Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
St000029: Permutations ⟶ ℤ
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 2
[1,1,1,0,0,0] => [3,2,1] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[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] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 4
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 4
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 4
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 5
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 4
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 4
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 4
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 5
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 5
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 5
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 6
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => 5
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Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called almost-increasing in [5].
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called almost-increasing in [5].
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
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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