- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000372: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1201 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 mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.