Identifier
Values
[1] => [1] => [1] => [-1] => 0
[1,2] => [2,1] => [2,1] => [-2,-1] => 0
[2,1] => [1,2] => [1,2] => [-1,-2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [-3,-1,-2] => 1
[2,1,3] => [2,3,1] => [2,3,1] => [-2,-3,-1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [-2,-1,-3] => 2
[3,1,2] => [1,3,2] => [1,3,2] => [-1,-3,-2] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [-1,-2,-3] => 3
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 2
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 2
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => 3
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 2
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 2
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 3
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 3
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => 4
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 2
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 3
[3,2,1,4] => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 3
[3,2,4,1] => [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 4
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 4
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 5
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 3
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => 4
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 4
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 5
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 5
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 6
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Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, St001866The nesting alignments of a signed permutation., an alignment of type EN, St001867The number of alignments of type EN of a signed permutation., or an alignment of type NE, St001868The number of alignments of type NE of a signed permutation..
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, St001862The number of crossings of a signed permutation., let \operatorname{wex} be the number of weak excedances, St001863The number of weak excedances of a signed permutation., and let \operatorname{neg} be the number of negative entries, St001429The number of negative entries in a signed permutation.. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
bar
Description
Return the signed permutation with all signs reversed.
Map
to signed permutation
Description
The signed permutation with all signs positive.