Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
St001461: Permutations ⟶ ℤ
Values
{{1}} => [1] => 1
{{1,2}} => [2,1] => 1
{{1},{2}} => [1,2] => 2
{{1,2,3}} => [2,3,1] => 1
{{1,2},{3}} => [2,1,3] => 2
{{1,3},{2}} => [3,2,1] => 2
{{1},{2,3}} => [1,3,2] => 2
{{1},{2},{3}} => [1,2,3] => 3
{{1,2,3,4}} => [2,3,4,1] => 1
{{1,2,3},{4}} => [2,3,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => 2
{{1,2},{3,4}} => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => 3
{{1,3,4},{2}} => [3,2,4,1] => 2
{{1,3},{2,4}} => [3,4,1,2] => 1
{{1,3},{2},{4}} => [3,2,1,4] => 3
{{1,4},{2,3}} => [4,3,2,1] => 2
{{1},{2,3,4}} => [1,3,4,2] => 2
{{1},{2,3},{4}} => [1,3,2,4] => 3
{{1,4},{2},{3}} => [4,2,3,1] => 3
{{1},{2,4},{3}} => [1,4,3,2] => 3
{{1},{2},{3,4}} => [1,2,4,3] => 3
{{1},{2},{3},{4}} => [1,2,3,4] => 4
{{1,2,3,4,5}} => [2,3,4,5,1] => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => 3
{{1,2,4,5},{3}} => [2,4,3,5,1] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => 3
{{1,2,5},{3},{4}} => [2,5,3,4,1] => 3
{{1,2},{3,5},{4}} => [2,1,5,4,3] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => 3
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => 4
{{1,3,4,5},{2}} => [3,2,4,5,1] => 2
{{1,3,4},{2,5}} => [3,5,4,1,2] => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => 3
{{1,3,5},{2,4}} => [3,4,5,2,1] => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => 3
{{1,3},{2,5},{4}} => [3,5,1,4,2] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => 3
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => 4
{{1,4,5},{2,3}} => [4,3,2,5,1] => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => 3
{{1,5},{2,3,4}} => [5,3,4,2,1] => 2
{{1},{2,3,4,5}} => [1,3,4,5,2] => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => 3
{{1,5},{2,3},{4}} => [5,3,2,4,1] => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => 3
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => 4
{{1,4,5},{2},{3}} => [4,2,3,5,1] => 3
{{1,4},{2,5},{3}} => [4,5,3,1,2] => 2
{{1,4},{2},{3,5}} => [4,2,5,1,3] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => 4
{{1,5},{2,4},{3}} => [5,4,3,2,1] => 3
{{1},{2,4,5},{3}} => [1,4,3,5,2] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => 4
{{1,5},{2},{3,4}} => [5,2,4,3,1] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => 3
{{1},{2},{3,4,5}} => [1,2,4,5,3] => 3
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => 4
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => 4
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => 4
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => 4
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 5
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 2
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 2
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 3
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 2
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => 3
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 2
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => 3
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 3
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 3
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 3
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => 4
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 2
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => 1
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => 3
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 1
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 2
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => 3
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => 4
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 2
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Description
The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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