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Your data matches 124 different statistics following compositions of up to 3 maps.
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Matching statistic: St001222
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St001222: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 2 - 2
Description
Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.
Matching statistic: St000884
(load all 71 compositions to match this statistic)
(load all 71 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 2 - 2
[1,0,1,0]
=> [1,2] => 0 = 2 - 2
[1,1,0,0]
=> [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [3,2,1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0 = 2 - 2
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000214
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => 0 = 2 - 2
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0 = 2 - 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 0 = 2 - 2
[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]]
=> [1,2,3,4,5] => 0 = 2 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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]]
=> [1,2,4,3,5] => 1 = 3 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[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]]
=> [1,3,2,4,5] => 1 = 3 - 2
[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]]
=> [1,3,2,5,4] => 2 = 4 - 2
[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]]
=> [1,2,4,3,5] => 1 = 3 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 0 = 2 - 2
[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]]
=> [2,1,3,4,5] => 1 = 3 - 2
[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]]
=> [2,1,3,5,4] => 2 = 4 - 2
[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]]
=> [2,1,4,3,5] => 2 = 4 - 2
[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]]
=> [2,1,3,5,4] => 2 = 4 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1 = 3 - 2
[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]]
=> [1,3,2,4,5] => 1 = 3 - 2
[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]]
=> [1,3,2,5,4] => 2 = 4 - 2
[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]]
=> [1,2,4,3,5] => 1 = 3 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 0 = 2 - 2
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => 0 = 2 - 2
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0 = 2 - 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 0 = 2 - 2
[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]]
=> [1,2,3,4,5] => 0 = 2 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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]]
=> [1,2,4,3,5] => 1 = 3 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[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]]
=> [1,3,2,4,5] => 1 = 3 - 2
[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]]
=> [1,3,2,5,4] => 2 = 4 - 2
[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]]
=> [1,2,4,3,5] => 1 = 3 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 0 = 2 - 2
[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]]
=> [2,1,3,4,5] => 1 = 3 - 2
[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]]
=> [2,1,3,5,4] => 2 = 4 - 2
[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]]
=> [2,1,4,3,5] => 2 = 4 - 2
[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]]
=> [2,1,3,5,4] => 2 = 4 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1 = 3 - 2
[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]]
=> [1,3,2,4,5] => 1 = 3 - 2
[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]]
=> [1,3,2,5,4] => 2 = 4 - 2
[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]]
=> [1,2,4,3,5] => 1 = 3 - 2
[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]]
=> [1,2,3,5,4] => 1 = 3 - 2
[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],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 0 = 2 - 2
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000658
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St001125
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St001216
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
Description
The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.
Matching statistic: St001230
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001274
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
Description
The number of indecomposable injective modules with projective dimension equal to two.
Matching statistic: St001276
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 3 - 2
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
The following 114 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001657The number of twos in an integer partition. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000011The number of touch points (or returns) of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001479The number of bridges of a graph. St001484The number of singletons of an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000932The number of occurrences of the pattern UDU in a Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001267The length of the Lyndon factorization of the binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001060The distinguishing index of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000648The number of 2-excedences of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000264The girth of a graph, which is not a tree. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001948The number of augmented double ascents of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001545The second Elser number of a connected graph. St000474Dyson's crank of a partition. St000993The multiplicity of the largest part of an integer partition. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St001432The order dimension of the partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St001096The size of the overlap set of a permutation. St000741The Colin de Verdière graph invariant. St001820The size of the image of the pop stack sorting operator. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000891The number of distinct diagonal sums of a permutation matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000640The rank of the largest boolean interval in a poset. St001964The interval resolution global dimension of a poset. St001621The number of atoms of a lattice. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000335The difference of lower and upper interactions. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000317The cycle descent number of a permutation. St000989The number of final rises of a permutation. St001423The number of distinct cubes in a binary word. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001624The breadth of a lattice. St000007The number of saliances of the permutation. St001115The number of even descents of a permutation. St000366The number of double descents of a permutation.
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