Identifier
Values
[1] => [1,0] => [2,1] => [2,1] => 1
[1,1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 1
[2] => [1,1,0,0] => [2,3,1] => [2,1,3] => 1
[1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 1
[1,2] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 2
[2,1] => [1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 2
[3] => [1,1,1,0,0,0] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,2,5,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,2,5,4] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,4,5,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,3,5,1,4] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,1,4,3,5] => 1
[3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,1,3,4] => 2
[4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,1,3,4,5] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,2,3,6,5] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,4,2,6,5] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,6,4] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,4,1,2,6,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,6,2] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,1,5,6,3,4] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,6,3,5] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,1,4,5,3,6] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,1,5,3,4,6] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,1,3,4,5] => 2
[5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,2,4,3,5,7,6] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,4,5,2,7,6] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [1,3,2,5,4,7,6] => 1
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Description
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.
In other words, this is $2^k$ where $k$ is the number of cycles of length at least three (St000486The number of cycles of length at least 3 of a permutation.) in its cycle decomposition.
The generating function for the number of equivalence classes, $f(n)$, is
$$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.