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Your data matches 585 different statistics following compositions of up to 3 maps.
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Matching statistic: St001088
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St001088: 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]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St001274
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001274: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
Description
The number of indecomposable injective modules with projective dimension equal to two.
Matching statistic: St000314
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000314: 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] => 1
[1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[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] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 3
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000996
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000996: 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] => 1
[1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[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] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 3
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001210
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001210: 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 = 1 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
Description
Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001231
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.
Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Matching statistic: St001234
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Matching statistic: St000007
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => 1
[1,0,1,0]
=> [3,1,2] => [2,1,3] => 1
[1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [2,1,3,4] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,2,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,2,4,1,5] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2,1,4,5] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,5,1,3,4] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,1,4,5,2] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [4,6,3,2,1,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,3,6,2,1,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [4,3,5,2,1,6] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [5,4,2,6,1,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [5,3,2,4,1,6] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,3,2,5,1,6] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [3,6,2,4,1,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,2,6,4,1,3] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,2,5,6,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,2,5,4,1,6] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,5,4,1,3] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,4,3,1,6,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,6,3,1,5,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,3,5,1,6,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,6,5,1,4,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [5,4,2,1,3,6] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,6,2,1,3,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [5,3,2,1,4,6] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,6,2,1,4,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,6,1,3,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,2,5,1,3,6] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,2,5,1,4,6] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [2,6,5,1,3,4] => 3
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000015
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 3
Description
The number of peaks of a Dyck path.
Matching statistic: St000031
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: 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] => 1
[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] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,4,2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,4,5,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,6,3,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,2,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,3,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,2,5,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,6,1,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,5,1,6,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,4,2,1,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,5,2,3,1,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,4,6,1,2,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,6,1,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,5,1,2,6,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,5,2,1,4,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,6,1,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,1,6,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,6,1,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,2,5,3,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,5,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,1,6,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,1,5,6,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,1,2,6,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,1,5,6,3,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,6,5,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,1,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,4,6,2,5,1] => 3
Description
The number of cycles in the cycle decomposition of a permutation.
The following 575 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000374The number of exclusive right-to-left minima of a permutation. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000237The number of small exceedances. St000441The number of successions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000444The length of the maximal rise of a Dyck path. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000239The number of small weak excedances. St000273The domination number of a graph. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000382The first part of an integer composition. St000383The last part of an integer composition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000544The cop number of a graph. St000619The number of cyclic descents of a permutation. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000740The last entry of a permutation. St000925The number of topologically connected components of a set partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000971The smallest closer of a set partition. St000990The first ascent of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000083The number of left oriented leafs of a binary tree except the first one. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000203The number of external nodes of a binary tree. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000338The number of pixed points of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000497The lcb statistic of a set partition. St000546The number of global descents of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000648The number of 2-excedences of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000711The number of big exceedences of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000917The open packing number of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001826The maximal number of leaves on a vertex of a graph. St000381The largest part of an integer composition. St000686The finitistic dominant dimension of a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St000504The cardinality of the first block of a set partition. St000502The number of successions of a set partitions. St000877The depth of the binary word interpreted as a path. St001061The number of indices that are both descents and recoils of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000906The length of the shortest maximal chain in a poset. St001948The number of augmented double ascents of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000025The number of initial rises of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000068The number of minimal elements in a poset. St000215The number of adjacencies of a permutation, zero appended. St001462The number of factors of a standard tableaux under concatenation. St000475The number of parts equal to 1 in a partition. St000287The number of connected components of a graph. St000439The position of the first down step of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St000352The Elizalde-Pak rank of a permutation. St001432The order dimension of the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000297The number of leading ones in a binary word. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000993The multiplicity of the largest part of an integer partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000939The number of characters of the symmetric group whose value on the partition is positive. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000742The number of big ascents of a permutation after prepending zero. St000145The Dyson rank of a partition. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001889The size of the connectivity set of a signed permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001568The smallest positive integer that does not appear twice in the partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000144The pyramid weight of the Dyck path. St000293The number of inversions of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000456The monochromatic index of a connected graph. St000519The largest length of a factor maximising the subword complexity. St000631The number of distinct palindromic decompositions of a binary word. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000922The minimal number such that all substrings of this length are unique. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000982The length of the longest constant subword. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001118The acyclic chromatic index of a graph. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001488The number of corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000668The least common multiple of the parts of the partition. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000528The height of a poset. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001645The pebbling number of a connected graph. St000080The rank of the poset. St001782The order of rowmotion on the set of order ideals of a poset. 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. St001060The distinguishing index of a graph. St000895The number of ones on the main diagonal of an alternating sign matrix. St000054The first entry of the permutation. St000649The number of 3-excedences of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001403The number of vertical separators in a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001566The length of the longest arithmetic progression in a permutation. St000717The number of ordinal summands of a poset. St000910The number of maximal chains of minimal length in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000770The major index of an integer partition when read from bottom to top. St000295The length of the border of a binary word. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000022The number of fixed points of a permutation. St000392The length of the longest run of ones in a binary word. St000505The biggest entry in the block containing the 1. St001618The cardinality of the Frattini sublattice of a lattice. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000702The number of weak deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000873The aix statistic of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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 $eA$. St001434The number of negative sum pairs of a signed permutation. St001545The second Elser number of a connected graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000051The size of the left subtree of a binary tree. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000461The rix statistic of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000894The trace of an alternating sign matrix. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001712The number of natural descents of a standard Young tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001372The length of a longest cyclic run of ones of a binary word. St001530The depth of a Dyck path. St000738The first entry in the last row of a standard tableau. St001686The order of promotion on a Gelfand-Tsetlin pattern. 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. St001621The number of atoms of a lattice. St000075The orbit size of a standard tableau under promotion. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St001096The size of the overlap set of a permutation. St000534The number of 2-rises of a permutation. St001115The number of even descents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000035The number of left outer peaks of a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000647The number of big descents of a permutation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000744The length of the path to the largest entry in a standard Young tableau. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000834The number of right outer peaks of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001964The interval resolution global dimension of a poset. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St001330The hat guessing number of a graph. St000883The number of longest increasing subsequences of a permutation. St001769The reflection length of a signed permutation. St000884The number of isolated descents of a permutation. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000492The rob statistic of a set partition. St000659The number of rises of length at least 2 of a Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001732The number of peaks visible from the left. St001864The number of excedances of a signed permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St000496The rcs statistic of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000665The number of rafts of a permutation. St000839The largest opener of a set partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001867The number of alignments of type EN of a signed permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St001052The length of the exterior of a permutation. St001820The size of the image of the pop stack sorting operator. St000359The number of occurrences of the pattern 23-1. St000731The number of double exceedences of a permutation. St001846The number of elements which do not have a complement in the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000074The number of special entries. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000617The number of global maxima of a Dyck path. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions 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. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000296The length of the symmetric border of a binary word. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000891The number of distinct diagonal sums of a permutation matrix. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000264The girth of a graph, which is not a tree. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000662The staircase size of the code of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000307The number of rowmotion orbits of a poset. St000909The number of maximal chains of maximal size in a poset. St001868The number of alignments of type NE of a signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000498The lcs statistic of a set partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000091The descent variation of a composition. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000133The "bounce" of a permutation. St000210Minimum over maximum difference of elements in cycles. St000553The number of blocks of a graph. St000657The smallest part of an integer composition. St000729The minimal arc length of a set partition. St000876The number of factors in the Catalan decomposition of a binary word. St000916The packing number of a graph. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001405The number of bonds in a permutation. St001468The smallest fixpoint of a permutation. St001471The magnitude of a Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001729The number of visible descents of a permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001806The upper middle entry of a permutation. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St000258The burning number of a graph. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000363The number of minimal vertex covers of a graph. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000469The distinguishing number of a graph. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000767The number of runs in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000779The tier of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001366The maximal multiplicity of a degree of a vertex of a graph. St001377The major index minus the number of inversions of a permutation. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001437The flex of a binary word. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001481The minimal height of a peak of a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001689The number of celebrities in a graph. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001810The number of fixed points of a permutation smaller than its largest moved point. St001822The number of alignments of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001866The nesting alignments of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000808The number of up steps of the associated bargraph. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000983The length of the longest alternating subword. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000735The last entry on the main diagonal of a standard tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001644The dimension of a graph. St000850The number of 1/2-balanced pairs in a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St001812The biclique partition number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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