Identifier
-
Mp00032:
Dyck paths
—inverse zeta map⟶
Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000029: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => 0
[1,0,1,0] => [1,1,0,0] => [2,1] => 1
[1,1,0,0] => [1,0,1,0] => [1,2] => 0
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [3,2,1] => 2
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [1,3,2] => 1
[1,1,0,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => 2
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => 3
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 2
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => 3
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 3
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 6
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 4
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 5
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 4
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 6
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 3
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 5
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 3
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 3
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 3
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 4
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 3
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 9
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => 7
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => 7
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 5
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => 7
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => 5
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => 8
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => 5
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,2,4,3,1,6] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => 7
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 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] => [2,5,4,3,6,1] => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => 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] => [2,1,3,6,5,4] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => 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] => [1,3,5,4,2,6] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 2
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Description
The depth of a permutation.
This is given by
dp(σ)=∑σi>i(σi−i)=|{i≤j:σi>j}|.
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] ∑i|σi−i|.
Permutations with depth at most 1 are called almost-increasing in [5].
This is given by
dp(σ)=∑σi>i(σi−i)=|{i≤j:σi>j}|.
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] ∑i|σi−i|.
Permutations with depth at most 1 are called almost-increasing in [5].
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 non-crossing permutation
Description
Sends a Dyck path D with valley at positions {(i1,j1),…,(ik,jk)} to the unique non-crossing permutation π having descents {i1,…,ik} and whose inverse has descents {j1,…,jk}.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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