Identifier
-
Mp00176:
Set partitions
—rotate decreasing⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000375: 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}} => {{1,3},{2}} => [3,2,1] => 0
{{1,3},{2}} => {{1},{2,3}} => [1,3,2] => 0
{{1},{2,3}} => {{1,2},{3}} => [2,1,3] => 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}} => {{1,2,4},{3}} => [2,4,3,1] => 0
{{1,2,4},{3}} => {{1,3,4},{2}} => [3,2,4,1] => 0
{{1,2},{3,4}} => {{1,4},{2,3}} => [4,3,2,1] => 1
{{1,2},{3},{4}} => {{1,4},{2},{3}} => [4,2,3,1] => 0
{{1,3,4},{2}} => {{1},{2,3,4}} => [1,3,4,2] => 0
{{1,3},{2,4}} => {{1,3},{2,4}} => [3,4,1,2] => 0
{{1,3},{2},{4}} => {{1},{2,4},{3}} => [1,4,3,2] => 0
{{1,4},{2,3}} => {{1,2},{3,4}} => [2,1,4,3] => 0
{{1},{2,3,4}} => {{1,2,3},{4}} => [2,3,1,4] => 0
{{1},{2,3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => 0
{{1,4},{2},{3}} => {{1},{2},{3,4}} => [1,2,4,3] => 0
{{1},{2,4},{3}} => {{1,3},{2},{4}} => [3,2,1,4] => 0
{{1},{2},{3,4}} => {{1},{2,3},{4}} => [1,3,2,4] => 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}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 0
{{1,2,3,5},{4}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 0
{{1,2,3},{4,5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 1
{{1,2,3},{4},{5}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 0
{{1,2,4,5},{3}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 0
{{1,2,4},{3,5}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 1
{{1,2,4},{3},{5}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 0
{{1,2,5},{3,4}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 1
{{1,2},{3,4,5}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 1
{{1,2},{3,4},{5}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 1
{{1,2,5},{3},{4}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 0
{{1,2},{3,5},{4}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 1
{{1,2},{3},{4,5}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 1
{{1,2},{3},{4},{5}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 0
{{1,3,4,5},{2}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 0
{{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 0
{{1,3,4},{2},{5}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 0
{{1,3,5},{2,4}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 0
{{1,3},{2,4,5}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 0
{{1,3},{2,4},{5}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 0
{{1,3,5},{2},{4}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 0
{{1,3},{2,5},{4}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 0
{{1,3},{2},{4,5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 1
{{1,3},{2},{4},{5}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 0
{{1,4,5},{2,3}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 0
{{1,4},{2,3,5}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 0
{{1,4},{2,3},{5}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 0
{{1,5},{2,3,4}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 0
{{1},{2,3,4,5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 0
{{1},{2,3,4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 0
{{1,5},{2,3},{4}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 0
{{1},{2,3,5},{4}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 0
{{1},{2,3},{4,5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 0
{{1},{2,3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 0
{{1,4,5},{2},{3}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 0
{{1,4},{2,5},{3}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 0
{{1,4},{2},{3,5}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 0
{{1,4},{2},{3},{5}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => 0
{{1,5},{2,4},{3}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 0
{{1},{2,4,5},{3}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 0
{{1},{2,4},{3,5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 0
{{1},{2,4},{3},{5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 0
{{1,5},{2},{3,4}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 0
{{1},{2,5},{3,4}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 1
{{1},{2},{3,4,5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 0
{{1},{2},{3,4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 0
{{1,5},{2},{3},{4}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 0
{{1},{2,5},{3},{4}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 0
{{1},{2},{3,5},{4}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => 0
{{1},{2},{3},{4,5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 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}} => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 0
{{1,2,3,4,6},{5}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 0
{{1,2,3,4},{5,6}} => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 1
{{1,2,3,4},{5},{6}} => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 0
{{1,2,3,5,6},{4}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 0
{{1,2,3,5},{4,6}} => {{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 1
{{1,2,3,5},{4},{6}} => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 0
{{1,2,3,6},{4,5}} => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 1
{{1,2,3},{4,5,6}} => {{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 1
{{1,2,3},{4,5},{6}} => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 1
{{1,2,3,6},{4},{5}} => {{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => 0
{{1,2,3},{4,6},{5}} => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 1
{{1,2,3},{4},{5,6}} => {{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => 1
{{1,2,3},{4},{5},{6}} => {{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => 0
{{1,2,4,5,6},{3}} => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 0
{{1,2,4,5},{3,6}} => {{1,3,4,6},{2,5}} => [3,5,4,6,2,1] => 1
{{1,2,4,5},{3},{6}} => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 0
{{1,2,4,6},{3,5}} => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => 1
{{1,2,4},{3,5,6}} => {{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => 1
{{1,2,4},{3,5},{6}} => {{1,3,6},{2,4},{5}} => [3,4,6,2,5,1] => 1
{{1,2,4,6},{3},{5}} => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 0
{{1,2,4},{3,6},{5}} => {{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => 1
{{1,2,4},{3},{5,6}} => {{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => 1
{{1,2,4},{3},{5},{6}} => {{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => 0
{{1,2,5,6},{3,4}} => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 1
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Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
rotate decreasing
Description
The rotation of the set partition obtained by subtracting 1 from each entry different from 1, and replacing 1 with the largest entry.
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