Identifier
Values
{{1}} => {{1}} => [1] => [1] => 0
{{1,2}} => {{1,2}} => [2,1] => [2,1] => 0
{{1},{2}} => {{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => {{1,2,3}} => [2,3,1] => [2,3,1] => 0
{{1,2},{3}} => {{1},{2,3}} => [1,3,2] => [1,3,2] => 0
{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => [3,2,1] => 0
{{1},{2,3}} => {{1,2},{3}} => [2,1,3] => [2,1,3] => 1
{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3},{4}} => {{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => 0
{{1,2,4},{3}} => {{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 0
{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}} => {{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 0
{{1,3,4},{2}} => {{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,4}} => {{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => 0
{{1,3},{2},{4}} => {{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 0
{{1,4},{2,3}} => {{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}} => {{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => 1
{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}} => {{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3,4}} => {{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 2
{{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}} => [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1,2,3},{4},{5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,4,5,3] => 0
{{1,2,4},{3},{5}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,4,3,5,2] => 0
{{1,2},{3,4},{5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1,2},{3},{4},{5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,3,4},{2},{5}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,5,2,3] => 0
{{1,3},{2,4},{5}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,3,5,4,2] => 0
{{1,3},{2},{4},{5}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,4,2,5] => 1
{{1},{2,3},{4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,3,4,2] => 0
{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1},{2},{3,4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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.