Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000722
(load all 7 compositions to match this statistic)
Mapping
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => 1
([(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => 5
([(1,4),(2,3),(3,4)],5)
 => 5
([(0,1),(2,4),(3,4)],5)
 => 4
([(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000189
(load all 3 compositions to match this statistic)
Mapping
Mp00243: Graphs —weak duplicate order⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => ([],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 3
([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2)],4)
 => ([],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 3
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5
([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 5
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => 4
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(2,3),(2,4)],5)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],5)
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => 3
Description
The number of elements in the poset.
Matching statistic: St000228
(load all 6 compositions to match this statistic)
Mapping
Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => 1
([],2)
 => ([],1)
 => [1]
 => 1
([(0,1)],2)
 => ([],2)
 => [1,1]
 => 2
([],3)
 => ([],1)
 => [1]
 => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(0,2),(1,2)],3)
 => ([],2)
 => [1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => [1,1,1]
 => 3
([],4)
 => ([],1)
 => [1]
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => [1,1]
 => 2
([(0,3),(1,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => [1,1]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => [1,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
([],5)
 => ([],1)
 => [1]
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => [1,1]
 => 2
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => 5
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => [1,1,1,1]
 => 4
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => [1,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => [1,1,1]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,1,1,1]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => [1,1,1]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000987
Mapping
Mp00203: Graphs —cone⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],2)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St000718
Mapping
Mp00203: Graphs —cone⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
([],2)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(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 = 5 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(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 = 5 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 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: St000293
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => 10 => 1
([],2)
 => ([],1)
 => [1]
 => 10 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 100 => 2
([],3)
 => ([],1)
 => [1]
 => 10 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => [2]
 => 100 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 3
([],4)
 => ([],1)
 => [1]
 => 10 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => 100 => 2
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => [2]
 => 100 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 4
([],5)
 => ([],1)
 => [1]
 => 10 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => 100 => 2
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 5
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 100010 => 5
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 4
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 1010 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => [2]
 => 100 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => 100000 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 3
Description
The number of inversions of a binary word.
Matching statistic: St001034
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => [1,0]
 => 1
([],2)
 => ([],1)
 => [1]
 => [1,0]
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([],3)
 => ([],1)
 => [1]
 => [1,0]
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],4)
 => ([],1)
 => [1]
 => [1,0]
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([],5)
 => ([],1)
 => [1]
 => [1,0]
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St001746
(load all 2 compositions to match this statistic)
Mapping
Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001746: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
Description
The coalition number of a graph. This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.
Matching statistic: St001342
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St001342: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],2)
 => 2 = 1 + 1
([],2)
 => ([],1)
 => ([],2)
 => 2 = 1 + 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 3 = 2 + 1
([],3)
 => ([],1)
 => ([],2)
 => 2 = 1 + 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([],4)
 => ([],1)
 => ([],2)
 => 2 = 1 + 1
([(2,3)],4)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => 5 = 4 + 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([],5)
 => ([],1)
 => ([],2)
 => 2 = 1 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(2,5),(3,4)],6)
 => 6 = 5 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(2,5),(3,4),(4,5)],6)
 => 6 = 5 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(1,4),(2,3)],5)
 => 5 = 4 + 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([(2,3)],4)
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 6 = 5 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 6 = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 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)
 => ([(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)
 => ([(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 + 1
Description
The number of vertices in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001723
Mapping
Mp00203: Graphs —cone⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001723: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([],2)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(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 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(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 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {6,6,6,6} - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {6,6,6,6} - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {6,6,6,6} - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {6,6,6,6} - 1
Description
The differential of a graph. The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$. The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$. The differential of a graph is the maximal differential of a set of vertices.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001724The 2-packing differential of a graph. St000171The degree of the graph. St001268The size of the largest ordinal summand in the poset. St001644The dimension of a graph. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001725The harmonious chromatic number of a graph. St001622The number of join-irreducible elements of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001742The difference of the maximal and the minimal degree in a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001458The rank of the adjacency matrix of a graph. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St000080The rank of the poset. St001782The order of rowmotion on the set of order ideals of a poset.