Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000799: Permutations ⟶ ℤ
Values
{{1,2}} => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => 1
{{1,3},{2}} => [3,2,1] => [2,3,1] => 0
{{1},{2,3}} => [1,3,2] => [3,1,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [3,2,4,1] => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => [4,2,3,1] => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}} => [3,2,4,1] => [2,3,4,1] => 0
{{1,3},{2,4}} => [3,4,1,2] => [4,3,1,2] => 0
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [3,4,2,1] => 0
{{1},{2,3,4}} => [1,3,4,2] => [3,1,4,2] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [3,1,2,4] => 2
{{1,4},{2},{3}} => [4,2,3,1] => [2,4,3,1] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [4,1,3,2] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [3,2,4,5,1] => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [3,2,4,1,5] => 3
{{1,2,3,5},{4}} => [2,3,5,4,1] => [3,2,5,4,1] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => 4
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => 4
{{1,2,4,5},{3}} => [2,4,3,5,1] => [4,2,3,5,1] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,2,5,1,3] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,2,3,1,5] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,2,4,3,1] => 0
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 3
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,2,3,4,1] => 0
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,4,5,3] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,5,3,4] => 3
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,3,4,5,1] => 0
{{1,3,4},{2,5}} => [3,5,4,1,2] => [5,3,4,1,2] => 0
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,3,4,1,5] => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => [4,3,5,2,1] => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => [4,3,1,5,2] => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,3,1,2,5] => 3
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,3,5,4,1] => 0
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [5,3,1,4,2] => 0
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,4,2,5,1] => 1
{{1,4},{2,3,5}} => [4,3,5,1,2] => [3,4,5,1,2] => 0
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [3,4,2,1,5] => 2
{{1,5},{2,3,4}} => [5,3,4,2,1] => [3,5,4,2,1] => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [3,1,4,5,2] => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [3,1,4,2,5] => 3
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,5,2,4,1] => 1
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [3,1,5,4,2] => 2
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [3,1,2,5,4] => 4
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [3,1,2,4,5] => 4
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [2,4,3,5,1] => 0
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [5,4,3,1,2] => 0
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,4,5,1,3] => 1
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [2,4,3,1,5] => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [4,5,3,2,1] => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [4,1,3,5,2] => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [4,1,5,2,3] => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [4,1,3,2,5] => 3
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,5,4,3,1] => 0
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [5,1,4,3,2] => 0
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,5,3,4,1] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [5,1,3,4,2] => 0
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => 0
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,5,3,4] => 0
{{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] => [3,2,4,5,6,1] => 3
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [3,2,4,5,1,6] => 4
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [3,2,4,6,5,1] => 3
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [3,2,4,1,6,5] => 5
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [3,2,4,1,5,6] => 5
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [3,2,5,4,6,1] => 3
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [3,2,5,6,1,4] => 4
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [3,2,5,4,1,6] => 4
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [3,2,6,5,4,1] => 3
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,2,1,5,6,4] => 6
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 6
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [3,2,6,4,5,1] => 3
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,2,1,6,5,4] => 6
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 6
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 6
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [4,2,3,5,6,1] => 4
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [4,2,6,5,1,3] => 2
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [4,2,3,5,1,6] => 5
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [4,2,5,6,3,1] => 2
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [4,2,5,1,6,3] => 3
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [4,2,5,1,3,6] => 4
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [4,2,3,6,5,1] => 4
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [4,2,6,1,5,3] => 3
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,2,3,1,6,5] => 6
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,2,3,1,5,6] => 6
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [5,2,4,3,6,1] => 3
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [5,2,4,6,1,3] => 2
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Description
The number of occurrences of the vincular pattern |213 in a permutation.
This is the number of occurrences of the pattern $(2,1,3)$, such that the letter matched by $2$ is the first entry of the permutation.
This is the number of occurrences of the pattern $(2,1,3)$, such that the letter matched by $2$ is the first entry of the permutation.
Map
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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