Identifier
-
Mp00174:
Set partitions
—dual major index to intertwining number⟶
Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001727: Permutations ⟶ ℤ
Values
{{1}} => {{1}} => {{1}} => [1] => 0
{{1,2}} => {{1,2}} => {{1,2}} => [2,1] => 0
{{1},{2}} => {{1},{2}} => {{1},{2}} => [1,2] => 0
{{1,2,3}} => {{1,2,3}} => {{1,2,3}} => [2,3,1] => 0
{{1,2},{3}} => {{1,2},{3}} => {{1,2},{3}} => [2,1,3] => 0
{{1,3},{2}} => {{1},{2,3}} => {{1},{2,3}} => [1,3,2] => 0
{{1},{2,3}} => {{1,3},{2}} => {{1,3},{2}} => [3,2,1] => 1
{{1},{2},{3}} => {{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => 0
{{1,2,3,4}} => {{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => 0
{{1,2,3},{4}} => {{1,2,3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => 0
{{1,2,4},{3}} => {{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => 0
{{1,2},{3,4}} => {{1,2,4},{3}} => {{1,2,4},{3}} => [2,4,3,1] => 1
{{1,2},{3},{4}} => {{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => 0
{{1,3,4},{2}} => {{1},{2,3,4}} => {{1},{2,3,4}} => [1,3,4,2] => 0
{{1,3},{2,4}} => {{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => 1
{{1,3},{2},{4}} => {{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => 0
{{1,4},{2,3}} => {{1,3,4},{2}} => {{1,3,4},{2}} => [3,2,4,1] => 1
{{1},{2,3,4}} => {{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => 2
{{1},{2,3},{4}} => {{1,3},{2},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => 1
{{1,4},{2},{3}} => {{1},{2},{3,4}} => {{1},{2},{3,4}} => [1,2,4,3] => 0
{{1},{2,4},{3}} => {{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => 1
{{1},{2},{3,4}} => {{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => 2
{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => 0
{{1,2,3,4,5}} => {{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}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 0
{{1,2,3,5},{4}} => {{1,2,3},{4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 0
{{1,2,3},{4,5}} => {{1,2,3,5},{4}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 1
{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 0
{{1,2,4,5},{3}} => {{1,2},{3,4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 0
{{1,2,4},{3,5}} => {{1,2,5},{3,4}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}} => {{1,2},{3,4},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 0
{{1,2,5},{3,4}} => {{1,2,4,5},{3}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 1
{{1,2},{3,4,5}} => {{1,2,4},{3,5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 2
{{1,2},{3,4},{5}} => {{1,2,4},{3},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 1
{{1,2,5},{3},{4}} => {{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 0
{{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}} => {{1,2,5},{3},{4}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 2
{{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 0
{{1,3,4,5},{2}} => {{1},{2,3,4,5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 0
{{1,3,4},{2,5}} => {{1,5},{2,3,4}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 1
{{1,3,4},{2},{5}} => {{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 0
{{1,3,5},{2,4}} => {{1,4,5},{2,3}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 1
{{1,3},{2,4,5}} => {{1,4},{2,3,5}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 2
{{1,3},{2,4},{5}} => {{1,4},{2,3},{5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 1
{{1,3,5},{2},{4}} => {{1},{2,3},{4,5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 0
{{1,3},{2,5},{4}} => {{1},{2,3,5},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 1
{{1,3},{2},{4,5}} => {{1,5},{2,3},{4}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 2
{{1,3},{2},{4},{5}} => {{1},{2,3},{4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 0
{{1,4,5},{2,3}} => {{1,3,4,5},{2}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 1
{{1,4},{2,3,5}} => {{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 2
{{1,4},{2,3},{5}} => {{1,3,4},{2},{5}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 1
{{1,5},{2,3,4}} => {{1,3},{2,4,5}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 2
{{1},{2,3,4,5}} => {{1,3,5},{2,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 3
{{1},{2,3,4},{5}} => {{1,3},{2,4},{5}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 2
{{1,5},{2,3},{4}} => {{1,3},{2},{4,5}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 1
{{1},{2,3,5},{4}} => {{1,3,5},{2},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 2
{{1},{2,3},{4,5}} => {{1,3},{2,5},{4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 3
{{1},{2,3},{4},{5}} => {{1,3},{2},{4},{5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 1
{{1,4,5},{2},{3}} => {{1},{2},{3,4,5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 0
{{1,4},{2,5},{3}} => {{1},{2,5},{3,4}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 1
{{1,4},{2},{3,5}} => {{1,5},{2},{3,4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 2
{{1,4},{2},{3},{5}} => {{1},{2},{3,4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 0
{{1,5},{2,4},{3}} => {{1},{2,4,5},{3}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 1
{{1},{2,4,5},{3}} => {{1},{2,4},{3,5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 2
{{1},{2,4},{3,5}} => {{1,5},{2,4},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 3
{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => 1
{{1,5},{2},{3,4}} => {{1,4,5},{2},{3}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 2
{{1},{2,5},{3,4}} => {{1,4},{2},{3,5}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 3
{{1},{2},{3,4,5}} => {{1,4},{2,5},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 4
{{1},{2},{3,4},{5}} => {{1,4},{2},{3},{5}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 2
{{1,5},{2},{3},{4}} => {{1},{2},{3},{4,5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 0
{{1},{2,5},{3},{4}} => {{1},{2},{3,5},{4}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => 1
{{1},{2},{3,5},{4}} => {{1},{2,5},{3},{4}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 2
{{1},{2},{3},{4,5}} => {{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 3
{{1},{2},{3},{4},{5}} => {{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}} => {{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}} => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 0
{{1,2,3,4,6},{5}} => {{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}} => {{1,2,3,4,6},{5}} => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 1
{{1,2,3,4},{5},{6}} => {{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},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 0
{{1,2,3,5},{4,6}} => {{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 1
{{1,2,3,5},{4},{6}} => {{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}} => {{1,2,3,5,6},{4}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 1
{{1,2,3},{4,5,6}} => {{1,2,3,5},{4,6}} => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 2
{{1,2,3},{4,5},{6}} => {{1,2,3,5},{4},{6}} => {{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => 1
{{1,2,3,6},{4},{5}} => {{1,2,3},{4},{5,6}} => {{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 0
{{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 1
{{1,2,3},{4},{5,6}} => {{1,2,3,6},{4},{5}} => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 2
{{1,2,3},{4},{5},{6}} => {{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},{3,4,5,6}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 0
{{1,2,4,5},{3,6}} => {{1,2,6},{3,4,5}} => {{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 1
{{1,2,4,5},{3},{6}} => {{1,2},{3,4,5},{6}} => {{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => 0
{{1,2,4,6},{3,5}} => {{1,2,5,6},{3,4}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 1
{{1,2,4},{3,5,6}} => {{1,2,5},{3,4,6}} => {{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => 2
{{1,2,4},{3,5},{6}} => {{1,2,5},{3,4},{6}} => {{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 1
{{1,2,4,6},{3},{5}} => {{1,2},{3,4},{5,6}} => {{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => 0
{{1,2,4},{3,6},{5}} => {{1,2},{3,4,6},{5}} => {{1,2},{3,4,6},{5}} => [2,1,4,6,5,3] => 1
{{1,2,4},{3},{5,6}} => {{1,2,6},{3,4},{5}} => {{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => 2
{{1,2,4},{3},{5},{6}} => {{1,2},{3,4},{5},{6}} => {{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => 0
{{1,2,5,6},{3,4}} => {{1,2,4,5,6},{3}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 1
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Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation π is a pair i<j such that π(j)≤min. Thus, an invisible inversion satisfies \pi(i) > \pi(j) > i.
A visible inversion of a permutation π is a pair i<j such that π(j)≤min. Thus, an invisible inversion satisfies \pi(i) > \pi(j) > i.
Map
dual major index to intertwining number
Description
A bijection sending the dual major index of a set partition to its intertwining number.
More precisely, St000493The los statistic of a set partition.(P) = St000490The intertwining number of a set partition.(\phi(P)) for all set partitions P.
This is the inverse of Mp00171intertwining number to dual major index.
More precisely, St000493The los statistic of a set partition.(P) = St000490The intertwining number of a set partition.(\phi(P)) for all set partitions P.
This is the inverse of Mp00171intertwining number to dual major index.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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.
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