Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001683: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [3,2,1] => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,1,2,3] => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,1,2,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [4,3,1,2] => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [4,1,3,2] => 1
{{1,3},{2,4}} => [3,4,1,2] => [2,4,1,3] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}} => [1,3,4,2] => [1,4,2,3] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [3,1,4,2] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,1,2,3,4] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,1,2,3,5] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,4,1,2,3] => 0
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,1,2,5,4] => 1
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,1,2,4,5] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [5,1,2,4,3] => 1
{{1,2,4},{3,5}} => [2,4,5,1,3] => [3,5,1,2,4] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,3,1,2,5] => 0
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,4,3,1,2] => 0
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,3,4] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 1
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [4,1,2,5,3] => 1
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [5,1,3,2,4] => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [2,5,4,1,3] => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [4,1,3,2,5] => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => [5,2,1,3,4] => 0
{{1,3},{2,4,5}} => [3,4,1,5,2] => [5,2,4,1,3] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [2,4,1,3,5] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [5,4,1,3,2] => 1
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [4,2,5,1,3] => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,4,5] => 0
{{1,4,5},{2,3}} => [4,3,2,5,1] => [5,1,4,3,2] => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [2,5,1,4,3] => 2
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => 0
{{1,5},{2,3,4}} => [5,3,4,2,1] => [4,2,1,5,3] => 1
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,2,3,4] => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,2,3,5] => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [4,1,5,3,2] => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,4,2,3] => 2
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [5,1,3,4,2] => 2
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [2,5,3,1,4] => 1
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [3,5,1,4,2] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [3,1,4,2,5] => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,2,4,3] => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,3,5,2,4] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => 2
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [4,3,1,5,2] => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => 3
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,3,4] => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [4,1,3,5,2] => 2
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,4,2,5,3] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 0
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,5,1,2,3,4] => 0
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [4,1,2,3,6,5] => 1
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [4,1,2,3,5,6] => 0
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [6,1,2,3,5,4] => 1
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [4,6,1,2,3,5] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [5,4,1,2,3,6] => 0
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,5,4,1,2,3] => 0
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,1,2,6,4,5] => 1
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,1,2,5,4,6] => 1
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [5,1,2,3,6,4] => 1
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,1,2,6,5,4] => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,1,2,4,6,5] => 1
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,1,2,4,5,6] => 0
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [6,1,2,4,3,5] => 1
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [3,6,5,1,2,4] => 2
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [5,1,2,4,3,6] => 1
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [6,3,1,2,4,5] => 0
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [6,3,5,1,2,4] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [3,5,1,2,4,6] => 1
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [6,5,1,2,4,3] => 1
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [5,3,6,1,2,4] => 1
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,3,1,2,6,5] => 1
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,3,1,2,5,6] => 0
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [6,1,2,5,4,3] => 2
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Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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