- FindStat

Identifier

Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions

Images

{{1}} => {{1}} => {{1}}

{{1,2}} => {{1,2}} => {{1,2}}

{{1},{2}} => {{1},{2}} => {{1},{2}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Map

Wachs-White-rho

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 $\rho$.
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-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.