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Your data matches 160 different statistics following compositions of up to 3 maps.
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Matching statistic: St000395
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 4
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> 7
[1,1,0,1,0,1,0,1,0,0]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> 7
[1,1,0,1,1,0,1,0,0,0]
=> 8
[1,1,0,1,1,1,0,0,0,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001034
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> 7
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> 7
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St001348
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> 8
[1,0,1,1,0,0,1,1,0,0]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,0,1,0,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> 8
[1,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,1,0,0]
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> 7
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> 7
[1,1,0,0,1,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> 5
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St000029
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 1
[1,0,1,0]
=> [3,1,2] => 2
[1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,1,0,0,1,0]
=> [2,4,1,3] => 3
[1,1,0,1,0,0]
=> [4,3,1,2] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 6
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 6
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 5
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 7
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 5
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 7
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 8
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 7
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 7
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 8
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 6
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].
Matching statistic: St000197
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000197: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 3
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 4
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 5
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 5
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 6
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 5
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 5
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 6
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 7
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 7
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 8
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 6
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St000229
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> {{1,2}}
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 3
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 3
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 4
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 4
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 5
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 4
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 4
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 5
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 6
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 5
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 4
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 6
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 5
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 7
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 6
Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called ''dimension index'' in [2]
Matching statistic: St000394
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 7
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 7
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 8
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000018
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 5
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 6
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => 7
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => 8
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => 7
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => 7
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => 8
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => 6
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000030
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => 3
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 3
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,4,1,2,3] => 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,1,2,5,3] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,5,2,3] => 6
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,1,5,4,2] => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,3,1,5,2] => 6
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,4,1,2,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [5,1,2,3,6,4] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,5,4,1,2,3] => 5
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,3,1,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,5,3,1,2,4] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,1,2,6,3,5] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,1,2,6,3,4] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,1,2,6,5,3] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,4,3,1,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,3,1,2,6,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,4,1,2,6,3] => 7
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,5,4,3,1,2] => 5
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,5,2,1,3,4] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,4,2,1,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,2,1,3,6,4] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,5,4,2,1,3] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,1,6,5,2,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,6,2,3,5] => 7
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,5,2,3,4] => 8
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,1,6,5,2,3] => 7
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,1,6,4,2,5] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,1,5,2,6,4] => 7
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,4,1,6,2,3] => 8
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,1,6,5,4,2] => 6
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 [[St000111]] and [[St000154]] 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.
Matching statistic: St000224
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000224: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000224: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 4
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 6
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 7
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 7
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 7
Description
The sorting index of a permutation.
The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols.
Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.
The following 150 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000228The size of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000004The major index of a permutation. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000246The number of non-inversions of a permutation. St000332The positive inversions of an alternating sign matrix. St000548The number of different non-empty partial sums of an integer partition. St000795The mad of a permutation. St000883The number of longest increasing subsequences of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001869The maximum cut size of a graph. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000479The Ramsey number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000171The degree of the graph. St001120The length of a longest path in a graph. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001725The harmonious chromatic number of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000031The number of cycles in the cycle decomposition of a permutation. St000680The Grundy value for Hackendot on posets. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000223The number of nestings in the permutation. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St001727The number of invisible inversions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000245The number of ascents of a permutation. St000670The reversal length of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001554The number of distinct nonempty subtrees of a binary tree. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000703The number of deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001343The dimension of the reduced incidence algebra of a poset. St000624The normalized sum of the minimal distances to a greater element. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000308The height of the tree associated to a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. St000730The maximal arc length of a set partition. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001286The annihilation number of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001405The number of bonds in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001821The sorting index of a signed permutation. St000133The "bounce" of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000519The largest length of a factor maximising the subword complexity. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000837The number of ascents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001298The number of repeated entries in the Lehmer code of a permutation. St001323The independence gap of a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001565The number of arithmetic progressions of length 2 in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001077The prefix exchange distance of a permutation. St001416The length of a longest palindromic factor of a binary word. St000806The semiperimeter of the associated bargraph. St000632The jump number of the poset. St001820The size of the image of the pop stack sorting operator. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000527The width of the poset. St000528The height of a poset. St000845The maximal number of elements covered by an element in a poset. St000906The length of the shortest maximal chain in a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St000462The major index minus the number of excedences of a permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000307The number of rowmotion orbits of a poset. St001644The dimension of a graph. St001812The biclique partition number of a graph. St001638The book thickness of a graph. St001555The order of a signed permutation.
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