Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: Permutations ⟶ ℤ
Values
{{1,2}} => [2,1] => [1,2] => [1,2] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}} => [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}} => [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}} => [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}} => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}} => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
{{1,3},{2,4}} => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,3,4}} => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,3,4,5,2] => [1,5,3,4,2] => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,4,5}} => [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 3
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 2
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,4,2,5,3] => [1,5,4,2,3] => 3
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,2,3] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,5,2,3,4] => [1,5,2,3,4] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,5,2,3,4] => [1,5,2,3,4] => 3
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 1
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 2
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 2
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => 2
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => 2
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => 2
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 2
>>> Load all 792 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
Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth St000029The depth of a permutation. minus the usual length St000018The number of inversions of a permutation..
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth St000029The depth of a permutation. minus the usual length St000018The number of inversions of a permutation..
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!