Identifier
Values
[1] => [1] => [1] => [-1] => 1
[1,2] => [2,1] => [2,1] => [1,-2] => 1
[2,1] => [1,2] => [1,2] => [2,-1] => 0
[1,2,3] => [2,3,1] => [2,3,1] => [1,2,-3] => 1
[1,3,2] => [2,1,3] => [2,1,3] => [1,3,-2] => 0
[2,1,3] => [3,2,1] => [3,2,1] => [2,1,-3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,-2] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,2,-1] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 1
[1,2,4,3] => [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 0
[1,3,2,4] => [2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => 1
[1,3,4,2] => [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 0
[1,4,2,3] => [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 0
[1,4,3,2] => [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 0
[2,1,3,4] => [3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => 1
[2,1,4,3] => [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 0
[2,3,1,4] => [3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => 0
[2,3,4,1] => [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 0
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => 0
[3,1,2,4] => [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 0
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 0
[3,2,1,4] => [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 3
[3,2,4,1] => [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 0
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => 0
[3,4,2,1] => [4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => 0
[4,1,2,3] => [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 0
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 1
[1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 0
[1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => 1
[1,2,4,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => 0
[1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => 0
[1,2,5,4,3] => [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 0
[1,3,2,4,5] => [2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => 1
[1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => 0
[1,3,4,2,5] => [2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => 0
[1,3,4,5,2] => [2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => 0
[1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => 0
[1,3,5,4,2] => [2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => 0
[1,4,2,3,5] => [2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => 0
[1,4,2,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => 0
[1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => 3
[1,4,3,5,2] => [2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => 0
[1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => 0
[1,4,5,3,2] => [2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => 0
[1,5,2,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => 0
[1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 0
[1,5,3,2,4] => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => 0
[1,5,3,4,2] => [2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => 0
[1,5,4,2,3] => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 0
[1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 0
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Description
The number of Hecke atoms of a signed permutation.
For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set
$$ \{ w\in\mathfrak H_n | w^{-1} \star w = z\}, $$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
Map
inverse Kreweras complement
Description
The inverse Kreweras complement of a signed permutation.
This is the signed permutation $c \pi^{-1}$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the inverse Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.