Identifier
Values
[1,0] => [2,1] => {{1,2}} => [2,1] => 1
[1,0,1,0] => [3,1,2] => {{1,3},{2}} => [3,2,1] => 2
[1,1,0,0] => [2,3,1] => {{1,2,3}} => [2,3,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => {{1,4},{2},{3}} => [4,2,3,1] => 3
[1,0,1,1,0,0] => [3,1,4,2] => {{1,3,4},{2}} => [3,2,4,1] => 2
[1,1,0,0,1,0] => [2,4,1,3] => {{1,2,4},{3}} => [2,4,3,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => {{1,4},{2,3}} => [4,3,2,1] => 2
[1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => {{1,3,4,5},{2}} => [3,2,4,5,1] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 3
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => {{1,2,4,5},{3}} => [2,4,3,5,1] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => {{1,4,5},{2,3}} => [4,3,2,5,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => {{1,2,3,5},{4}} => [2,3,5,4,1] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => {{1,2,5},{3,4}} => [2,5,4,3,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => {{1,5},{2,3,4}} => [5,3,4,2,1] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => 4
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}} => [6,2,4,5,3,1] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}} => [5,3,2,4,6,1] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}} => [5,4,3,2,6,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}} => [4,3,2,6,5,1] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => {{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => {{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => {{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => {{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => {{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[] => [1] => {{1}} => [1] => 1
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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.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.