Identifier
Values
[1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,1,2] => 1
[1,1,0,0] => [2,3,1] => [3,2,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [4,1,3,2] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,1,2,4,3] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,1,3,2,4] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,5,3,2] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,1,3,4,2] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,2,1,4,3] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,2,1,4] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,5,2,3,1] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,4,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,5,3,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,5,4,2,1] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,2,6,4,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,6,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,1,6,3,4,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,6,3,5,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,1,3,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,1,3,6,4,2] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [4,1,6,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,1,3,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,2,1,4,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,1,6,4,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,2,1,4,5,3] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,6,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,6,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,6,2,3,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,5,2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,6,2,3,5,1] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,6,2,4,1,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,2,5,3,1] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,4,5,1] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,2,3,1,5,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,6,3,1,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,6,3,4,1] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,6,3,5,1] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,6,4,2,1,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,6,4,2,5,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,6,4,1] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [4,2,6,5,3,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,6,4,5,2,1] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 5
[] => [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
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.