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Your data matches 278 different statistics following compositions of up to 3 maps.
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Matching statistic: St001065
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
St001065: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2
[1,0,1,0]
=> 3
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 7
[1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 6
[1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 5
Description
Number of indecomposable reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001726
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 5
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 5
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 5
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 6
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 6
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 6
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 7
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
Description
The number of visible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Matching statistic: St000539
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000539: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000539: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 0 = 2 - 2
[1,0,1,0]
=> [3,1,2] => [2,1,3] => [2,1,3] => 1 = 3 - 2
[1,1,0,0]
=> [2,3,1] => [1,3,2] => [3,1,2] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 2 = 4 - 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2 = 4 - 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => [4,3,1,2] => 3 = 5 - 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 4 = 6 - 2
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,2,1,4] => [3,2,5,1,4] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [2,4,1,5,3] => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,2,4,1,5] => [3,2,4,1,5] => 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => [2,5,1,4,3] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [4,3,5,1,2] => 4 = 6 - 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,5,1,3,4] => [2,1,3,5,4] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [4,5,3,1,2] => 4 = 6 - 2
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,1,4,5,2] => [3,4,1,5,2] => 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => [4,2,1,3,5] => 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [5,4,3,1,2] => 5 = 7 - 2
Description
The number of odd inversions of a permutation.
An inversion $i < j$ of a permutation is odd if $i \not\equiv j\ (\operatorname{mod} 2)$. See [[St000538]] for even inversions.
Matching statistic: St000777
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,2] => ([],2)
=> ? = 2
[1,0,1,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3
[1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> ? = 3
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {4,5}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {4,5}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {5,6,6,6,7}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {5,6,6,6,7}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,6,6,6,7}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,6,6,6,7}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {5,6,6,6,7}
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000718
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5} + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,5} + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6 = 5 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,6,6,6,7} + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St000499
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St000499: Set partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St000499: Set partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 1 = 2 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,2,4},{3}}
=> 2 = 3 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> {{1,2,5},{3,4,6}}
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> {{1,2,3,5},{4,6}}
=> 4 = 5 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> {{1,2,5,6},{3,4}}
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> {{1,2,4},{3,5,6}}
=> {{1,2,3,5,6},{4}}
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> {{1,2,3},{4,5,6}}
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> {{1,2,5,6,8},{3,4,7}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> {{1,2,3,5,6,8},{4,7}}
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> {{1,2,6,8},{3,4,5,7}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> {{1,2,3,4,6,8},{5,7}}
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> {{1,2,6},{3,4,5,7,8}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> {{1,2,4,5},{3,6,7,8}}
=> {{1,2,3,4,6},{5,7,8}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> {{1,2,3,6,7,8},{4,5}}
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> {{1,2,3,5},{4,6,7,8}}
=> {{1,2,3,4,6,7,8},{5}}
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? ∊ {3,4,5,5,5,5,5,5,6,7} - 1
Description
The rcb statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcb''' (right-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St001645
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 2 - 2
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 3 - 2
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> ? ∊ {3,5} - 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> ? ∊ {3,5} - 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2 = 4 - 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {3,4,4,5,5,5,7} - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4 = 6 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 5 - 2
Description
The pebbling number of a connected graph.
Matching statistic: St001621
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {3,5}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {3,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001623
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001623: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001623: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {3,5}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {3,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
Description
The number of doubly irreducible elements of a lattice.
An element $d$ of a lattice $L$ is '''doubly irreducible''' if it is both join and meet irreducible. That means, $d$ is neither the least nor the greatest element of $L$ and if $d=x\vee y$ or $d=x\wedge y$, then $d\in\{x,y\}$ for all $x,y\in L$.
In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.
Matching statistic: St001626
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {3,5}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {3,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {3,4,4,5,5,6,6,6,7}
Description
The number of maximal proper sublattices of a lattice.
The following 268 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000670The reversal length of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000028The number of stack-sorts needed to sort a permutation. St000259The diameter of a connected graph. St001090The number of pop-stack-sorts needed to sort a permutation. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St000693The modular (standard) major index of a standard tableau. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001330The hat guessing number of a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001485The modular major index of a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000062The length of the longest increasing subsequence of the permutation. St000089The absolute variation of a composition. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000239The number of small weak excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000482The (zero)-forcing number of a graph. St000495The number of inversions of distance at most 2 of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000702The number of weak deficiencies of a permutation. St000730The maximal arc length of a set partition. St000778The metric dimension of a graph. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St001424The number of distinct squares in a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001823The Stasinski-Voll length of a signed permutation. St001894The depth of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000624The normalized sum of the minimal distances to a greater element. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001565The number of arithmetic progressions of length 2 in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000527The width of the poset. St000845The maximal number of elements covered by an element in a poset. St000080The rank of the poset. St000619The number of cyclic descents of a permutation. St000632The jump number of the poset. St001875The number of simple modules with projective dimension at most 1. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. 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. St001510The number of self-evacuating linear extensions of a finite poset. St001782The order of rowmotion on the set of order ideals 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000307The number of rowmotion orbits of a poset. St001555The order of a signed permutation. St001644The dimension of a graph. St001812The biclique partition number of a graph. St001638The book thickness of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000422The energy 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. St000306The bounce count of a Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000260The radius of a connected graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000004The major index of a permutation. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000075The orbit size of a standard tableau under promotion. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000443The number of long tunnels of a Dyck path. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000493The los statistic of a set partition. St000504The cardinality of the first block of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000653The last descent of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001117The game chromatic index of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001517The length of a longest pair of twins in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St000116The major index of a semistandard tableau obtained by standardizing. St000133The "bounce" of a permutation. St000135The number of lucky cars of the parking function. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000245The number of ascents of a permutation. St000325The width of the tree associated to a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000470The number of runs in a permutation. St000521The number of distinct subtrees of an ordered tree. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000736The last entry in the first row of a semistandard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000988The orbit size of a permutation under Foata's bijection. St000989The number of final rises of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001480The number of simple summands of the module J^2/J^3. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001570The minimal number of edges to add to make a graph Hamiltonian. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001596The number of two-by-two squares inside a skew partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001684The reduced word complexity of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001768The number of reduced words of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001928The number of non-overlapping descents in a permutation. St001946The number of descents in a parking function. St000002The number of occurrences of the pattern 123 in a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001114The number of odd descents of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001520The number of strict 3-descents. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St001964The interval resolution global dimension of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000735The last entry on the main diagonal of a standard tableau.
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