- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [2,3,1] => 1
[1,3,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
[1,2,4,3] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 2
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[1,4,2,3] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,3,1,4] => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[2,4,1,3] => [3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,2,3,1,4] => [1,4,3,5,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[5,4,1,2,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[5,4,1,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[5,4,2,1,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 14 + 8\ q + 2\ q^{2}$

Description

The number of crossings of a signed permutation.
A crossing of a signed permutation

$\pi$ is a pair

$(i, j)$ of indices such that

$i < j \leq \pi(i) < \pi(j)$ , or
$-i < j \leq -\pi(i) < \pi(j)$ , or
$i > j > \pi(i) > \pi(j)$ .

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

to signed permutation

Description

The signed permutation with all signs positive.

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.