Identifier
-
Mp00221:
Set partitions
—conjugate⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000724: Permutations ⟶ ℤ
Values
{{1,2}} => {{1},{2}} => [1,2] => [1,2] => 2
{{1},{2}} => {{1,2}} => [2,1] => [2,1] => 2
{{1,2,3}} => {{1},{2},{3}} => [1,2,3] => [1,3,2] => 3
{{1,2},{3}} => {{1,2},{3}} => [2,1,3] => [2,1,3] => 2
{{1,3},{2}} => {{1},{2,3}} => [1,3,2] => [1,3,2] => 3
{{1},{2,3}} => {{1,3},{2}} => [3,2,1] => [3,2,1] => 3
{{1},{2},{3}} => {{1,2,3}} => [2,3,1] => [2,3,1] => 3
{{1,2,3,4}} => {{1},{2},{3},{4}} => [1,2,3,4] => [1,4,3,2] => 4
{{1,2,3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => [2,1,4,3] => 2
{{1,2,4},{3}} => {{1},{2,3},{4}} => [1,3,2,4] => [1,4,3,2] => 4
{{1,2},{3,4}} => {{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 3
{{1,2},{3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => [2,4,1,3] => 4
{{1,3,4},{2}} => {{1},{2},{3,4}} => [1,2,4,3] => [1,4,3,2] => 4
{{1,3},{2,4}} => {{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => 2
{{1,3},{2},{4}} => {{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}} => {{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 4
{{1},{2,3,4}} => {{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => 3
{{1},{2,3},{4}} => {{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => 4
{{1,4},{2},{3}} => {{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => 4
{{1},{2,4},{3}} => {{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 4
{{1},{2},{3,4}} => {{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 3
{{1},{2},{3},{4}} => {{1,2,3,4}} => [2,3,4,1] => [2,4,3,1] => 4
{{1,2,3,4,5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,5,4,3,2] => 5
{{1,2,3,4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,5,4,3] => 2
{{1,2,3,5},{4}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,5,4,3,2] => 5
{{1,2,3},{4,5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,5,4] => 3
{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [2,5,1,4,3] => 5
{{1,2,4,5},{3}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,5,4,3,2] => 5
{{1,2,4},{3,5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => [3,5,1,4,2] => 4
{{1,2,4},{3},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,5,4,3] => 2
{{1,2,5},{3,4}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,5,4,3,2] => 5
{{1,2},{3,4,5}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => [4,2,5,1,3] => 4
{{1,2},{3,4},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [2,5,4,1,3] => 5
{{1,2,5},{3},{4}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,5,4,3,2] => 5
{{1,2},{3,5},{4}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => 4
{{1,2},{3},{4,5}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [3,2,5,1,4] => 3
{{1,2},{3},{4},{5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,5,4,1,3] => 5
{{1,3,4,5},{2}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,5,4,3,2] => 5
{{1,3,4},{2,5}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,5,3,1,2] => 2
{{1,3,4},{2},{5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,5,4,3] => 2
{{1,3,5},{2,4}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,4,3,2] => 5
{{1,3},{2,4,5}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => [4,2,5,1,3] => 4
{{1,3},{2,4},{5}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => [2,5,4,1,3] => 5
{{1,3,5},{2},{4}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,5,4,3,2] => 5
{{1,3},{2,5},{4}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => [4,3,5,1,2] => 2
{{1,3},{2},{4,5}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => 3
{{1,3},{2},{4},{5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => [2,5,1,4,3] => 5
{{1,4,5},{2,3}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,5,4,3,2] => 5
{{1,4},{2,3,5}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => [3,5,1,4,2] => 4
{{1,4},{2,3},{5}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => 2
{{1,5},{2,3,4}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,4,3,2] => 5
{{1},{2,3,4,5}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,2,4,3,1] => 4
{{1},{2,3,4},{5}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [2,5,4,3,1] => 5
{{1,5},{2,3},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,4,3,2] => 5
{{1},{2,3,5},{4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => [5,3,2,4,1] => 5
{{1},{2,3},{4,5}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [3,2,5,4,1] => 3
{{1},{2,3},{4},{5}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,5,4,3,1] => 5
{{1,4,5},{2},{3}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,5,4,3,2] => 5
{{1,4},{2,5},{3}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => [3,5,4,1,2] => 2
{{1,4},{2},{3,5}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => [3,5,1,4,2] => 4
{{1,4},{2},{3},{5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => 2
{{1,5},{2,4},{3}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => 5
{{1},{2,4,5},{3}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,2,4,3,1] => 4
{{1},{2,4},{3,5}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => [3,5,4,2,1] => 5
{{1},{2,4},{3},{5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => [2,5,4,3,1] => 5
{{1,5},{2},{3,4}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,4,3,2] => 5
{{1},{2,5},{3,4}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => 5
{{1},{2},{3,4,5}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => [4,2,5,3,1] => 5
{{1},{2},{3,4},{5}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,5,4,3,1] => 5
{{1,5},{2},{3},{4}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,4,3,2] => 5
{{1},{2,5},{3},{4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => [5,3,4,2,1] => 4
{{1},{2},{3,5},{4}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => [4,3,2,5,1] => 4
{{1},{2},{3},{4,5}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,5,4,1] => 3
{{1},{2},{3},{4},{5}} => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,5,4,3,1] => 5
{{1,2,3,4,5,6}} => {{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 6
{{1,2,3,4,5},{6}} => {{1,2},{3},{4},{5},{6}} => [2,1,3,4,5,6] => [2,1,6,5,4,3] => 2
{{1,2,3,4,6},{5}} => {{1},{2,3},{4},{5},{6}} => [1,3,2,4,5,6] => [1,6,5,4,3,2] => 6
{{1,2,3,4},{5,6}} => {{1,3},{2},{4},{5},{6}} => [3,2,1,4,5,6] => [3,2,1,6,5,4] => 3
{{1,2,3,4},{5},{6}} => {{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [2,6,1,5,4,3] => 6
{{1,2,3,5,6},{4}} => {{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => [1,6,5,4,3,2] => 6
{{1,2,3,5},{4,6}} => {{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => [3,6,1,5,4,2] => 5
{{1,2,3,5},{4},{6}} => {{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => [2,1,6,5,4,3] => 2
{{1,2,3,6},{4,5}} => {{1},{2,4},{3},{5},{6}} => [1,4,3,2,5,6] => [1,6,5,4,3,2] => 6
{{1,2,3},{4,5,6}} => {{1,4},{2},{3},{5},{6}} => [4,2,3,1,5,6] => [4,2,6,1,5,3] => 4
{{1,2,3},{4,5},{6}} => {{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [2,6,5,1,4,3] => 6
{{1,2,3,6},{4},{5}} => {{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => [1,6,5,4,3,2] => 6
{{1,2,3},{4,6},{5}} => {{1,4},{2,3},{5},{6}} => [4,3,2,1,5,6] => [4,3,2,1,6,5] => 4
{{1,2,3},{4},{5,6}} => {{1,3,4},{2},{5},{6}} => [3,2,4,1,5,6] => [3,2,6,1,5,4] => 3
{{1,2,3},{4},{5},{6}} => {{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [2,6,5,1,4,3] => 6
{{1,2,4,5,6},{3}} => {{1},{2},{3},{4,5},{6}} => [1,2,3,5,4,6] => [1,6,5,4,3,2] => 6
{{1,2,4,5},{3,6}} => {{1,4},{2,5},{3},{6}} => [4,5,3,1,2,6] => [4,6,3,1,5,2] => 5
{{1,2,4,5},{3},{6}} => {{1,2},{3},{4,5},{6}} => [2,1,3,5,4,6] => [2,1,6,5,4,3] => 2
{{1,2,4,6},{3,5}} => {{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => [1,6,5,4,3,2] => 6
{{1,2,4},{3,5,6}} => {{1,4},{2},{3,5},{6}} => [4,2,5,1,3,6] => [4,2,6,1,5,3] => 4
{{1,2,4},{3,5},{6}} => {{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [2,6,5,1,4,3] => 6
{{1,2,4,6},{3},{5}} => {{1},{2,3},{4,5},{6}} => [1,3,2,5,4,6] => [1,6,5,4,3,2] => 6
{{1,2,4},{3,6},{5}} => {{1,4},{2,3,5},{6}} => [4,3,5,1,2,6] => [4,3,6,1,5,2] => 5
{{1,2,4},{3},{5,6}} => {{1,3},{2},{4,5},{6}} => [3,2,1,5,4,6] => [3,2,1,6,5,4] => 3
{{1,2,4},{3},{5},{6}} => {{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [2,6,1,5,4,3] => 6
{{1,2,5,6},{3,4}} => {{1},{2},{3,5},{4},{6}} => [1,2,5,4,3,6] => [1,6,5,4,3,2] => 6
{{1,2,5},{3,4,6}} => {{1,3},{2,5},{4},{6}} => [3,5,1,4,2,6] => [3,6,1,5,4,2] => 5
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Description
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.
Associate an increasing binary tree to the permutation using Mp00061to increasing tree. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1].
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum (St000060The greater neighbor of the maximum.), see also [3].
Associate an increasing binary tree to the permutation using Mp00061to increasing tree. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1].
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum (St000060The greater neighbor of the maximum.), see also [3].
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
conjugate
Description
The conjugate of a set partition.
This is an involution exchanging singletons and circular adjacencies due to [1].
This is an involution exchanging singletons and circular adjacencies due to [1].
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