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Identifier

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 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] => 0
[2,1,3] => [2,1,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 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] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 2
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [1,3,4,2] => 0
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [3,4,2,1] => 1
[3,1,4,2] => [4,3,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 1
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[2,3,5,4,1] => [4,2,3,5,1] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[2,4,3,5,1] => [5,2,4,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[2,4,5,3,1] => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[2,5,3,4,1] => [4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[2,5,4,3,1] => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,2,4,5,1] => [5,3,2,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,4,5,2,1] => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[3,5,2,4,1] => [4,5,3,2,1] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,5,4,2,1] => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[4,2,3,5,1] => [5,4,2,3,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[4,2,5,3,1] => [3,5,4,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[4,3,2,5,1] => [5,3,4,2,1] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[4,3,5,2,1] => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[4,5,2,3,1] => [3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[4,5,3,2,1] => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,3,4,1] => [4,5,2,3,1] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[5,2,4,3,1] => [3,4,5,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,3,2,4,1] => [4,3,5,2,1] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[5,3,4,2,1] => [2,4,5,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[5,4,2,3,1] => [3,4,2,5,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,4,3,2,1] => [2,3,4,5,1] => [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 nesting alignments of a signed permutation.
A nesting alignment of a signed permutation

$\pi\in\mathfrak H_n$ is a pair

$1\leq i, j \leq n$ such that

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

Map

to signed permutation

Description

The signed permutation with all signs positive.

Map

Clarke-Steingrimsson-Zeng

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map

$\Phi$ in [1, sec.3]. In particular, it satisfies

$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$
where

$des$ is the number of descents, St000021The number of descents of a permutation.,
$exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
$Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
$Ebot$ is the sum of the excedance bottoms,
$Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
$Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
$Res$ is the sum of the (right) embracing numbers,
$Ine$ is the sum of the side numbers.

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.