Identifier
Values
[[1]] => [1] => [1] => 1
[[1,2]] => [1,2] => [1,2] => 2
[[1],[2]] => [2,1] => [1,2] => 2
[[1,2,3]] => [1,2,3] => [1,2,3] => 3
[[1,3],[2]] => [2,1,3] => [1,2,3] => 3
[[1,2],[3]] => [3,1,2] => [1,3,2] => 2
[[1],[2],[3]] => [3,2,1] => [1,3,2] => 2
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 4
[[1,3,4],[2]] => [2,1,3,4] => [1,2,3,4] => 4
[[1,2,4],[3]] => [3,1,2,4] => [1,3,2,4] => 3
[[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => 3
[[1,3],[2,4]] => [2,4,1,3] => [1,2,4,3] => 3
[[1,2],[3,4]] => [3,4,1,2] => [1,3,2,4] => 3
[[1,4],[2],[3]] => [3,2,1,4] => [1,3,2,4] => 3
[[1,3],[2],[4]] => [4,2,1,3] => [1,4,3,2] => 3
[[1,2],[3],[4]] => [4,3,1,2] => [1,4,2,3] => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [1,4,2,3] => 2
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[[1,3,4,5],[2]] => [2,1,3,4,5] => [1,2,3,4,5] => 5
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,3,2,4,5] => 4
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,4,3,2,5] => 4
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[[1,3,5],[2,4]] => [2,4,1,3,5] => [1,2,4,3,5] => 4
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,3,2,4,5] => 4
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,2,5,4,3] => 4
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,3,2,5,4] => 3
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,4,2,5,3] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,3,2,4,5] => 4
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,4,3,2,5] => 4
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,4,2,3,5] => 3
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,5,4,2,3] => 2
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,5,3,2,4] => 3
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,3,5,4,2] => 3
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,4,2,3,5] => 3
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,4,2,3,5] => 3
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,5,3,4,2] => 4
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,5,2,3,4] => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,4,2,3,5] => 3
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,5,3,2,4] => 3
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,5,2,4,3] => 3
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,5,2,4,3] => 3
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => 6
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,3,2,4,5,6] => 5
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,4,3,2,5,6] => 5
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,5,4,3,2,6] => 4
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 4
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [1,2,4,3,5,6] => 5
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => 5
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,2,5,4,3,6] => 5
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,3,2,5,4,6] => 4
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,4,2,5,3,6] => 3
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 4
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,3,2,6,5,4] => 4
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,4,2,6,5,3] => 3
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 3
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,3,2,4,5,6] => 5
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,4,3,2,5,6] => 5
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,4,2,3,5,6] => 4
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,5,4,3,2,6] => 4
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,5,4,2,3,6] => 3
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,5,3,2,4,6] => 4
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,6,5,4,3,2] => 4
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,6,5,4,2,3] => 3
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => 2
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,6,4,2,5,3] => 3
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [1,2,4,3,6,5] => 4
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,3,6,5,2,4] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,2,5,3,6,4] => 3
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,3,6,4,2,5] => 3
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => 3
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [1,3,5,4,2,6] => 4
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,4,2,3,5,6] => 4
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,4,2,3,5,6] => 4
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,5,3,4,2,6] => 5
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,5,2,3,4,6] => 3
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,3,6,5,4,2] => 3
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,4,2,3,6,5] => 3
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,4,2,3,6,5] => 3
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,5,3,6,4,2] => 3
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,5,2,3,6,4] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,5,2,4,3,6] => 4
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,6,5,3,4,2] => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,6,5,2,3,4] => 2
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,6,4,2,3,5] => 2
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,6,4,2,3,5] => 2
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,6,3,5,2,4] => 3
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,4,2,3,5,6] => 4
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,5,4,2,3,6] => 3
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,5,3,2,4,6] => 4
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,5,2,4,3,6] => 4
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,6,5,4,2,3] => 3
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,6,5,3,2,4] => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,6,5,2,4,3] => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,6,4,2,5,3] => 3
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,6,4,2,5,3] => 3
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,6,3,4,2,5] => 4
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [1,3,2,6,4,5] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [1,4,5,2,6,3] => 2
>>> Load all 173 entries. <<<
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [1,4,5,2,6,3] => 2
[[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [1,5,2,6,3,4] => 2
[[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,5,2,6,3,4] => 2
[[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [1,4,6,5,2,3] => 2
[[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [1,5,2,3,4,6] => 3
[[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [1,5,2,4,6,3] => 3
[[1,2],[3,6],[4],[5]] => [5,4,3,6,1,2] => [1,5,2,4,6,3] => 3
[[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [1,6,4,5,2,3] => 2
[[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [1,6,3,2,4,5] => 3
[[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [1,6,2,4,5,3] => 4
[[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [1,6,3,2,5,4] => 4
[[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,6,2,5,3,4] => 2
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => 4
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [1,6,5,2,4,3] => 2
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [1,6,4,2,5,3] => 3
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [1,6,3,4,2,5] => 4
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,6,2,5,3,4] => 2
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => 2
[[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[[1,2,4,5,6,7],[3]] => [3,1,2,4,5,6,7] => [1,3,2,4,5,6,7] => 6
[[1,2,3,5,6,7],[4]] => [4,1,2,3,5,6,7] => [1,4,3,2,5,6,7] => 6
[[1,3,5,6,7],[2,4]] => [2,4,1,3,5,6,7] => [1,2,4,3,5,6,7] => 6
[[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => [1,3,2,4,5,6,7] => 6
[[1,3,4,6,7],[2,5]] => [2,5,1,3,4,6,7] => [1,2,5,4,3,6,7] => 6
[[1,2,4,6,7],[3,5]] => [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => 5
[[1,2,3,6,7],[4,5]] => [4,5,1,2,3,6,7] => [1,4,2,5,3,6,7] => 4
[[1,3,4,5,7],[2,6]] => [2,6,1,3,4,5,7] => [1,2,6,5,4,3,7] => 5
[[1,2,4,5,7],[3,6]] => [3,6,1,2,4,5,7] => [1,3,2,6,5,4,7] => 5
[[1,2,3,5,7],[4,6]] => [4,6,1,2,3,5,7] => [1,4,2,6,5,3,7] => 4
[[1,3,4,5,6],[2,7]] => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => 5
[[1,2,4,5,6],[3,7]] => [3,7,1,2,4,5,6] => [1,3,2,7,6,5,4] => 4
[[1,2,3,5,6],[4,7]] => [4,7,1,2,3,5,6] => [1,4,2,7,6,5,3] => 3
[[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [1,3,2,4,5,6,7] => 6
[[1,3,5,6,7],[2],[4]] => [4,2,1,3,5,6,7] => [1,4,3,2,5,6,7] => 6
[[1,2,5,6,7],[3],[4]] => [4,3,1,2,5,6,7] => [1,4,2,3,5,6,7] => 5
[[1,3,5,7],[2,4,6]] => [2,4,6,1,3,5,7] => [1,2,4,3,6,5,7] => 5
[[1,2,5,7],[3,4,6]] => [3,4,6,1,2,5,7] => [1,3,6,5,2,4,7] => 3
[[1,3,4,7],[2,5,6]] => [2,5,6,1,3,4,7] => [1,2,5,3,6,4,7] => 4
[[1,2,4,7],[3,5,6]] => [3,5,6,1,2,4,7] => [1,3,6,4,2,5,7] => 4
[[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => [1,4,2,5,3,6,7] => 4
[[1,3,5,6],[2,4,7]] => [2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => 5
[[1,2,5,6],[3,4,7]] => [3,4,7,1,2,5,6] => [1,3,7,6,5,2,4] => 3
[[1,3,4,6],[2,5,7]] => [2,5,7,1,3,4,6] => [1,2,5,3,7,6,4] => 4
[[1,2,4,6],[3,5,7]] => [3,5,7,1,2,4,6] => [1,3,7,6,4,2,5] => 2
[[1,2,3,6],[4,5,7]] => [4,5,7,1,2,3,6] => [1,4,2,5,3,7,6] => 3
[[1,3,4,5],[2,6,7]] => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 4
[[1,2,4,5],[3,6,7]] => [3,6,7,1,2,4,5] => [1,3,7,5,2,6,4] => 3
[[1,2,3,5],[4,6,7]] => [4,6,7,1,2,3,5] => [1,4,2,6,3,7,5] => 2
[[1,4,6,7],[2,5],[3]] => [3,2,5,1,4,6,7] => [1,3,5,4,2,6,7] => 5
[[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => [1,4,2,3,5,6,7] => 5
[[1,2,6,7],[3,5],[4]] => [4,3,5,1,2,6,7] => [1,4,2,3,5,6,7] => 5
[[1,4,5,7],[2,6],[3]] => [3,2,6,1,4,5,7] => [1,3,6,5,4,2,7] => 4
[[1,3,5,7],[2,6],[4]] => [4,2,6,1,3,5,7] => [1,4,2,3,6,5,7] => 4
[[1,2,5,7],[3,6],[4]] => [4,3,6,1,2,5,7] => [1,4,2,3,6,5,7] => 4
[[1,4,5,6],[2,7],[3]] => [3,2,7,1,4,5,6] => [1,3,7,6,5,4,2] => 4
[[1,3,5,6],[2,7],[4]] => [4,2,7,1,3,5,6] => [1,4,2,3,7,6,5] => 4
[[1,2,5,6],[3,7],[4]] => [4,3,7,1,2,5,6] => [1,4,2,3,7,6,5] => 4
[[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [1,4,2,3,5,6,7] => 5
[[1,4,6],[2,5,7],[3]] => [3,2,5,7,1,4,6] => [1,3,5,2,4,7,6] => 3
[[1,4,5],[2,6,7],[3]] => [3,2,6,7,1,4,5] => [1,3,6,4,7,5,2] => 3
[[1,4,7],[2,5],[3,6]] => [3,6,2,5,1,4,7] => [1,3,2,6,4,5,7] => 4
[[1,3,7],[2,5],[4,6]] => [4,6,2,5,1,3,7] => [1,4,5,2,6,3,7] => 3
[[1,2,7],[3,5],[4,6]] => [4,6,3,5,1,2,7] => [1,4,5,2,6,3,7] => 3
[[1,4,6],[2,5],[3,7]] => [3,7,2,5,1,4,6] => [1,3,2,7,6,4,5] => 3
[[1,3,6],[2,5],[4,7]] => [4,7,2,5,1,3,6] => [1,4,5,2,7,6,3] => 3
[[1,2,6],[3,5],[4,7]] => [4,7,3,5,1,2,6] => [1,4,5,2,7,6,3] => 3
[[1,4,5],[2,6],[3,7]] => [3,7,2,6,1,4,5] => [1,3,2,7,5,4,6] => 4
[[1,3,5],[2,6],[4,7]] => [4,7,2,6,1,3,5] => [1,4,6,3,2,7,5] => 2
[[1,2,5],[3,6],[4,7]] => [4,7,3,6,1,2,5] => [1,4,6,2,7,5,3] => 2
[[1,5,7],[2,6],[3],[4]] => [4,3,2,6,1,5,7] => [1,4,6,5,2,3,7] => 3
[[1,5],[2,6],[3,7],[4]] => [4,3,7,2,6,1,5] => [1,4,2,3,7,5,6] => 3
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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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.