- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

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

$F_{5} = 42 + 45\ q + 25\ q^{2} + 7\ q^{3} + q^{4}$

$F_{6} = 132 + 220\ q + 198\ q^{2} + 112\ q^{3} + 44\ q^{4} + 12\ q^{5} + 2\ q^{6}$

Description

The number of nestings in the permutation.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$