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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
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(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 3
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> 3
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> 2
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 3
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 4
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 2
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
{{1,2,4,7},{3,6},{5}}
=> [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> 3
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [1,6,3,2,4,5,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 2
{{1,2,6,7},{3,4,5}}
=> [2,6,4,5,3,7,1] => [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> 2
{{1,2,7},{3,4,5,6}}
=> [2,7,4,5,6,3,1] => [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> 1
{{1,2,7},{3,4,5},{6}}
=> [2,7,4,5,3,6,1] => [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> 2
{{1,2,6,7},{3,4},{5}}
=> [2,6,4,3,5,7,1] => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 3
{{1,2,7},{3,4,6},{5}}
=> [2,7,4,6,5,3,1] => [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000528
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(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,5,2,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,5,3,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ? = 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 2
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,3,4,2,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,5,4,3,2,6] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,3,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 2
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,3,4,5,2,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 7
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 7
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 3
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ([(0,1),(0,4),(0,5),(0,6),(1,18),(2,8),(2,9),(2,21),(3,2),(3,11),(3,12),(3,20),(4,10),(4,14),(4,18),(5,10),(5,13),(5,18),(6,3),(6,13),(6,14),(6,18),(8,17),(8,19),(9,17),(9,19),(10,15),(10,20),(11,8),(11,16),(11,21),(12,9),(12,16),(12,21),(13,11),(13,15),(13,20),(14,12),(14,15),(14,20),(15,16),(15,21),(16,17),(16,19),(17,7),(18,20),(19,7),(20,21),(21,19)],22)
=> ? = 6
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 7
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 5
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ? = 3
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,6,4,5,3,7] => ?
=> ? = 2
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,5,4,6,3,7] => ?
=> ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 6
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,6,5,4,3,7] => ([(0,4),(0,5),(0,6),(1,17),(2,7),(2,11),(3,1),(3,10),(3,12),(4,13),(4,14),(5,3),(5,14),(5,15),(6,2),(6,13),(6,15),(7,19),(9,20),(9,21),(10,17),(10,18),(11,9),(11,19),(12,9),(12,17),(12,18),(13,7),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,18),(16,19),(17,21),(18,20),(18,21),(19,20),(20,8),(21,8)],22)
=> ? = 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,4,6,5,3,7] => ?
=> ? = 3
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 5
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 7
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ?
=> ? = 4
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(1,17),(1,18),(2,11),(2,12),(2,18),(3,10),(3,12),(3,17),(4,7),(4,8),(4,10),(4,18),(5,7),(5,9),(5,11),(5,17),(7,14),(7,21),(7,22),(8,14),(8,21),(8,23),(9,14),(9,22),(9,23),(10,21),(10,24),(11,22),(11,24),(12,24),(13,6),(14,15),(14,16),(15,13),(15,19),(16,13),(16,19),(17,21),(17,23),(17,24),(18,22),(18,23),(18,24),(19,6),(20,19),(21,15),(21,20),(22,16),(22,20),(23,15),(23,16),(23,20),(24,20)],25)
=> ? = 7
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 5
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ?
=> ? = 2
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ?
=> ? = 6
{{1,2,4,7},{3,6},{5}}
=> [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ?
=> ? = 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [1,4,3,2,5,6,7] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 3
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [1,6,3,2,4,5,7] => ?
=> ? = 2
{{1,2,6,7},{3,4,5}}
=> [2,6,4,5,3,7,1] => [1,5,3,4,2,6,7] => ?
=> ? = 2
{{1,2,7},{3,4,5,6}}
=> [2,7,4,5,6,3,1] => [1,6,3,4,5,2,7] => ?
=> ? = 1
{{1,2,7},{3,4,5},{6}}
=> [2,7,4,5,3,6,1] => [1,5,3,4,6,2,7] => ?
=> ? = 2
{{1,2,6,7},{3,4},{5}}
=> [2,6,4,3,5,7,1] => [1,4,3,5,2,6,7] => ?
=> ? = 3
{{1,2,7},{3,4,6},{5}}
=> [2,7,4,6,5,3,1] => [1,6,3,5,4,2,7] => ?
=> ? = 1
{{1,2,7},{3,4},{5},{6}}
=> [2,7,4,3,5,6,1] => [1,4,3,5,6,2,7] => ?
=> ? = 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [1,3,4,2,5,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,18),(2,8),(2,9),(2,21),(3,2),(3,11),(3,12),(3,20),(4,10),(4,14),(4,18),(5,10),(5,13),(5,18),(6,3),(6,13),(6,14),(6,18),(8,17),(8,19),(9,17),(9,19),(10,15),(10,20),(11,8),(11,16),(11,21),(12,9),(12,16),(12,21),(13,11),(13,15),(13,20),(14,12),(14,15),(14,20),(15,16),(15,21),(16,17),(16,19),(17,7),(18,20),(19,7),(20,21),(21,19)],22)
=> ? = 6
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [1,6,4,2,3,5,7] => ?
=> ? = 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [1,3,6,2,4,5,7] => ?
=> ? = 6
{{1,2,6,7},{3,5},{4}}
=> [2,6,5,4,3,7,1] => [1,5,4,3,2,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,7),(2,11),(3,1),(3,10),(3,12),(4,13),(4,14),(5,3),(5,14),(5,15),(6,2),(6,13),(6,15),(7,19),(9,20),(9,21),(10,17),(10,18),(11,9),(11,19),(12,9),(12,17),(12,18),(13,7),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,18),(16,19),(17,21),(18,20),(18,21),(19,20),(20,8),(21,8)],22)
=> ? = 2
{{1,2,7},{3,5,6},{4}}
=> [2,7,5,4,6,3,1] => [1,6,4,3,5,2,7] => ?
=> ? = 1
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000454
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
{{1,2,3}}
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
{{1,2,3,4}}
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
{{1,2,4},{3}}
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
{{1,2,3,5},{4}}
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
{{1,2,4,5},{3}}
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
{{1,2,5},{3,4}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
{{1,2,3,4,5,6}}
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
{{1,2,3,4,6},{5}}
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
{{1,2,3,5,6},{4}}
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
{{1,2,3,6},{4,5}}
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [3,1,1,1] => ([(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)
=> ? = 5 - 1
{{1,2,4,5,6},{3}}
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
{{1,2,4,6},{3,5}}
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
{{1,2,5,6},{3,4}}
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,6},{3,4,5}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,5,6},{3},{4}}
=> [4,1,1] => [3,1,1,1] => ([(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)
=> ? = 5 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,6},{3},{4,5}}
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
{{1,2,3,4,5,6,7}}
=> [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 7 - 1
{{1,2,3,4,5,7},{6}}
=> [6,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
{{1,2,3,4,6,7},{5}}
=> [6,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 1
{{1,2,3,5,6,7},{4}}
=> [6,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 1
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
{{1,2,4,5,6,7},{3}}
=> [6,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
{{1,2,4,5,7},{3,6}}
=> [5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
{{1,2,4,5,7},{3},{6}}
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
{{1,2,4,6,7},{3,5}}
=> [5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
{{1,2,4,7},{3,5,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,4,7},{3,5},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 1
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,5,6,7},{3,4}}
=> [5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
{{1,2,5,7},{3,4,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,6,7},{3,4,5}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,7},{3,4,5,6}}
=> [3,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,7},{3,4,5},{6}}
=> [3,3,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,6,7},{3,4},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
{{1,2,7},{3,4,6},{5}}
=> [3,3,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,5,6,7},{3},{4}}
=> [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 1
{{1,2,5,7},{3,6},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,5,7},{3},{4,6}}
=> [4,1,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 1
{{1,2,6,7},{3,5},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,7},{3,5,6},{4}}
=> [3,3,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
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.
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