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Your data matches 92 different statistics following compositions of up to 3 maps.
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Matching statistic: St001499
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000062
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => 1
[1,1,0,0]
=> [2,1] => [1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000314
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => 1
[1,1,0,0]
=> [2,1] => [1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => 2
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: St000155
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1 = 2 - 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000245
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => 1 = 2 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => 1 = 2 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000996
(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
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1 = 2 - 1
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: St001036
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000007
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => [1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [1,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [3,2,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [2,3,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [2,3,1] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [4,3,1,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,4,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [4,1,3,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,4,3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,4,3,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [1,3,4,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => [4,1,2,3] => 2
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: St000010
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => [1]
=> 1
[1,1,0,0]
=> [2,1] => [1] => [1]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,1]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [2]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [2]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [2]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [2,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [2,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [2,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [2,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [3]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [3]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [3]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [3,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [3,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [2,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [3,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [3,1]
=> 2
Description
The length of the partition.
The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000204The number of internal nodes of a binary tree. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000288The number of ones in a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in 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. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils 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. St000829The Ulam distance of a permutation to the identity permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001315The dissociation number of a graph. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000159The number of distinct parts of the integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000836The number of descents of distance 2 of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000312The number of leaves in a graph. St000742The number of big ascents of a permutation after prepending zero. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000837The number of ascents of distance 2 of a permutation. St000236The number of cyclical small weak excedances. St000024The number of double up and double down steps of a Dyck path. St000636The hull number of a graph. St000710The number of big deficiencies of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001864The number of excedances of a signed permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001737The number of descents of type 2 in a permutation. 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 $eA$. St001960The number of descents of a permutation minus one if its first entry is not one. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000443The number of long tunnels of a Dyck path. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001964The interval resolution global dimension of a poset. St001488The number of corners of a skew partition. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St001556The number of inversions of the third entry of a permutation.
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