Identifier
-
Mp00216:
Set partitions
—inverse Wachs-White⟶
Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001727: Permutations ⟶ ℤ
Values
{{1}} => {{1}} => {{1}} => [1] => 0
{{1,2}} => {{1,2}} => {{1,2}} => [2,1] => 0
{{1},{2}} => {{1},{2}} => {{1},{2}} => [1,2] => 0
{{1,2,3}} => {{1,2,3}} => {{1,2,3}} => [2,3,1] => 0
{{1,2},{3}} => {{1},{2,3}} => {{1},{2,3}} => [1,3,2] => 0
{{1,3},{2}} => {{1,3},{2}} => {{1,3},{2}} => [3,2,1] => 1
{{1},{2,3}} => {{1,2},{3}} => {{1,2},{3}} => [2,1,3] => 0
{{1},{2},{3}} => {{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => 0
{{1,2,3,4}} => {{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => 0
{{1,2,3},{4}} => {{1},{2,3,4}} => {{1},{2,3,4}} => [1,3,4,2] => 0
{{1,2,4},{3}} => {{1,3,4},{2}} => {{1,3,4},{2}} => [3,2,4,1] => 1
{{1,2},{3,4}} => {{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => 0
{{1,2},{3},{4}} => {{1},{2},{3,4}} => {{1},{2},{3,4}} => [1,2,4,3] => 0
{{1,3,4},{2}} => {{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => 2
{{1,3},{2,4}} => {{1,2,4},{3}} => {{1,2,4},{3}} => [2,4,3,1] => 1
{{1,3},{2},{4}} => {{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => 1
{{1,4},{2,3}} => {{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => 1
{{1},{2,3,4}} => {{1,2,3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => 0
{{1},{2,3},{4}} => {{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => 0
{{1,4},{2},{3}} => {{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => 2
{{1},{2,4},{3}} => {{1,3},{2},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => 1
{{1},{2},{3,4}} => {{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => 0
{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => 0
{{1,2,3,4,5}} => {{1,2,3,4,5}} => {{1,2,3,4,5}} => [2,3,4,5,1] => 0
{{1,2,3,4},{5}} => {{1},{2,3,4,5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 0
{{1,2,3,5},{4}} => {{1,3,4,5},{2}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 1
{{1,2,3},{4,5}} => {{1,2},{3,4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 0
{{1,2,3},{4},{5}} => {{1},{2},{3,4,5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 0
{{1,2,4,5},{3}} => {{1,3},{2,4,5}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 2
{{1,2,4},{3,5}} => {{1,2,4,5},{3}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 1
{{1,2,4},{3},{5}} => {{1},{2,4,5},{3}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 1
{{1,2,5},{3,4}} => {{1,4,5},{2,3}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 1
{{1,2},{3,4,5}} => {{1,2,3},{4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 0
{{1,2},{3,4},{5}} => {{1},{2,3},{4,5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 0
{{1,2,5},{3},{4}} => {{1,4,5},{2},{3}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 2
{{1,2},{3,5},{4}} => {{1,3},{2},{4,5}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 1
{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}} => {{1},{2},{3},{4,5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 0
{{1,3,4,5},{2}} => {{1,3,5},{2,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 3
{{1,3,4},{2,5}} => {{1,2,4},{3,5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 2
{{1,3,4},{2},{5}} => {{1},{2,4},{3,5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 2
{{1,3,5},{2,4}} => {{1,4},{2,3,5}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 2
{{1,3},{2,4,5}} => {{1,2,3,5},{4}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 1
{{1,3},{2,4},{5}} => {{1},{2,3,5},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 1
{{1,3,5},{2},{4}} => {{1,4},{2},{3,5}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 3
{{1,3},{2,5},{4}} => {{1,3,5},{2},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 2
{{1,3},{2},{4,5}} => {{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 1
{{1,3},{2},{4},{5}} => {{1},{2},{3,5},{4}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => 1
{{1,4,5},{2,3}} => {{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 2
{{1,4},{2,3,5}} => {{1,2,5},{3,4}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 1
{{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 1
{{1,5},{2,3,4}} => {{1,5},{2,3,4}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 1
{{1},{2,3,4,5}} => {{1,2,3,4},{5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 0
{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 0
{{1,5},{2,3},{4}} => {{1,5},{2},{3,4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 2
{{1},{2,3,5},{4}} => {{1,3,4},{2},{5}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 1
{{1},{2,3},{4,5}} => {{1,2},{3,4},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 0
{{1},{2,3},{4},{5}} => {{1},{2},{3,4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 0
{{1,4,5},{2},{3}} => {{1,4},{2,5},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 4
{{1,4},{2,5},{3}} => {{1,3},{2,5},{4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 3
{{1,4},{2},{3,5}} => {{1,2,5},{3},{4}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 2
{{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 2
{{1,5},{2,4},{3}} => {{1,5},{2,4},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 3
{{1},{2,4,5},{3}} => {{1,3},{2,4},{5}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 2
{{1},{2,4},{3,5}} => {{1,2,4},{3},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 1
{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => 1
{{1,5},{2},{3,4}} => {{1,5},{2,3},{4}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 2
{{1},{2,5},{3,4}} => {{1,4},{2,3},{5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 1
{{1},{2},{3,4,5}} => {{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 0
{{1},{2},{3,4},{5}} => {{1},{2,3},{4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 0
{{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 3
{{1},{2,5},{3},{4}} => {{1,4},{2},{3},{5}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 2
{{1},{2},{3,5},{4}} => {{1,3},{2},{4},{5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 1
{{1},{2},{3},{4,5}} => {{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 0
{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 0
{{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 0
{{1,2,3,4,6},{5}} => {{1,3,4,5,6},{2}} => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 1
{{1,2,3,4},{5,6}} => {{1,2},{3,4,5,6}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 0
{{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 0
{{1,2,3,5,6},{4}} => {{1,3},{2,4,5,6}} => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 2
{{1,2,3,5},{4,6}} => {{1,2,4,5,6},{3}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 1
{{1,2,3,5},{4},{6}} => {{1},{2,4,5,6},{3}} => {{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => 1
{{1,2,3,6},{4,5}} => {{1,4,5,6},{2,3}} => {{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => 1
{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 0
{{1,2,3},{4,5},{6}} => {{1},{2,3},{4,5,6}} => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 0
{{1,2,3,6},{4},{5}} => {{1,4,5,6},{2},{3}} => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 2
{{1,2,3},{4,6},{5}} => {{1,3},{2},{4,5,6}} => {{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => 1
{{1,2,3},{4},{5,6}} => {{1,2},{3},{4,5,6}} => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 0
{{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => 0
{{1,2,4,5,6},{3}} => {{1,3,5,6},{2,4}} => {{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => 3
{{1,2,4,5},{3,6}} => {{1,2,4},{3,5,6}} => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 2
{{1,2,4,5},{3},{6}} => {{1},{2,4},{3,5,6}} => {{1},{2,5,6},{3,4}} => [1,5,4,3,6,2] => 2
{{1,2,4,6},{3,5}} => {{1,4},{2,3,5,6}} => {{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => 2
{{1,2,4},{3,5,6}} => {{1,2,3,5,6},{4}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 1
{{1,2,4},{3,5},{6}} => {{1},{2,3,5,6},{4}} => {{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => 1
{{1,2,4,6},{3},{5}} => {{1,4},{2},{3,5,6}} => {{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 3
{{1,2,4},{3,6},{5}} => {{1,3,5,6},{2},{4}} => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 2
{{1,2,4},{3},{5,6}} => {{1,2},{3,5,6},{4}} => {{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => 1
{{1,2,4},{3},{5},{6}} => {{1},{2},{3,5,6},{4}} => {{1},{2},{3,5,6},{4}} => [1,2,5,4,6,3] => 1
{{1,2,5,6},{3,4}} => {{1,3,4},{2,5,6}} => {{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => 2
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Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Map
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution.
This is defined in [1] and yields the set partition with the number of nestings and crossings exchanged.
This is defined in [1] and yields the set partition with the number of nestings and crossings exchanged.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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.
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.
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