Identifier
-
Mp00115:
Set partitions
—Kasraoui-Zeng⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001513: Permutations ⟶ ℤ
Values
{{1}} => {{1}} => [1] => 0
{{1,2}} => {{1,2}} => [2,1] => 0
{{1},{2}} => {{1},{2}} => [1,2] => 0
{{1,2,3}} => {{1,2,3}} => [2,3,1] => 0
{{1,2},{3}} => {{1,2},{3}} => [2,1,3] => 0
{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => 0
{{1},{2,3}} => {{1},{2,3}} => [1,3,2] => 0
{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => 0
{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => 0
{{1,2,3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => 0
{{1,2,4},{3}} => {{1,2,4},{3}} => [2,4,3,1] => 0
{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => 0
{{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => 0
{{1,3,4},{2}} => {{1,3,4},{2}} => [3,2,4,1] => 0
{{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => 1
{{1,3},{2},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => 0
{{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => 0
{{1},{2,3,4}} => {{1},{2,3,4}} => [1,3,4,2] => 0
{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => 0
{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => 0
{{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => 0
{{1},{2},{3,4}} => {{1},{2},{3,4}} => [1,2,4,3] => 0
{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => 0
{{1,2,3,4,5}} => {{1,2,3,4,5}} => [2,3,4,5,1] => 0
{{1,2,3,4},{5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 0
{{1,2,3,5},{4}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 0
{{1,2,3},{4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 0
{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 0
{{1,2,4,5},{3}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 0
{{1,2,4},{3,5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 1
{{1,2,4},{3},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 0
{{1,2,5},{3,4}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 0
{{1,2},{3,4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 0
{{1,2},{3,4},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 0
{{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 0
{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 0
{{1,3,4,5},{2}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 0
{{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 1
{{1,3,4},{2},{5}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 0
{{1,3,5},{2,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 2
{{1,3},{2,4,5}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 1
{{1,3},{2,4},{5}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 1
{{1,3,5},{2},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 0
{{1,3},{2,5},{4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 1
{{1,3},{2},{4,5}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 0
{{1,4,5},{2,3}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 0
{{1,4},{2,3,5}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 1
{{1,4},{2,3},{5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 0
{{1,5},{2,3,4}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 0
{{1},{2,3,4,5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 0
{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 0
{{1,5},{2,3},{4}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 0
{{1},{2,3,5},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 0
{{1},{2,3},{4,5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 0
{{1,4,5},{2},{3}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 0
{{1,4},{2,5},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 1
{{1,4},{2},{3,5}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 1
{{1,4},{2},{3},{5}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 0
{{1,5},{2,4},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 0
{{1},{2,4,5},{3}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 0
{{1},{2,4},{3,5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 1
{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => 0
{{1,5},{2},{3,4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 0
{{1},{2,5},{3,4}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 0
{{1},{2},{3,4,5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 0
{{1},{2},{3,4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 0
{{1},{2,5},{3},{4}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 0
{{1},{2},{3,5},{4}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 0
{{1,2,3,4,5},{6}} => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 0
{{1,2,3,4,6},{5}} => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 0
{{1,2,3,4},{5,6}} => {{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 0
{{1,2,3,4},{5},{6}} => {{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 0
{{1,2,3,5,6},{4}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 0
{{1,2,3,5},{4,6}} => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 1
{{1,2,3,5},{4},{6}} => {{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => 0
{{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 0
{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 0
{{1,2,3},{4,5},{6}} => {{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => 0
{{1,2,3,6},{4},{5}} => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 0
{{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 0
{{1,2,3},{4},{5,6}} => {{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 0
{{1,2,3},{4},{5},{6}} => {{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => 0
{{1,2,4,5,6},{3}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 0
{{1,2,4,5},{3,6}} => {{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => 1
{{1,2,4,5},{3},{6}} => {{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => 0
{{1,2,4,6},{3,5}} => {{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 2
{{1,2,4},{3,5,6}} => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 1
{{1,2,4},{3,5},{6}} => {{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => 1
{{1,2,4,6},{3},{5}} => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 0
{{1,2,4},{3,6},{5}} => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 1
{{1,2,4},{3},{5,6}} => {{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => 0
{{1,2,4},{3},{5},{6}} => {{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => 0
{{1,2,5,6},{3,4}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 0
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Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see St000155The number of exceedances (also excedences) of a permutation..
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see St000155The number of exceedances (also excedences) of a permutation..
Map
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution.
This is defined in [1] and yields the set partition with the number of nestings and crossings exchanged.
This is defined in [1] and yields the set partition with the number of nestings and crossings exchanged.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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