- FindStat

Identifier

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000018: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,2,4,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => [5,2,3,4,1] => 7

[2,3,5,1,4] => [5,2,3,1,4] => [4,1,3,2,5] => [2,5,3,4,1] => 6

[2,3,5,4,1] => [4,2,3,5,1] => [1,5,3,2,4] => [5,1,3,4,2] => 6

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

[2,4,1,5,3] => [5,2,4,1,3] => [3,1,4,2,5] => [3,5,2,4,1] => 7

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

[2,4,3,5,1] => [5,2,4,3,1] => [1,3,4,2,5] => [5,3,2,4,1] => 8

[2,4,5,1,3] => [5,2,1,4,3] => [3,4,1,2,5] => [3,2,5,4,1] => 6

[2,4,5,3,1] => [3,2,5,4,1] => [1,4,5,2,3] => [5,2,1,4,3] => 6

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

[2,5,1,4,3] => [4,2,5,1,3] => [3,1,5,2,4] => [3,5,1,4,2] => 6

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

[2,5,3,4,1] => [4,2,5,3,1] => [1,3,5,2,4] => [5,3,1,4,2] => 7

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

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

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

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

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

[3,1,4,5,2] => [5,3,1,4,2] => [2,4,1,3,5] => [4,2,5,3,1] => 7

[3,1,5,2,4] => [5,3,1,2,4] => [4,2,1,3,5] => [2,4,5,3,1] => 6

[3,1,5,4,2] => [4,3,1,5,2] => [2,5,1,3,4] => [4,1,5,3,2] => 6

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

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

[3,2,4,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => [1,5,4,3,2] => 6

[3,2,4,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => [5,2,4,3,1] => 8

[3,2,5,1,4] => [5,3,2,1,4] => [4,1,2,3,5] => [2,5,4,3,1] => 7

[3,2,5,4,1] => [4,3,2,5,1] => [1,5,2,3,4] => [5,1,4,3,2] => 7

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

[3,4,1,5,2] => [5,4,3,1,2] => [2,1,3,4,5] => [4,5,3,2,1] => 9

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

[3,4,2,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 10

[3,4,5,1,2] => [5,1,3,4,2] => [2,4,3,1,5] => [4,2,3,5,1] => 6

[3,4,5,2,1] => [2,5,3,4,1] => [1,4,3,5,2] => [5,2,3,1,4] => 6

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

[3,5,1,4,2] => [4,5,3,1,2] => [2,1,3,5,4] => [4,5,3,1,2] => 8

>>> Load all 1982 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

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