Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001083: 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] => 1
{{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] => 0
{{1},{2},{3}} => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,2,1,3] => 0
{{1,2,3},{4}} => [2,3,1,4] => [1,4,2,3] => 1
{{1,2,4},{3}} => [2,4,3,1] => [1,4,3,2] => 2
{{1,2},{3,4}} => [2,1,4,3] => [4,1,3,2] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [4,1,2,3] => 0
{{1,3,4},{2}} => [3,2,4,1] => [2,1,4,3] => 2
{{1,3},{2,4}} => [3,4,1,2] => [2,3,1,4] => 0
{{1,3},{2},{4}} => [3,2,1,4] => [1,2,4,3] => 1
{{1,4},{2,3}} => [4,3,2,1] => [1,3,2,4] => 1
{{1},{2,3,4}} => [1,3,4,2] => [2,3,4,1] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [2,4,1,3] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [3,1,4,2] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [3,1,2,4] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [4,3,1,2] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [4,2,3,1] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,2,3,1,4] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [5,2,1,3,4] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,2,1,4,3] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,5,2,4,3] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,5,2,3,4] => 1
{{1,2,4,5},{3}} => [2,4,3,5,1] => [5,3,2,1,4] => 0
{{1,2,4},{3,5}} => [2,4,5,1,3] => [5,3,4,2,1] => 0
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,5,3,2,4] => 2
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,5,4,3,2] => 3
{{1,2},{3,4,5}} => [2,1,4,5,3] => [5,1,3,4,2] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [5,1,3,2,4] => 1
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,4,2,1,3] => 0
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [5,1,4,3,2] => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [5,1,2,4,3] => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [5,1,2,3,4] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,5,3,1,4] => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [2,4,3,1,5] => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,1,5,3,4] => 2
{{1,3,5},{2,4}} => [3,4,5,2,1] => [2,3,1,4,5] => 0
{{1,3},{2,4,5}} => [3,4,1,5,2] => [2,1,3,4,5] => 0
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [2,3,5,4,1] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,1,5,4,3] => 4
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [2,4,3,5,1] => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,2,5,4,3] => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,2,5,3,4] => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,2,1,5,4] => 3
{{1,4},{2,3,5}} => [4,3,5,1,2] => [3,2,4,1,5] => 0
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,3,2,5,4] => 3
{{1,5},{2,3,4}} => [5,3,4,2,1] => [4,2,1,3,5] => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [2,3,4,5,1] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [2,3,5,1,4] => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [4,2,1,5,3] => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [2,4,1,3,5] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [2,5,1,4,3] => 3
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [2,5,1,3,4] => 1
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [3,5,2,1,4] => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [3,4,2,1,5] => 0
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [3,5,4,2,1] => 1
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [3,1,5,2,4] => 2
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,4,3,2,5] => 2
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [3,1,2,4,5] => 0
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [3,4,5,1,2] => 0
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [3,1,2,5,4] => 2
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [4,1,5,3,2] => 2
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [4,1,3,2,5] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [5,3,4,1,2] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [5,3,1,2,4] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [4,5,2,1,3] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [4,1,2,3,5] => 0
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [5,4,1,3,2] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [5,2,4,1,3] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [5,2,3,4,1] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,2,3,4,1,5] => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [6,2,3,1,4,5] => 0
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,2,3,1,5,4] => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [6,2,1,3,5,4] => 1
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [6,2,1,3,4,5] => 0
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [6,2,4,3,1,5] => 1
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [6,2,4,5,3,1] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [6,2,1,4,3,5] => 2
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,2,1,5,4,3] => 4
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,6,2,4,5,3] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,6,2,4,3,5] => 2
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [6,2,5,3,1,4] => 1
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [1,6,2,5,4,3] => 3
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [1,6,2,3,5,4] => 2
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,6,2,3,4,5] => 1
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [6,3,2,4,1,5] => 0
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [6,3,5,4,2,1] => 1
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [6,3,2,1,4,5] => 0
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [6,3,4,1,5,2] => 1
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [6,3,1,4,5,2] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [6,3,4,2,5,1] => 0
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [6,3,2,1,5,4] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [6,3,5,4,1,2] => 1
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [1,6,3,2,5,4] => 4
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,6,3,2,4,5] => 2
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [6,4,3,2,1,5] => 0
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Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
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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