Identifier
-
Mp00217:
Set partitions
—Wachs-White-rho ⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001520: Permutations ⟶ ℤ
Values
{{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] => [3,2,1] => 0
{{1,2},{3}} => {{1,2},{3}} => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => [3,1,2] => 0
{{1},{2,3}} => {{1},{2,3}} => [1,3,2] => [1,3,2] => 0
{{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] => [4,2,3,1] => 1
{{1,2,3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}} => {{1,2,4},{3}} => [2,4,3,1] => [4,2,1,3] => 1
{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}} => {{1,3,4},{2}} => [3,2,4,1] => [4,3,2,1] => 1
{{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => [4,1,2,3] => 1
{{1,3},{2},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => [3,1,2,4] => 0
{{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => [2,4,3,1] => 1
{{1},{2,3,4}} => {{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => 0
{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => [4,3,1,2] => 1
{{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => [1,4,2,3] => 0
{{1},{2},{3,4}} => {{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 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}} => [2,3,4,5,1] => [5,2,3,4,1] => 2
{{1,2,3,4},{5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => [4,2,3,1,5] => 1
{{1,2,3,5},{4}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => [5,2,3,1,4] => 1
{{1,2,3},{4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => 0
{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => 0
{{1,2,4,5},{3}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => [5,2,4,3,1] => 2
{{1,2,4},{3,5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => [5,2,1,3,4] => 1
{{1,2,4},{3},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,2,1,3,5] => 1
{{1,2,5},{3,4}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => [3,2,5,4,1] => 1
{{1,2},{3,4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => 0
{{1,2},{3,4},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,2,4,1,3] => 1
{{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,3,4] => 0
{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => [5,3,2,4,1] => 2
{{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => [2,5,4,3,1] => 1
{{1,3,4},{2},{5}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [4,3,2,1,5] => 1
{{1,3,5},{2,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => [5,1,4,2,3] => 1
{{1,3},{2,4,5}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => [5,4,2,3,1] => 2
{{1,3},{2,4},{5}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,1,2,3,5] => 1
{{1,3,5},{2},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [5,3,2,1,4] => 1
{{1,3},{2,5},{4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => [5,4,2,1,3] => 2
{{1,3},{2},{4,5}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,1,2,5,4] => 0
{{1,3},{2},{4},{5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,1,2,4,5] => 0
{{1,4,5},{2,3}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => [4,5,3,1,2] => 2
{{1,4},{2,3,5}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => [5,1,3,4,2] => 1
{{1,4},{2,3},{5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => [2,4,3,1,5] => 1
{{1,5},{2,3,4}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => [2,5,3,1,4] => 2
{{1},{2,3,4,5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,3,4,2] => 1
{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => 0
{{1,5},{2,3},{4}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => [4,5,3,2,1] => 2
{{1},{2,3,5},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,3,2,4] => 1
{{1},{2,3},{4,5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,4,5},{2},{3}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => [5,3,4,2,1] => 2
{{1,4},{2,5},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,1,2,3,4] => 1
{{1,4},{2},{3,5}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,4,1,3,2] => 2
{{1,4},{2},{3},{5}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => [4,3,1,2,5] => 1
{{1,5},{2,4},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => [2,5,1,4,3] => 1
{{1},{2,4,5},{3}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,4,3,2] => 1
{{1},{2,4},{3,5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,2,3,4] => 1
{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,2,3,5] => 0
{{1,5},{2},{3,4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => [3,4,5,2,1] => 2
{{1},{2,5},{3,4}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,3,5,4,2] => 1
{{1},{2},{3,4,5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => 0
{{1},{2},{3,4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,3,4,1,2] => 2
{{1},{2,5},{3},{4}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,4,2,3] => 1
{{1},{2},{3,5},{4}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,3,4] => 0
{{1},{2},{3},{4,5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{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 strict 3-descents.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
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.
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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