Identifier
-
Mp00218:
Set partitions
—inverse Wachs-White-rho⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤ
Values
{{1,2}} => {{1,2}} => [2,1] => ([(0,1)],2) => 1
{{1},{2}} => {{1},{2}} => [1,2] => ([(0,1)],2) => 1
{{1,2,3}} => {{1,2,3}} => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
{{1,2},{3}} => {{1,2},{3}} => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => ([(0,2),(2,1)],3) => 1
{{1},{2,3}} => {{1},{2,3}} => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => ([(0,2),(2,1)],3) => 1
{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
{{1,2,3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
{{1,2,4},{3}} => {{1,2,4},{3}} => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
{{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
{{1,3,4},{2}} => {{1,3,4},{2}} => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
{{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
{{1,3},{2},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
{{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
{{1},{2,3,4}} => {{1},{2,3,4}} => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
{{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
{{1},{2},{3,4}} => {{1},{2},{3,4}} => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
{{1,4},{2,5},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
{{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
{{1},{2},{3},{4},{5},{6}} => {{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
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Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
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.
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
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