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Your data matches 85 different statistics following compositions of up to 3 maps.
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Matching statistic: St001180
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St001180: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2
[1,0,1,0]
=> 2
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 4
Description
Number of indecomposable injective modules with projective dimension at most 1.
Matching statistic: St000213
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 3 = 4 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000314
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
=> [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 3 = 4 - 1
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: St000007
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => 3 = 4 - 1
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)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
=> [1] => [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000031
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 3 = 4 - 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000157
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> [[1],[2]]
=> 1 = 2 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4],[3,5,6]]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3,4],[2,5,6]]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,3,5],[2,4,6]]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6],[3,5,7,8]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,7],[3,5,6,8]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,5,6],[3,4,7,8]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5],[3,6,7,8]]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,5,7],[3,4,6,8]]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,3,4,6],[2,5,7,8]]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,3,4,7],[2,5,6,8]]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,6],[4,5,7,8]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5],[4,6,7,8]]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,7],[4,5,6,8]]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,3,5,6],[2,4,7,8]]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 3 = 4 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000443
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St000542
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1,3] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,2,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,4,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,1,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,3,1,2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [4,1,3,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2,5,1,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,5,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 3 = 4 - 1
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000991
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 3 = 4 - 1
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000245The number of ascents of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000507The number of ascents of a standard tableau. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. 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. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000239The number of small weak excedances. St000288The number of ones in a binary word. St000291The number of descents of a binary word. 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. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000676The number of odd rises of a Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000021The number of descents of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000374The number of exclusive right-to-left minima of a permutation. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000702The number of weak deficiencies 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. St000061The number of nodes on the left branch of a binary tree. St000925The number of topologically connected components of a set partition. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000068The number of minimal elements in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001820The size of the image of the pop stack sorting operator. St001863The number of weak excedances of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function.
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