- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000039: 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] => [3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,5,1,4,2] => [2,4,1,5,3] => [5,2,4,1,3] => [2,4,5,3,1] => 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 crossings of a permutation.
A crossing of a permutation

$\pi$ is given by a pair

$(i,j)$ such that either

$i < j \leq \pi(i) \leq \pi(j)$ or

$\pi(i) < \pi(j) < i < j$ .
Pictorially, the diagram of a permutation is obtained by writing the numbers from

$1$ to

$n$ in this order on a line, and connecting

$i$ and

$\pi(i)$ with an arc above the line if

$i\leq\pi(i)$ and with an arc below the line if

$i > \pi(i)$ . Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.

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

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.