3,4,7},{5}} => [3,1,3,1] => 1
[(1,6),(2,4),(3,8),(5,7)] => {{1,6},{2,4},{3,8},{5,7}} => {{1,6,8},{2,4},{3,7},{5}} => [3,2,2,1] => 1
[(1,5),(2,4),(3,8),(6,7)] => {{1,5},{2,4},{3,8},{6,7}} => {{1,6},{2,3,8},{4,7},{5}} => [2,3,2,1] => 1
[(1,3),(2,5),(4,8),(6,7)] => {{1,3},{2,5},{4,8},{6,7}} => {{1,5},{2,3,7},{4,8},{6}} => [2,3,2,1] => 1
[(1,2),(3,5),(4,8),(6,7)] => {{1,2},{3,5},{4,8},{6,7}} => {{1,5},{2,3,6},{4},{7,8}} => [2,3,1,2] => 2
[(1,2),(3,6),(4,8),(5,7)] => {{1,2},{3,6},{4,8},{5,7}} => {{1,5,6},{2,4},{3},{7,8}} => [3,2,1,2] => 2
[(1,3),(2,6),(4,8),(5,7)] => {{1,3},{2,6},{4,8},{5,7}} => {{1,5,7},{2,4},{3,8},{6}} => [3,2,2,1] => 1
[(1,4),(2,6),(3,8),(5,7)] => {{1,4},{2,6},{3,8},{5,7}} => {{1,6,7},{2,4,8},{3},{5}} => [3,3,1,1] => 1
[(1,6),(2,5),(3,8),(4,7)] => {{1,6},{2,5},{3,8},{4,7}} => {{1,6,8},{2,5,7},{3},{4}} => [3,3,1,1] => 1
[(1,7),(2,5),(3,8),(4,6)] => {{1,7},{2,5},{3,8},{4,6}} => {{1,6,8},{2,7},{3,5},{4}} => [3,2,2,1] => 1
[(1,8),(2,5),(3,7),(4,6)] => {{1,8},{2,5},{3,7},{4,6}} => {{1,8},{2,6,7},{3,5},{4}} => [2,3,2,1] => 1
[(1,8),(2,6),(3,7),(4,5)] => {{1,8},{2,6},{3,7},{4,5}} => {{1,8},{2,6,7},{3},{4,5}} => [2,3,1,2] => 2
[(1,7),(2,6),(3,8),(4,5)] => {{1,7},{2,6},{3,8},{4,5}} => {{1,6,8},{2,7},{3},{4,5}} => [3,2,1,2] => 2
>>> Load all 112 entries. <<<
[(1,5),(2,7),(3,8),(4,6)] => {{1,5},{2,7},{3,8},{4,6}} => {{1,6,7,8},{2},{3,5},{4}} => [4,1,2,1] => 1
[(1,4),(2,7),(3,8),(5,6)] => {{1,4},{2,7},{3,8},{5,6}} => {{1,6,7},{2,8},{3,4},{5}} => [3,2,2,1] => 1
[(1,3),(2,7),(4,8),(5,6)] => {{1,3},{2,7},{4,8},{5,6}} => {{1,5,7},{2},{3,4,8},{6}} => [3,1,3,1] => 1
[(1,2),(3,7),(4,8),(5,6)] => {{1,2},{3,7},{4,8},{5,6}} => {{1,5,6},{2},{3,4},{7,8}} => [3,1,2,2] => 2
[(1,2),(3,8),(4,7),(5,6)] => {{1,2},{3,8},{4,7},{5,6}} => {{1,6},{2,5},{3,4},{7,8}} => [2,2,2,2] => 2
[(1,3),(2,8),(4,7),(5,6)] => {{1,3},{2,8},{4,7},{5,6}} => {{1,7},{2,5},{3,4,8},{6}} => [2,2,3,1] => 1
[(1,4),(2,8),(3,7),(5,6)] => {{1,4},{2,8},{3,7},{5,6}} => {{1,7},{2,6,8},{3,4},{5}} => [2,3,2,1] => 1
[(1,5),(2,8),(3,7),(4,6)] => {{1,5},{2,8},{3,7},{4,6}} => {{1,7,8},{2,6},{3,5},{4}} => [3,2,2,1] => 1
[(1,6),(2,8),(3,7),(4,5)] => {{1,6},{2,8},{3,7},{4,5}} => {{1,7,8},{2,6},{3},{4,5}} => [3,2,1,2] => 2
[(1,7),(2,8),(3,6),(4,5)] => {{1,7},{2,8},{3,6},{4,5}} => {{1,7,8},{2},{3,6},{4,5}} => [3,1,2,2] => 2
[(1,8),(2,7),(3,6),(4,5)] => {{1,8},{2,7},{3,6},{4,5}} => {{1,8},{2,7},{3,6},{4,5}} => [2,2,2,2] => 2
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The last part of an integer composition.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
Map
Wachs-White
Description
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 $\gamma$.
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.
Map
to set partition
Description
Return the set partition corresponding to the perfect matching.