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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000240
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St000240: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 3 - 2
[1,2] => 2 = 4 - 2
[2,1] => 1 = 3 - 2
[1,2,3] => 3 = 5 - 2
[1,3,2] => 2 = 4 - 2
[2,1,3] => 2 = 4 - 2
[2,3,1] => 1 = 3 - 2
[3,1,2] => 3 = 5 - 2
[3,2,1] => 3 = 5 - 2
[1,2,3,4] => 4 = 6 - 2
[1,2,4,3] => 3 = 5 - 2
[1,3,2,4] => 3 = 5 - 2
[1,3,4,2] => 2 = 4 - 2
[1,4,2,3] => 4 = 6 - 2
[1,4,3,2] => 4 = 6 - 2
[2,1,3,4] => 3 = 5 - 2
[2,1,4,3] => 2 = 4 - 2
[2,3,1,4] => 2 = 4 - 2
[2,3,4,1] => 1 = 3 - 2
[2,4,1,3] => 3 = 5 - 2
[2,4,3,1] => 3 = 5 - 2
[3,1,2,4] => 4 = 6 - 2
[3,1,4,2] => 3 = 5 - 2
[3,2,1,4] => 4 = 6 - 2
[3,2,4,1] => 3 = 5 - 2
[3,4,1,2] => 4 = 6 - 2
[3,4,2,1] => 4 = 6 - 2
[4,1,2,3] => 4 = 6 - 2
[4,1,3,2] => 4 = 6 - 2
[4,2,1,3] => 4 = 6 - 2
[4,2,3,1] => 4 = 6 - 2
[4,3,1,2] => 3 = 5 - 2
[4,3,2,1] => 3 = 5 - 2
[1,2,3,4,5] => 5 = 7 - 2
[1,2,3,5,4] => 4 = 6 - 2
[1,2,4,3,5] => 4 = 6 - 2
[1,2,4,5,3] => 3 = 5 - 2
[1,2,5,3,4] => 5 = 7 - 2
[1,2,5,4,3] => 5 = 7 - 2
[1,3,2,4,5] => 4 = 6 - 2
[1,3,2,5,4] => 3 = 5 - 2
[1,3,4,2,5] => 3 = 5 - 2
[1,3,4,5,2] => 2 = 4 - 2
[1,3,5,2,4] => 4 = 6 - 2
[1,3,5,4,2] => 4 = 6 - 2
[1,4,2,3,5] => 5 = 7 - 2
[1,4,2,5,3] => 4 = 6 - 2
[1,4,3,2,5] => 5 = 7 - 2
[1,4,3,5,2] => 4 = 6 - 2
[1,4,5,2,3] => 5 = 7 - 2
Description
The number of indices that are not small excedances.
A small excedance is an index $i$ for which $\pi_i = i+1$.
Matching statistic: St001504
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2 = 3 - 1
[1,2] => [1,0,1,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> 3 = 4 - 1
[1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> 4 = 5 - 1
[2,1,3] => [1,1,0,0,1,0]
=> 3 = 4 - 1
[2,3,1] => [1,1,0,1,0,0]
=> 3 = 4 - 1
[3,1,2] => [1,1,1,0,0,0]
=> 4 = 5 - 1
[3,2,1] => [1,1,1,0,0,0]
=> 4 = 5 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 5 = 6 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 6 = 7 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 6 = 7 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 5 = 6 - 1
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001182
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,4,2,3] => [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]
=> 6
[1,4,3,2] => [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]
=> 6
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 6
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 5
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 7
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 6
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 6
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 6
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 6
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000209
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> [1] => 0 = 3 - 3
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> [2,1] => 1 = 4 - 3
[2,1] => [[.,.],.]
=> [[.,.],.]
=> [1,2] => 0 = 3 - 3
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 2 = 5 - 3
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 5 - 3
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 4 - 3
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [3,1,2] => 2 = 5 - 3
[3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1 = 4 - 3
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => 0 = 3 - 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 6 - 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 6 - 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3 = 6 - 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3 = 6 - 3
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3 = 6 - 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3 = 6 - 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 5 - 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 5 - 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 6 - 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 6 - 3
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 6 - 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 5 - 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 5 - 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 5 - 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 4 - 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2 = 5 - 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 6 - 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 6 - 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 5 - 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 5 - 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 4 - 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2 = 5 - 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 4 - 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 3 - 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4 = 7 - 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4 = 7 - 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 4 = 7 - 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4 = 7 - 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 4 = 7 - 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 4 = 7 - 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 4 = 7 - 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 4 = 7 - 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 7 - 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 7 - 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 7 - 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 4 = 7 - 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 4 = 7 - 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 4 = 7 - 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 4 = 7 - 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 4 = 7 - 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 7 - 3
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St001115
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 0 = 3 - 3
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0 = 3 - 3
[2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 1 = 4 - 3
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 3 - 3
[1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 1 = 4 - 3
[2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 1 = 4 - 3
[2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2 = 5 - 3
[3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 2 = 5 - 3
[3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 2 = 5 - 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0 = 3 - 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => 1 = 4 - 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => 1 = 4 - 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 2 = 5 - 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => 2 = 5 - 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => 2 = 5 - 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => 1 = 4 - 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => 2 = 5 - 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 2 = 5 - 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => 3 = 6 - 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => 3 = 6 - 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => 3 = 6 - 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 2 = 5 - 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => 3 = 6 - 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 2 = 5 - 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => 3 = 6 - 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 2 = 5 - 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 2 = 5 - 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 3 = 6 - 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 3 = 6 - 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 3 = 6 - 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 3 = 6 - 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 3 = 6 - 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 3 = 6 - 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 3 - 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => 1 = 4 - 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => 1 = 4 - 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => 2 = 5 - 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => 2 = 5 - 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => 2 = 5 - 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => 1 = 4 - 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => 2 = 5 - 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => 2 = 5 - 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => 3 = 6 - 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => 3 = 6 - 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => 3 = 6 - 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => 2 = 5 - 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => 3 = 6 - 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => 2 = 5 - 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => 3 = 6 - 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => 2 = 5 - 3
Description
The number of even descents of a permutation.
Matching statistic: St000653
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 3 - 3
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 3 - 3
[2,1] => [2,1] => [2,1] => [2,1] => 1 = 4 - 3
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 3 - 3
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 4 - 3
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 4 - 3
[2,3,1] => [2,3,1] => [2,3,1] => [3,2,1] => 2 = 5 - 3
[3,1,2] => [2,3,1] => [2,3,1] => [3,2,1] => 2 = 5 - 3
[3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 2 = 5 - 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 3 - 3
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1 = 4 - 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1 = 4 - 3
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 3 = 6 - 3
[1,4,2,3] => [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 3 = 6 - 3
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 3 = 6 - 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 4 - 3
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [4,2,1,3] => 2 = 5 - 3
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 2 = 5 - 3
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3 = 6 - 3
[2,4,1,3] => [4,3,1,2] => [1,4,3,2] => [1,3,4,2] => 3 = 6 - 3
[2,4,3,1] => [2,4,3,1] => [4,2,3,1] => [2,3,4,1] => 3 = 6 - 3
[3,1,2,4] => [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 2 = 5 - 3
[3,1,4,2] => [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3 = 6 - 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 5 - 3
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => [2,4,3,1] => 3 = 6 - 3
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [3,4,1,2] => 2 = 5 - 3
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2 = 5 - 3
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3 = 6 - 3
[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => [2,3,4,1] => 3 = 6 - 3
[4,2,1,3] => [3,2,4,1] => [3,2,4,1] => [2,4,3,1] => 3 = 6 - 3
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2 = 5 - 3
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2 = 5 - 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [3,4,1,2] => 2 = 5 - 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 3 - 3
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 1 = 4 - 3
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1 = 4 - 3
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [5,4,2,3,1] => 4 = 7 - 3
[1,2,5,3,4] => [1,2,4,5,3] => [4,5,1,2,3] => [5,4,2,3,1] => 4 = 7 - 3
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [4,5,2,3,1] => 4 = 7 - 3
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 1 = 4 - 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [5,3,2,1,4] => 3 = 6 - 3
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 3 = 6 - 3
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [5,4,3,2,1] => 4 = 7 - 3
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [1,4,5,3,2] => 4 = 7 - 3
[1,3,5,4,2] => [1,3,5,4,2] => [5,3,4,1,2] => [3,5,4,2,1] => 4 = 7 - 3
[1,4,2,3,5] => [1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 3 = 6 - 3
[1,4,2,5,3] => [1,3,4,5,2] => [3,4,5,1,2] => [5,4,3,2,1] => 4 = 7 - 3
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 3 = 6 - 3
[1,4,3,5,2] => [1,4,3,5,2] => [4,3,5,1,2] => [4,5,3,2,1] => 4 = 7 - 3
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => [3,4,5,2,1] => 4 = 7 - 3
[1,4,5,3,2] => [1,4,5,3,2] => [4,5,3,1,2] => [4,3,5,2,1] => 4 = 7 - 3
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000831
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => ? = 3 - 3
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0 = 3 - 3
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1 = 4 - 3
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 3 - 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 5 - 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 4 - 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 4 - 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 5 - 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 5 - 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 3 - 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 5 - 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 5 - 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 5 - 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 6 - 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 6 - 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 4 - 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 6 - 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 4 - 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 4 - 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3 = 6 - 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3 = 6 - 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 5 - 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 5 - 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 5 - 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 5 - 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 5 - 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 5 - 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 6 - 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 6 - 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 6 - 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 6 - 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 6 - 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 6 - 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 3 - 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 5 - 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 5 - 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 5 - 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 4 = 7 - 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 4 = 7 - 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 5 - 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 3 = 6 - 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 5 - 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 5 - 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3 = 6 - 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3 = 6 - 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 3 = 6 - 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3 = 6 - 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 3 = 6 - 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3 = 6 - 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 3 = 6 - 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 3 = 6 - 3
Description
The number of indices that are either descents or recoils.
This is, for a permutation $\pi$ of length $n$, this statistics counts the set
$$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$
Matching statistic: St001232
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 0 = 3 - 3
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 4 - 3
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 3 - 3
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 5 - 3
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 5 - 3
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 4 - 3
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 5 - 3
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 4 - 3
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 3 - 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6} - 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6} - 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6} - 3
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 6 - 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6} - 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 5 - 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 5 - 3
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 6 - 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6} - 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 5 - 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 4 - 3
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 5 - 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 6 - 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6} - 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 5 - 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 4 - 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 5 - 3
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 4 - 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 7 - 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 7 - 3
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 7 - 3
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 7 - 3
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 6 - 3
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 6 - 3
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 7 - 3
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 6 - 3
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 7 - 3
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,1,3,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000777
(load all 49 compositions to match this statistic)
(load all 49 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1 = 3 - 2
[1,2] => [1,2] => ([],2)
=> ? = 3 - 2
[2,1] => [2,1] => ([(0,1)],2)
=> 2 = 4 - 2
[1,2,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {3,4,5} - 2
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {3,4,5} - 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {3,4,5} - 2
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 5 - 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 5 - 2
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
[1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[2,1,3,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
[2,3,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
[2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 6 - 2
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,5,5,5,5,6} - 2
[3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
[3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 5 - 2
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[4,1,2,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
[4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
[4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 6 - 2
[4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[1,2,3,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,2,3,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,2,4,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,2,4,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,2,5,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,2,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,3,2,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,3,4,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,3,5,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,4,2,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,4,2,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,4,3,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,5,2,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,1,3,4,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,1,3,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,1,4,3,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[2,3,1,4,5] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,3,1,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,3,4,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,3,4,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[2,3,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,3,5,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[2,4,1,3,5] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,4,1,5,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,4,3,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[2,4,5,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,4,5,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[2,5,1,3,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,5,1,4,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,5,3,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,5,3,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[2,5,4,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[2,5,4,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,1,2,4,5] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,1,2,5,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,1,4,2,5] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,1,4,5,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,1,5,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,1,5,4,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,2,1,4,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
[3,2,4,1,5] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,2,4,5,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,2,5,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,2,5,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[3,4,1,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 7 - 2
[4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7} - 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1] => ([],1)
=> 0 = 3 - 3
[1,2] => ([],2)
=> [2] => ([],2)
=> 0 = 3 - 3
[2,1] => ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 4 - 3
[1,2,3] => ([],3)
=> [3] => ([],3)
=> 0 = 3 - 3
[1,3,2] => ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1 = 4 - 3
[2,1,3] => ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1 = 4 - 3
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 5 - 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 5 - 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 5 - 3
[1,2,3,4] => ([],4)
=> [4] => ([],4)
=> 0 = 3 - 3
[1,2,4,3] => ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1 = 4 - 3
[1,3,2,4] => ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1 = 4 - 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 5 - 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 5 - 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[2,1,3,4] => ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1 = 4 - 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 5 - 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 6 - 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 6 - 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 5 - 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 6 - 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 6 - 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 6 - 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 6 - 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {5,5,5,5,5,6,6,6,6,6} - 3
[1,2,3,4,5] => ([],5)
=> [5] => ([],5)
=> 0 = 3 - 3
[1,2,3,5,4] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1 = 4 - 3
[1,2,4,3,5] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1 = 4 - 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 5 - 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 5 - 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,2,4,5] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1 = 4 - 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 5 - 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 5 - 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1 = 4 - 3
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 5 - 3
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 7 - 3
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 3
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000259The diameter of a connected graph. St001645The pebbling number of a connected graph. St001769The reflection length of a signed permutation. St001330The hat guessing number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St000422The energy of a graph, if it is integral. St001626The number of maximal proper sublattices of a lattice. St000264The girth of a graph, which is not a tree. St001733The number of weak left to right maxima of a Dyck path. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset.
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