- FindStat

Identifier

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000155: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $den$ is the Denert index of a permutation, see St000156The Denert index of a permutation..

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

inverse

Description

Sends a permutation to its inverse.

Map

to labelling permutation

Description

Returns the labelling of the support Dyck path of the parking function.