Identifier
-
Mp00238:
Permutations
—Clarke-Steingrimsson-Zeng⟶
Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed 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] => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 2
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [1,3,4,2] => 0
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [3,4,2,1] => 1
[3,1,4,2] => [4,3,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 1
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[2,3,5,4,1] => [4,2,3,5,1] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[2,4,3,5,1] => [5,2,4,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[2,4,5,3,1] => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[2,5,3,4,1] => [4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[2,5,4,3,1] => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,2,4,5,1] => [5,3,2,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,4,5,2,1] => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[3,5,2,4,1] => [4,5,3,2,1] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,5,4,2,1] => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[4,2,3,5,1] => [5,4,2,3,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[4,2,5,3,1] => [3,5,4,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[4,3,2,5,1] => [5,3,4,2,1] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[4,3,5,2,1] => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[4,5,2,3,1] => [3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[4,5,3,2,1] => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,3,4,1] => [4,5,2,3,1] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[5,2,4,3,1] => [3,4,5,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,3,2,4,1] => [4,3,5,2,1] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[5,3,4,2,1] => [2,4,5,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[5,4,2,3,1] => [3,4,2,5,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,4,3,2,1] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation π∈Hn is a pair 1≤i,j≤n such that
A nesting alignment of a signed permutation π∈Hn is a pair 1≤i,j≤n such that
- −i<−j<−π(j)<−π(i), or
- −i<j≤π(j)<−π(i), or
- i<j≤π(j)<π(i).
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map Φ in [1, sec.3]. In particular, it satisfies
(des,Dbot,Ddif,Res)π=(exc,Ebot,Edif,Ine)Φ(π),
where
This is the map Φ in [1, sec.3]. In particular, it satisfies
(des,Dbot,Ddif,Res)π=(exc,Ebot,Edif,Ine)Φ(π),
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
Kreweras complement
Description
Sends the permutation π∈Sn to the permutation π−1c where c=(1,…,n) is the long cycle.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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