Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000030: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,3,1] => [3,2,1] => 2
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,2,1,3] => 3
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,3,1,2] => 3
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,1,3,2] => 4
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,2,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [5,3,2,1,4] => 4
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [5,4,2,1,3] => 4
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [5,2,1,4,3] => 5
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,4,3,1,2] => 4
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,3,1,2,5] => 3
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [5,4,1,3,2] => 5
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,1,4,3,2] => 6
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [4,1,3,2,5] => 4
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [5,3,1,2,4] => 4
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [5,1,3,2,4] => 5
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,4,2,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [5,4,1,2,3] => 4
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [5,1,2,4,3] => 5
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [5,1,4,2,3] => 6
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => 3
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 3
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,5,4,3,2,1] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,4,3,2,1,6] => 4
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,3,2,1,6,5] => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [6,4,3,2,1,5] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,3,2,1,5,6] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [6,5,3,2,1,4] => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [6,3,2,1,5,4] => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,3,2,1,4,6] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [3,2,1,6,4,5] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [6,3,2,1,4,5] => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,5,4,3] => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [2,1,6,4,3,5] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [6,5,4,2,1,3] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [5,4,2,1,3,6] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [6,5,2,1,4,3] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [6,2,1,5,4,3] => 7
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [5,2,1,4,3,6] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [6,4,2,1,3,5] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [6,2,1,4,3,5] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [4,2,1,3,5,6] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,6,5,3,4] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [2,1,6,3,5,4] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [6,5,2,1,3,4] => 5
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [6,2,1,3,5,4] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [6,2,1,5,3,4] => 7
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [5,2,1,3,4,6] => 4
>>> Load all 196 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 sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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!