- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001715: 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] => [1,3,2] => [3,2,1] => [2,3,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of non-records in a permutation.
A record in a permutation $\pi$ is a value $\pi(j)$ which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum.
For example, in the permutation $\pi = [1, 4, 3, 2, 5]$, the values $1$ is a left-to-right minimum, $1, 4, 5$ are left-to-right maxima, $5, 2, 1$ are right-to-left minima and $5$ is a right-to-left maximum. Hence, $3$ is the unique non-record.
Permutations without non-records are called square [1].

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].

Map

Inverse Kreweras complement

Description

Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.

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.