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