Identifier
-
Mp00223:
Permutations
—runsort⟶
Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤ
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[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] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,5,3,2,4] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[2,1,3,5,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[2,1,4,5,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[2,3,1,4,5] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[2,4,1,3,5] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[2,4,3,1,5] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[2,5,1,4,3] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[2,5,3,1,4] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[3,1,2,5,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[3,1,4,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,1,4,5,2] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[3,1,5,2,4] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,2,1,4,5] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[3,2,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,2,4,1,5] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[3,2,5,1,4] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,2,5,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[3,4,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[3,5,1,4,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
>>> Load all 153 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 nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
- $-i < -j < -\pi(j) < -\pi(i)$, or
- $-i < j \leq \pi(j) < -\pi(i)$, or
- $i < j \leq \pi(j) < \pi(i)$.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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!