Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000039: 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] => [3,2,1] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [2,3,1] => 1
{{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] => 0
{{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] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [2,3,4,1] => 2
{{1,3},{2,4}} => [3,4,1,2] => [4,3,1,2] => 1
{{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] => 1
{{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] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,4,3,1] => 1
{{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] => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [3,2,5,4,1] => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => 0
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [4,2,3,5,1] => 1
{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,2,5,1,3] => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,2,3,1,5] => 0
{{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] => 0
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{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] => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,5,3,4] => 0
{{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] => [2,3,4,5,1] => 3
{{1,3,4},{2,5}} => [3,5,4,1,2] => [5,3,4,1,2] => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,3,4,1,5] => 2
{{1,3,5},{2,4}} => [3,4,5,2,1] => [4,3,5,2,1] => 2
{{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] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,3,5,4,1] => 2
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [5,3,1,4,2] => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,4,2,5,1] => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [3,4,5,1,2] => 4
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [3,4,2,1,5] => 1
{{1,5},{2,3,4}} => [5,3,4,2,1] => [3,5,4,2,1] => 2
{{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] => 1
{{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] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [3,1,2,5,4] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [3,1,2,4,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [2,4,3,5,1] => 2
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [5,4,3,1,2] => 1
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,4,5,1,3] => 3
{{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] => 1
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [4,1,3,5,2] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [4,1,5,2,3] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [4,1,3,2,5] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,5,4,3,1] => 1
{{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] => 1
{{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] => 1
{{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] => 2
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [3,2,4,6,5,1] => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [3,2,4,1,6,5] => 1
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [3,2,4,1,5,6] => 1
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [3,2,5,4,6,1] => 2
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [3,2,5,6,1,4] => 3
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [3,2,5,4,1,6] => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [3,2,6,5,4,1] => 1
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,2,1,5,6,4] => 1
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 0
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [3,2,6,4,5,1] => 1
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,2,1,6,5,4] => 0
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 0
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 0
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [4,2,3,5,6,1] => 2
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [4,2,6,5,1,3] => 3
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [4,2,3,5,1,6] => 1
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [4,2,5,6,3,1] => 3
{{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] => 2
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [4,2,3,6,5,1] => 1
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [4,2,6,1,5,3] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,2,3,1,6,5] => 0
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,2,3,1,5,6] => 0
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [5,2,4,3,6,1] => 1
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Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
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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