Identifier
-
Mp00112:
Set partitions
—complement⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001511: Permutations ⟶ ℤ
Values
{{1}} => {{1}} => [1] => [1] => 0
{{1,2}} => {{1,2}} => [2,1] => [1,2] => 0
{{1},{2}} => {{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => {{1,2,3}} => [2,3,1] => [1,2,3] => 0
{{1,2},{3}} => {{1},{2,3}} => [1,3,2] => [1,2,3] => 0
{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => [1,3,2] => 1
{{1},{2,3}} => {{1,2},{3}} => [2,1,3] => [1,2,3] => 0
{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}} => {{1},{2,3,4}} => [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}} => {{1,3,4},{2}} => [3,2,4,1] => [1,3,4,2] => 2
{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}} => {{1},{2},{3,4}} => [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}} => {{1,2,4},{3}} => [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}} => {{1,3},{2,4}} => [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}} => {{1},{2,4},{3}} => [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}} => {{1,4},{2,3}} => [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}} => {{1,2,3},{4}} => [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}} => {{1,3},{2},{4}} => [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}} => {{1,2},{3},{4}} => [2,1,3,4] => [1,2,3,4] => 0
{{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}} => [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => [1,3,4,5,2] => 3
{{1,2,3},{4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,2,4},{3},{5}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,2,4,5,3] => 2
{{1,2,5},{3,4}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => [1,4,5,2,3] => 4
{{1,2},{3,4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,4,5,2,3] => 4
{{1,2},{3,5},{4}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => [1,4,2,3,5] => 2
{{1,3,4},{2},{5}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,2,3,5,4] => 1
{{1,3,5},{2,4}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,3},{2,4},{5}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,2,4,3,5] => 1
{{1,3,5},{2},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,4,2,3,5] => 2
{{1,3},{2},{4,5}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,3},{2},{4},{5}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,3,5,4] => 1
{{1,4,5},{2,3}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,4},{2,3,5}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,2,5,3,4] => 2
{{1,5},{2,3,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => [1,5,2,3,4] => 3
{{1},{2,3,4,5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,2,3,4,5] => 0
{{1,5},{2,3},{4}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,5,2,3,4] => 3
{{1},{2,3,5},{4}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1},{2,3},{4,5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1},{2,3},{4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,3,4,5] => 0
{{1,4,5},{2},{3}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,4},{2,5},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,4,2,5,3] => 3
{{1,4},{2},{3,5}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,2,5,3,4] => 2
{{1,5},{2,4},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,5,2,4,3] => 4
{{1},{2,4,5},{3}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1},{2,4},{3,5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,4,3,5] => 1
{{1,5},{2},{3,4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,5,2,3,4] => 3
{{1},{2,5},{3,4}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,4,2,3,5] => 2
{{1},{2},{3,4,5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3,4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,2,3,4,5] => 0
{{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,5,2,3,4] => 3
{{1},{2,5},{3},{4}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,4,2,3,5] => 2
{{1},{2},{3,5},{4}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1},{2},{3},{4,5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,2,3,4,5] => 0
{{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}} => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
{{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => [1,2,3,4,5,6] => 0
{{1,2,3,4,6},{5}} => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => [1,3,4,5,6,2] => 4
{{1,2,3,4},{5,6}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => [1,2,3,4,5,6] => 0
{{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => [1,2,3,4,5,6] => 0
{{1,2,3,5,6},{4}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,2,4,5,6,3] => 3
{{1,2,3,5},{4,6}} => {{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => [1,3,2,4,5,6] => 1
{{1,2,3,5},{4},{6}} => {{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => [1,2,4,5,6,3] => 3
{{1,2,3,6},{4,5}} => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => [1,4,5,6,2,3] => 6
{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,2,3,4,5,6] => 0
{{1,2,3},{4,5},{6}} => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => [1,2,3,4,5,6] => 0
{{1,2,3,6},{4},{5}} => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => [1,4,5,6,2,3] => 6
{{1,2,3},{4,6},{5}} => {{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => [1,3,2,4,5,6] => 1
{{1,2,3},{4},{5,6}} => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => [1,2,3,4,5,6] => 0
{{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => [1,2,3,4,5,6] => 0
{{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,2,3,5,6,4] => 2
{{1,2,4,5},{3,6}} => {{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => [1,4,2,3,5,6] => 2
{{1,2,4,5},{3},{6}} => {{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => [1,2,3,5,6,4] => 2
{{1,2,4,6},{3,5}} => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => [1,3,5,6,2,4] => 5
{{1,2,4},{3,5,6}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [1,2,4,3,5,6] => 1
{{1,2,4},{3,5},{6}} => {{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => [1,2,4,3,5,6] => 1
{{1,2,4,6},{3},{5}} => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => [1,3,5,6,2,4] => 5
{{1,2,4},{3,6},{5}} => {{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => [1,4,2,3,5,6] => 2
{{1,2,4},{3},{5,6}} => {{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => [1,2,3,5,6,4] => 2
{{1,2,4},{3},{5},{6}} => {{1},{2},{3,5,6},{4}} => [1,2,5,4,6,3] => [1,2,3,5,6,4] => 2
{{1,2,5,6},{3,4}} => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,2,5,6,3,4] => 4
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Description
The minimal number of transpositions needed to sort a permutation in either direction.
For a permutation $\sigma$, this is $\min\{ \operatorname{inv}(\sigma),\operatorname{inv}(\tau)\}$ where $\tau$ is the reverse permutation sending $i$ to $\sigma(n+1-i)$.
For a permutation $\sigma$, this is $\min\{ \operatorname{inv}(\sigma),\operatorname{inv}(\tau)\}$ where $\tau$ is the reverse permutation sending $i$ to $\sigma(n+1-i)$.
Map
complement
Description
The complement of a set partition obtained by replacing $i$ with $n+1-i$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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