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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001240
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
St001240: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
Matching statistic: St000725
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => 3
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 3
[1,1,1,0,0,0]
=> [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 4
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 4
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 5
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 5
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St001004
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => 3
[1,1,0,0,1,0]
=> [2,1,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => 3
[1,1,1,0,0,0]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St000314
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 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: St001654
Values
[1,0]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> 2
[1,1,0,0]
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 3
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The monophonic hull number of a graph.
The monophonic hull of a set of vertices $M$ of a graph $G$ is the set of vertices that lie on at least one induced path between vertices in $M$. The monophonic hull number is the size of the smallest set $M$ such that the monophonic hull of $M$ is all of $G$.
For example, the monophonic hull number of a graph $G$ with $n$ vertices is $n$ if and only if $G$ is a disjoint union of complete graphs.
Matching statistic: St000011
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000007
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
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] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [3,2,1] => 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [3,2,1] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [3,2,1] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [2,4,3,1] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 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
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ 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%
Mp00090: Permutations —cycle-as-one-line notation⟶ 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%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
Description
The number of peaks of a Dyck path.
Matching statistic: St000031
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,2,4,5,3] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,4,5,3] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000056
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 3
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => [1,2,3] => 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 4
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [1,3,2,4] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [1,2,4,3] => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => [1,2,3,4] => 4
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 4
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 4
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [1,4,5,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 5
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,4,2,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [1,3,5,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [1,3,4,2,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [1,2,3,4,5] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [1,3,4,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [1,2,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [1,3,2,5,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [1,3,2,4,5] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => [1,4,2,5,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [1,2,5,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => [1,2,4,3,5] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [1,2,3,4,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => [1,3,2,4,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => [1,2,4,5,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 5
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000160The multiplicity of the smallest part of a partition. St000203The number of external nodes of a binary tree. St000326The position of the first one in a binary word after appending a 1 at the end. St000475The number of parts equal to 1 in a partition. 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. St000025The number of initial rises of a Dyck path. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000331The number of upper interactions of a Dyck path. St000505The biggest entry in the block containing the 1. St000678The number of up steps after the last double rise 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000439The position of the first down step of a Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000654The first descent of a permutation. St000060The greater neighbor of the maximum. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001570The minimal number of edges to add to make a graph Hamiltonian. St001267The length of the Lyndon factorization of the binary word. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000469The distinguishing number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001691The number of kings in a graph. St000907The number of maximal antichains of minimal length in a poset. St000315The number of isolated vertices of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000259The diameter of a connected graph. St000054The first entry of the permutation. St000050The depth or height of a binary tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives 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. St001231The 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000474Dyson's crank of a partition. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001645The pebbling number of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000717The number of ordinal summands of a poset. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000942The number of critical left to right maxima of the parking functions. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000481The number of upper covers of a partition in dominance order. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001372The length of a longest cyclic run of ones of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation.
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