- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: 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] => [3,2,1] => [2,3,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a 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.