Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000039: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [1,3,2] => 0
{{1,2},{3}} => [2,1,3] => [3,1,2] => 0
{{1,3},{2}} => [3,2,1] => [1,2,3] => 0
{{1},{2,3}} => [1,3,2] => [2,3,1] => 1
{{1},{2},{3}} => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,3,4,2] => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,4,1,2] => 2
{{1,2,4},{3}} => [2,4,3,1] => [3,1,2,4] => 0
{{1,2},{3,4}} => [2,1,4,3] => [3,2,4,1] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [3,4,2,1] => 1
{{1,3,4},{2}} => [3,2,4,1] => [4,1,3,2] => 0
{{1,3},{2,4}} => [3,4,1,2] => [1,4,2,3] => 0
{{1,3},{2},{4}} => [3,2,1,4] => [4,3,1,2] => 1
{{1,4},{2,3}} => [4,3,2,1] => [2,4,1,3] => 1
{{1},{2,3,4}} => [1,3,4,2] => [4,2,1,3] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [4,3,2,1] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,2,3,4] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [2,4,3,1] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [3,2,1,4] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [3,1,4,2] => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,3,4,5,2] => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [3,4,5,1,2] => 4
{{1,2,3,5},{4}} => [2,3,5,4,1] => [3,4,1,2,5] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,4,2,5,1] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,4,5,2,1] => 3
{{1,2,4,5},{3}} => [2,4,3,5,1] => [3,5,1,4,2] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,3,5,2,4] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,5,4,1,2] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => [3,2,5,1,4] => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [3,5,2,1,4] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [3,5,4,2,1] => 2
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,1,2,4,5] => 0
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [3,2,5,4,1] => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [3,4,2,1,5] => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [3,4,1,5,2] => 3
{{1,3,4,5},{2}} => [3,2,4,5,1] => [4,1,3,5,2] => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [4,1,2,5,3] => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [4,3,5,1,2] => 3
{{1,3,5},{2,4}} => [3,4,5,2,1] => [4,5,1,2,3] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [4,5,2,3,1] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,4,5,3,2] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [4,3,1,2,5] => 1
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,4,2,5,3] => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [4,3,2,5,1] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [4,3,5,2,1] => 2
{{1,4,5},{2,3}} => [4,3,2,5,1] => [5,4,1,3,2] => 1
{{1,4},{2,3,5}} => [4,3,5,1,2] => [5,1,4,2,3] => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [5,4,3,1,2] => 1
{{1,5},{2,3,4}} => [5,3,4,2,1] => [2,4,5,1,3] => 3
{{1},{2,3,4,5}} => [1,3,4,5,2] => [4,1,5,2,3] => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [4,1,5,3,2] => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [2,4,1,3,5] => 1
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [4,2,1,5,3] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [4,3,2,1,5] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [4,3,1,5,2] => 2
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [5,1,3,4,2] => 0
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [5,1,2,4,3] => 0
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [5,1,3,2,4] => 0
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [5,3,4,1,2] => 2
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [2,5,4,1,3] => 2
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [5,4,2,1,3] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [5,2,1,3,4] => 0
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [5,4,3,2,1] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,3,5,1,4] => 2
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [2,5,4,3,1] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [3,1,5,2,4] => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [3,1,5,4,2] => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,2,3,4,5] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [2,4,1,5,3] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [3,2,1,5,4] => 0
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [3,1,4,2,5] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [3,1,4,5,2] => 2
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,3,4,5,6,2] => 3
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [3,4,5,6,1,2] => 6
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [3,4,5,1,2,6] => 4
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [3,4,5,2,6,1] => 4
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [3,4,5,6,2,1] => 5
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [3,4,6,1,5,2] => 4
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [1,3,4,6,2,5] => 2
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [3,4,6,5,1,2] => 5
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [3,4,2,6,1,5] => 2
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,4,6,2,1,5] => 3
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,4,6,5,2,1] => 4
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [3,4,1,2,5,6] => 2
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,4,2,6,5,1] => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,4,5,2,1,6] => 3
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,4,5,1,6,2] => 5
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [3,5,1,4,6,2] => 3
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [3,5,1,2,6,4] => 3
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [3,5,4,6,1,2] => 5
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [3,5,6,1,2,4] => 5
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [3,5,6,2,4,1] => 3
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,3,5,6,4,2] => 2
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [3,5,4,1,2,6] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [1,3,5,2,6,4] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [3,5,4,2,6,1] => 3
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [3,5,4,6,2,1] => 4
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [3,6,5,1,4,2] => 3
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Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
inverse toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
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