- FindStat

Identifier

Mp00188: Signed permutations —complement⟶ Signed permutations
St001870: Signed permutations ⟶ ℤ

Values

[1] => [1] => 0

[-1] => [-1] => 1

[1,2] => [2,1] => 0

[1,-2] => [2,-1] => 2

[-1,2] => [-2,1] => 1

[-1,-2] => [-2,-1] => 1

[2,1] => [1,2] => 0

[2,-1] => [1,-2] => 2

[-2,1] => [-1,2] => 1

[-2,-1] => [-1,-2] => 1

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

[1,2,-3] => [3,2,-1] => 2

[1,-2,3] => [3,-2,1] => 2

[1,-2,-3] => [3,-2,-1] => 2

[-1,2,3] => [-3,2,1] => 1

[-1,2,-3] => [-3,2,-1] => 3

[-1,-2,3] => [-3,-2,1] => 1

[-1,-2,-3] => [-3,-2,-1] => 1

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

[1,3,-2] => [3,1,-2] => 2

[1,-3,2] => [3,-1,2] => 2

[1,-3,-2] => [3,-1,-2] => 2

[-1,3,2] => [-3,1,2] => 1

[-1,3,-2] => [-3,1,-2] => 3

[-1,-3,2] => [-3,-1,2] => 1

[-1,-3,-2] => [-3,-1,-2] => 1

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

[2,1,-3] => [2,3,-1] => 2

[2,-1,3] => [2,-3,1] => 2

[2,-1,-3] => [2,-3,-1] => 2

[-2,1,3] => [-2,3,1] => 1

[-2,1,-3] => [-2,3,-1] => 3

[-2,-1,3] => [-2,-3,1] => 1

[-2,-1,-3] => [-2,-3,-1] => 1

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

[2,3,-1] => [2,1,-3] => 2

[2,-3,1] => [2,-1,3] => 2

[2,-3,-1] => [2,-1,-3] => 2

[-2,3,1] => [-2,1,3] => 1

[-2,3,-1] => [-2,1,-3] => 3

[-2,-3,1] => [-2,-1,3] => 1

[-2,-3,-1] => [-2,-1,-3] => 1

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

[3,1,-2] => [1,3,-2] => 2

[3,-1,2] => [1,-3,2] => 2

[3,-1,-2] => [1,-3,-2] => 2

[-3,1,2] => [-1,3,2] => 1

[-3,1,-2] => [-1,3,-2] => 3

[-3,-1,2] => [-1,-3,2] => 1

[-3,-1,-2] => [-1,-3,-2] => 1

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

[3,2,-1] => [1,2,-3] => 2

[3,-2,1] => [1,-2,3] => 2

[3,-2,-1] => [1,-2,-3] => 2

[-3,2,1] => [-1,2,3] => 1

[-3,2,-1] => [-1,2,-3] => 3

[-3,-2,1] => [-1,-2,3] => 1

[-3,-2,-1] => [-1,-2,-3] => 1

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

[1,2,3,-4] => [4,3,2,-1] => 2

[1,2,-3,4] => [4,3,-2,1] => 2

[1,2,-3,-4] => [4,3,-2,-1] => 2

[1,-2,3,4] => [4,-3,2,1] => 2

[1,-2,3,-4] => [4,-3,2,-1] => 4

[1,-2,-3,4] => [4,-3,-2,1] => 2

[1,-2,-3,-4] => [4,-3,-2,-1] => 2

[-1,2,3,4] => [-4,3,2,1] => 1

[-1,2,3,-4] => [-4,3,2,-1] => 3

[-1,2,-3,4] => [-4,3,-2,1] => 3

[-1,2,-3,-4] => [-4,3,-2,-1] => 3

[-1,-2,3,4] => [-4,-3,2,1] => 1

[-1,-2,3,-4] => [-4,-3,2,-1] => 3

[-1,-2,-3,4] => [-4,-3,-2,1] => 1

[-1,-2,-3,-4] => [-4,-3,-2,-1] => 1

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

[1,2,4,-3] => [4,3,1,-2] => 2

[1,2,-4,3] => [4,3,-1,2] => 2

[1,2,-4,-3] => [4,3,-1,-2] => 2

[1,-2,4,3] => [4,-3,1,2] => 2

[1,-2,4,-3] => [4,-3,1,-2] => 4

[1,-2,-4,3] => [4,-3,-1,2] => 2

[1,-2,-4,-3] => [4,-3,-1,-2] => 2

[-1,2,4,3] => [-4,3,1,2] => 1

[-1,2,4,-3] => [-4,3,1,-2] => 3

[-1,2,-4,3] => [-4,3,-1,2] => 3

[-1,2,-4,-3] => [-4,3,-1,-2] => 3

[-1,-2,4,3] => [-4,-3,1,2] => 1

[-1,-2,4,-3] => [-4,-3,1,-2] => 3

[-1,-2,-4,3] => [-4,-3,-1,2] => 1

[-1,-2,-4,-3] => [-4,-3,-1,-2] => 1

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

[1,3,2,-4] => [4,2,3,-1] => 2

[1,3,-2,4] => [4,2,-3,1] => 2

[1,3,-2,-4] => [4,2,-3,-1] => 2

[1,-3,2,4] => [4,-2,3,1] => 2

[1,-3,2,-4] => [4,-2,3,-1] => 4

[1,-3,-2,4] => [4,-2,-3,1] => 2

[1,-3,-2,-4] => [4,-2,-3,-1] => 2

[-1,3,2,4] => [-4,2,3,1] => 1

[-1,3,2,-4] => [-4,2,3,-1] => 3

[-1,3,-2,4] => [-4,2,-3,1] => 3

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$F_{1} = 1 + q$

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

$F_{3} = 6 + 18\ q + 18\ q^{2} + 6\ q^{3}$

$F_{4} = 24 + 96\ q + 144\ q^{2} + 96\ q^{3} + 24\ q^{4}$

Description

The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation

$\pi\in\mathfrak H_n$ , this is the number of positive entries followed by a negative entry in

$\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$ .

Map

complement

Description

The complement of a signed permutation.
This is the signed permutation obtained by replacing

$i$ with

$n+1-i$ and

$-i$ with

$-(n+1-i)$ , for

$1 \leq i \leq n$ .