Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[2,1,3] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
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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
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 $\pi$.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
to signed permutation
Description
The signed permutation with all signs positive.