Identifier
Values
{{1}} => [1] => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [1,3,2] => [1,3,2] => 0
{{1,2},{3}} => [2,1,3] => [3,1,2] => [3,1,2] => 0
{{1,3},{2}} => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}} => [1,3,2] => [2,3,1] => [2,3,1] => 1
{{1},{2},{3}} => [1,2,3] => [3,2,1] => [3,2,1] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,3,4,2] => [1,3,4,2] => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,4,1,2] => [3,4,1,2] => 2
{{1,2,4},{3}} => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2},{3,4}} => [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 1
{{1,3,4},{2}} => [3,2,4,1] => [4,1,3,2] => [4,1,3,2] => 0
{{1,3},{2,4}} => [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 0
{{1,3},{2},{4}} => [3,2,1,4] => [4,3,1,2] => [4,3,1,2] => 1
{{1,4},{2,3}} => [4,3,2,1] => [2,4,1,3] => [2,4,1,3] => 1
{{1},{2,3,4}} => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [4,3,2,1] => [4,3,2,1] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [2,4,3,1] => [2,4,3,1] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [3,2,1,4] => [3,2,1,4] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [3,1,4,2] => [3,1,4,2] => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 1
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,4,2,5,3] => [1,4,2,5,3] => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
  • $i < j \leq \pi(i) < \pi(j)$, or
  • $-i < j \leq -\pi(i) < \pi(j)$, or
  • $i > j > \pi(i) > \pi(j)$.
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 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 $n-1$.