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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St001861
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Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001861: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => [3,4,1,2] => 4
[3,4,2,1] => [3,4,2,1] => 3
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [4,2,3,1] => 4
[4,3,1,2] => [4,3,1,2] => 3
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => 3
[1,3,5,2,4] => [1,3,5,2,4] => 3
[1,3,5,4,2] => [1,3,5,4,2] => 3
[1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,4,3,5,2] => 3
[1,4,5,2,3] => [1,4,5,2,3] => 4
Description
The number of Bruhat lower covers of a permutation.
This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.
Matching statistic: St000030
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> {{1}}
=> [1] => 0
[1,2] => {{1},{2}}
=> {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => 1
[2,1,3] => {{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => 1
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 2
[3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 2
[3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,4,2,3] => {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 2
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,1,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 3
[3,1,2,4] => {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 2
[3,1,4,2] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 4
[3,4,2,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,3,2] => {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 3
[4,2,1,3] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 4
[4,3,1,2] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 3
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 3
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 4
Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000081
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2,1] => ([],2)
=> ([],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => ([],3)
=> ([],3)
=> 0
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [4,3,2,1] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
Description
The number of edges of a graph.
Matching statistic: St001869
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001869: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001869: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2,1] => ([],2)
=> ([],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => ([],3)
=> ([],3)
=> 0
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [4,3,2,1] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
Description
The maximum cut size of a graph.
A '''cut''' is a set of edges which connect different sides of a vertex partition $V = A \sqcup B$.
Matching statistic: St000957
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
St000957: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 4
[4,3,1,2] => 3
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 4
[1,4,5,3,2] => 3
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000327
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ? = 0
[1,2] => [2,1] => ([],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => ([],3)
=> 0
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 3
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 3
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 3
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St000728
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => [3,2,1] => [2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => [3,1,2] => [3,1,2] => {{1,2,3}}
=> 2
[3,2,1] => [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => [4,3,2,1] => [3,2,4,1] => {{1,3,4},{2}}
=> 3
[2,4,1,3] => [4,2,1,3] => [2,4,1,3] => {{1,2,3,4}}
=> 3
[2,4,3,1] => [3,4,2,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => [4,3,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 3
[3,2,1,4] => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,2,4,1] => [2,4,3,1] => [3,4,2,1] => {{1,2,3,4}}
=> 3
[3,4,1,2] => [4,1,3,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 4
[3,4,2,1] => [4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 3
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 3
[4,1,3,2] => [3,4,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 3
[4,2,3,1] => [2,3,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 4
[4,3,1,2] => [3,1,4,2] => [4,3,1,2] => {{1,2,3,4}}
=> 3
[4,3,2,1] => [3,2,4,1] => [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 3
[1,3,5,4,2] => [1,4,5,3,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,4,3,5,2] => [1,3,5,4,2] => [1,4,5,3,2] => {{1},{2,3,4,5}}
=> 3
[1,4,5,2,3] => [1,5,2,4,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 4
[1,4,5,3,2] => [1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
Description
The dimension of a set partition.
This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks.
A slightly shifted definition of the dimension is [[St000572]].
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 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: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Values
[1] => ([],1)
=> [1] => ([],1)
=> 0
[1,2] => ([],2)
=> [2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> [3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 2
[1,2,3,4] => ([],4)
=> [4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,1,3,4] => ([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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
[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
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[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
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[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
[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)
=> ? = 4
[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)
=> ? = 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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
[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
[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)
=> ? = 4
[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)
=> ? = 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)
=> ? = 3
[1,2,3,4,5] => ([],5)
=> [5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[1,3,2,4,5] => ([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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
[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
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[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
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[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
[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)
=> ? = 4
[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)
=> ? = 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)
=> ? = 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
[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
[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)
=> ? = 4
[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)
=> ? = 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)
=> ? = 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.
Matching statistic: St001596
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 60%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0
[2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[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]
=> [[6],[]]
=> 0
[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]
=> [[5,5],[3]]
=> 1
[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]
=> [[5,4],[2]]
=> 1
[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]
=> [[4,4,4],[2,2]]
=> ? = 2
[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]
=> [[5,5],[2]]
=> ? = 2
[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]
=> [[5,5],[2]]
=> ? = 2
[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]
=> [[5,3],[1]]
=> 1
[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]
=> [[4,4,3],[2,1]]
=> ? = 2
[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]
=> [[4,3,3],[1,1]]
=> ? = 2
[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]
=> [[3,3,3,3],[1,1,1]]
=> ? = 3
[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]
=> [[4,4,3],[1,1]]
=> ? = 3
[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]
=> [[4,4,3],[1,1]]
=> ? = 3
[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]
=> [[5,4],[1]]
=> ? = 2
[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]
=> [[4,4,4],[2,1]]
=> ? = 3
[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]
=> [[5,4],[1]]
=> ? = 2
[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]
=> [[4,4,4],[2,1]]
=> ? = 3
[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]
=> [[5,5],[1]]
=> ? = 4
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[5,5],[1]]
=> ? = 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 4
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 3
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001633
Values
[1] => ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 2
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 3
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 4
[3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 3
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 3
[4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 4
[4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 3
[4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 4
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 3
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 3
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 3
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 4
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 3
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 3
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001645The pebbling number of a connected graph. St000264The girth of a graph, which is not a tree. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000091The descent variation of a composition. St000173The segment statistic of a semistandard tableau. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001330The hat guessing number of a graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
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