Identifier
-
Mp00032:
Dyck paths
—inverse zeta map⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000154: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => 0
[1,0,1,0] => [1,1,0,0] => [1,2] => 0
[1,1,0,0] => [1,0,1,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [2,3,1] => 1
[1,1,0,0,1,0] => [1,1,0,1,0,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [3,1,2] => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [3,2,1] => 3
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 1
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 3
[1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 3
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 4
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 6
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 6
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 6
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 3
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 4
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 6
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 6
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 6
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 7
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 6
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 6
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 7
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 8
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 10
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => 5
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 8
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Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see St000111The sum of the descent tops (or Genocchi descents) of a permutation..
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see St000111The sum of the descent tops (or Genocchi descents) of a permutation..
Map
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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