Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001579: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => 1
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [2,1,3] => 1
{{1,2},{3}} => [2,1,3] => [2,3,1] => 2
{{1,3},{2}} => [3,2,1] => [3,2,1] => 1
{{1},{2,3}} => [1,3,2] => [3,1,2] => 2
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [2,1,3,4] => 1
{{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => [4,2,1,3] => 4
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,3,4,1] => 3
{{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 2
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => 4
{{1,3},{2},{4}} => [3,2,1,4] => [3,4,2,1] => 3
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 2
{{1},{2,3,4}} => [1,3,4,2] => [3,1,2,4] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [4,3,1,2] => 3
{{1},{2},{3,4}} => [1,2,4,3] => [4,1,2,3] => 3
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [2,1,3,4,5] => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,1,4,5] => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,2,1,3,4] => 5
{{1,2,3},{4,5}} => [2,3,1,5,4] => [2,1,5,3,4] => 3
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [2,3,4,1,5] => 3
{{1,2,4,5},{3}} => [2,4,3,5,1] => [4,2,3,1,5] => 5
{{1,2,4},{3,5}} => [2,4,5,1,3] => [2,4,1,3,5] => 3
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [2,4,3,1,5] => 4
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,4,2,1,3] => 6
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,4,5,3] => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,4,1,5,3] => 4
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,2,3,1,4] => 6
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [5,2,1,4,3] => 4
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,3,4,5,1] => 4
{{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 3
{{1,3,4},{2,5}} => [3,5,4,1,2] => [3,5,4,1,2] => 7
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [3,4,2,5,1] => 4
{{1,3,5},{2,4}} => [3,4,5,2,1] => [3,2,1,4,5] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [3,1,4,2,5] => 3
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [3,4,5,1,2] => 6
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [3,2,1,5,4] => 4
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [5,3,4,1,2] => 4
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,5,4,1] => 4
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,4,5,2,1] => 5
{{1,4,5},{2,3}} => [4,3,2,5,1] => [4,3,5,2,1] => 4
{{1,4},{2,3,5}} => [4,3,5,1,2] => [4,5,1,3,2] => 5
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,5,3,2,1] => 5
{{1,5},{2,3,4}} => [5,3,4,2,1] => [5,3,2,4,1] => 2
{{1},{2,3,4,5}} => [1,3,4,5,2] => [3,1,2,4,5] => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,2,4,5] => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [5,3,4,2,1] => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [5,3,1,2,4] => 6
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [3,1,5,2,4] => 4
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,4,2,5] => 2
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [4,2,3,5,1] => 2
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,5,3,1,2] => 6
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [4,5,2,3,1] => 4
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [2,4,3,5,1] => 5
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [4,1,3,2,5] => 4
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [4,5,1,2,3] => 6
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => 3
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,4,2,3,1] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [5,4,3,1,2] => 5
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [4,1,2,3,5] => 3
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,4,2,3,5] => 2
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,2,3,4,1] => 1
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [5,1,3,2,4] => 5
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [5,4,1,2,3] => 5
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [5,1,2,3,4] => 4
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [2,3,1,4,5,6] => 2
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,2,1,3,4,5] => 6
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [2,1,6,3,4,5] => 4
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [2,3,4,1,5,6] => 3
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [5,2,3,1,4,6] => 6
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [2,5,1,3,4,6] => 4
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [2,5,3,1,4,6] => 5
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,5,2,1,3,4] => 8
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [2,1,3,5,4,6] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [2,3,1,5,4,6] => 3
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [6,2,3,1,4,5] => 7
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [6,2,1,5,3,4] => 6
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [2,1,3,6,4,5] => 3
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [2,3,4,5,1,6] => 4
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [4,2,3,5,1,6] => 6
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [2,6,4,1,3,5] => 7
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [2,4,3,5,1,6] => 5
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [4,2,1,3,5,6] => 4
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [2,1,4,3,5,6] => 2
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [2,4,5,1,3,6] => 5
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [4,2,1,6,3,5] => 6
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [6,2,4,1,3,5] => 8
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,2,6,3,1,5] => 8
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [2,4,5,3,1,6] => 6
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [5,2,4,3,1,6] => 8
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Description
The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
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