Identifier
Values
[(1,2)] => {{1,2}} => {{1,2}} => {{1,2}} => 0
[(1,2),(3,4)] => {{1,2},{3,4}} => {{1,2},{3,4}} => {{1,2},{3,4}} => 0
[(1,3),(2,4)] => {{1,3},{2,4}} => {{1,2,4},{3}} => {{1,2,4},{3}} => 0
[(1,4),(2,3)] => {{1,4},{2,3}} => {{1,4},{2,3}} => {{1,3},{2,4}} => 0
[(1,2),(3,4),(5,6)] => {{1,2},{3,4},{5,6}} => {{1,2},{3,4},{5,6}} => {{1,2},{3,4},{5,6}} => 0
[(1,3),(2,4),(5,6)] => {{1,3},{2,4},{5,6}} => {{1,2},{3,4,6},{5}} => {{1,2},{3,4,6},{5}} => 0
[(1,4),(2,3),(5,6)] => {{1,4},{2,3},{5,6}} => {{1,2},{3,6},{4,5}} => {{1,2},{3,5},{4,6}} => 0
[(1,5),(2,3),(4,6)] => {{1,5},{2,3},{4,6}} => {{1,2,6},{3},{4,5}} => {{1,2,5},{3},{4,6}} => 0
[(1,6),(2,3),(4,5)] => {{1,6},{2,3},{4,5}} => {{1,6},{2,3},{4,5}} => {{1,3},{2,5},{4,6}} => 0
[(1,6),(2,4),(3,5)] => {{1,6},{2,4},{3,5}} => {{1,6},{2,3,5},{4}} => {{1,3,6},{2,5},{4}} => 0
[(1,5),(2,4),(3,6)] => {{1,5},{2,4},{3,6}} => {{1,2,6},{3,5},{4}} => {{1,2,5},{3,6},{4}} => 0
[(1,4),(2,5),(3,6)] => {{1,4},{2,5},{3,6}} => {{1,2,4},{3,6},{5}} => {{1,2,6},{3,4},{5}} => 1
[(1,3),(2,5),(4,6)] => {{1,3},{2,5},{4,6}} => {{1,2,4,6},{3},{5}} => {{1,2,4,6},{3},{5}} => 0
[(1,2),(3,5),(4,6)] => {{1,2},{3,5},{4,6}} => {{1,2,4},{3},{5,6}} => {{1,2,4},{3},{5,6}} => 0
[(1,2),(3,6),(4,5)] => {{1,2},{3,6},{4,5}} => {{1,4},{2,3},{5,6}} => {{1,3},{2,4},{5,6}} => 0
[(1,3),(2,6),(4,5)] => {{1,3},{2,6},{4,5}} => {{1,4,6},{2,3},{5}} => {{1,3},{2,4,6},{5}} => 0
[(1,4),(2,6),(3,5)] => {{1,4},{2,6},{3,5}} => {{1,4},{2,3,6},{5}} => {{1,3,4},{2,6},{5}} => 1
[(1,5),(2,6),(3,4)] => {{1,5},{2,6},{3,4}} => {{1,3,4},{2,6},{5}} => {{1,6},{2,3,4},{5}} => 2
[(1,6),(2,5),(3,4)] => {{1,6},{2,5},{3,4}} => {{1,6},{2,5},{3,4}} => {{1,4},{2,5},{3,6}} => 0
[(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}} => 0
[(1,4),(2,3),(5,6),(7,8)] => {{1,4},{2,3},{5,6},{7,8}} => {{1,2},{3,4},{5,8},{6,7}} => {{1,2},{3,4},{5,7},{6,8}} => 0
[(1,6),(2,3),(4,5),(7,8)] => {{1,6},{2,3},{4,5},{7,8}} => {{1,2},{3,8},{4,5},{6,7}} => {{1,2},{3,5},{4,7},{6,8}} => 0
[(1,8),(2,3),(4,5),(6,7)] => {{1,8},{2,3},{4,5},{6,7}} => {{1,8},{2,3},{4,5},{6,7}} => {{1,3},{2,5},{4,7},{6,8}} => 0
[(1,2),(3,6),(4,5),(7,8)] => {{1,2},{3,6},{4,5},{7,8}} => {{1,2},{3,6},{4,5},{7,8}} => {{1,2},{3,5},{4,6},{7,8}} => 0
[(1,6),(2,5),(3,4),(7,8)] => {{1,6},{2,5},{3,4},{7,8}} => {{1,2},{3,8},{4,7},{5,6}} => {{1,2},{3,6},{4,7},{5,8}} => 0
[(1,8),(2,5),(3,4),(6,7)] => {{1,8},{2,5},{3,4},{6,7}} => {{1,8},{2,3},{4,7},{5,6}} => {{1,3},{2,6},{4,7},{5,8}} => 0
[(1,2),(3,8),(4,5),(6,7)] => {{1,2},{3,8},{4,5},{6,7}} => {{1,6},{2,3},{4,5},{7,8}} => {{1,3},{2,5},{4,6},{7,8}} => 0
[(1,8),(2,7),(3,4),(5,6)] => {{1,8},{2,7},{3,4},{5,6}} => {{1,8},{2,7},{3,4},{5,6}} => {{1,4},{2,6},{3,7},{5,8}} => 0
[(1,2),(3,4),(5,8),(6,7)] => {{1,2},{3,4},{5,8},{6,7}} => {{1,4},{2,3},{5,6},{7,8}} => {{1,3},{2,4},{5,6},{7,8}} => 0
[(1,4),(2,3),(5,8),(6,7)] => {{1,4},{2,3},{5,8},{6,7}} => {{1,4},{2,3},{5,8},{6,7}} => {{1,3},{2,4},{5,7},{6,8}} => 0
[(1,8),(2,3),(4,7),(5,6)] => {{1,8},{2,3},{4,7},{5,6}} => {{1,8},{2,5},{3,4},{6,7}} => {{1,4},{2,5},{3,7},{6,8}} => 0
[(1,2),(3,8),(4,7),(5,6)] => {{1,2},{3,8},{4,7},{5,6}} => {{1,6},{2,5},{3,4},{7,8}} => {{1,4},{2,5},{3,6},{7,8}} => 0
[(1,8),(2,7),(3,6),(4,5)] => {{1,8},{2,7},{3,6},{4,5}} => {{1,8},{2,7},{3,6},{4,5}} => {{1,5},{2,6},{3,7},{4,8}} => 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},{5,6},{7,8},{9,10}} => {{1,2},{3,4},{5,6},{7,8},{9,10}} => 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}} => {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}} => {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}} => 0
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Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Map
inverse Wachs-White-rho
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 $\rho^{-1}$.
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.
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.
Map
to set partition
Description
Return the set partition corresponding to the perfect matching.