Your data matches 95 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000722
(load all 11 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),(2,3)],4)
 => 4
([(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
([],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
([(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3),(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
([(0,3),(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
([],6)
 => 1
([(4,5)],6)
 => 3
([(3,5),(4,5)],6)
 => 3
([(2,5),(3,5),(4,5)],6)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(3,4),(3,5),(4,5)],6)
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000189
(load all 4 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),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => 3
([],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
([(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
([(1,3),(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
([(0,3),(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => 3
([],6)
 => ([],1)
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],2)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],2)
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],3)
 => 3
Description
The number of elements in the poset.
Matching statistic: St001636
(load all 4 compositions to match this statistic)
Mapping
Mp00243: Graphs —weak duplicate order⟶ Posets
St001636: 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),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => 3
([],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
([(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
([(1,3),(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
([(0,3),(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => 3
([],6)
 => ([],1)
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],2)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],2)
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],3)
 => 3
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
Matching statistic: St001746
(load all 5 compositions to match this statistic)
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
St001746: Graphs ⟶ 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)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => 3
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(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
([],6)
 => ([],1)
 => 1
([(4,5)],6)
 => ([(1,2)],3)
 => 3
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2
([(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
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 2
([(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
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: St001391
(load all 4 compositions to match this statistic)
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
St001391: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,2)],3)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,3)],4)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],5)
 => ([],1)
 => 0 = 1 - 1
([(3,4)],5)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 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)
 => 2 = 3 - 1
([],6)
 => ([],1)
 => 0 = 1 - 1
([(4,5)],6)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => 2 = 3 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(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 = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(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 = 3 - 1
Description
The disjunction number of a graph. Let $V_n$ be the power set of $\{1,\dots,n\}$ and let $E_n=\{(a,b)| a,b\in V_n, a\neq b, a\cap b=\emptyset\}$. Then the disjunction number of a graph $G$ is the smallest integer $n$ such that $(V_n, E_n)$ has an induced subgraph isomorphic to $G$.
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),(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
([(0,2),(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
([],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
([(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
([(1,3),(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
([(0,3),(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => [1,1,1]
 => 3
([],6)
 => ([],1)
 => [1]
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],2)
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => [1,1]
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => [1,1,1]
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],2)
 => [1,1]
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],3)
 => [1,1,1]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000459
(load all 4 compositions to match this statistic)
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000459: 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)
 => ([(0,1)],2)
 => [2]
 => 2
([],3)
 => ([],1)
 => [1]
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => ([],1)
 => [1]
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],5)
 => ([],1)
 => [1]
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([(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]
 => 3
([],6)
 => ([],1)
 => [1]
 => 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(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]
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(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]
 => 3
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
(load all 3 compositions to match this statistic)
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000460: 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)
 => ([(0,1)],2)
 => [2]
 => 2
([],3)
 => ([],1)
 => [1]
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => ([],1)
 => [1]
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],5)
 => ([],1)
 => [1]
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([(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]
 => 3
([],6)
 => ([],1)
 => [1]
 => 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(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]
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(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]
 => 3
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
(load all 3 compositions to match this statistic)
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000870: 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)
 => ([(0,1)],2)
 => [2]
 => 2
([],3)
 => ([],1)
 => [1]
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => ([],1)
 => [1]
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],5)
 => ([],1)
 => [1]
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([(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]
 => 3
([],6)
 => ([],1)
 => [1]
 => 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(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]
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => [2]
 => 2
([(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]
 => 3
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001318
(load all 2 compositions to match this statistic)
Mapping
Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001318: Graphs ⟶ 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)
 => ([],2)
 => 2
([],3)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => ([],2)
 => ([],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],3)
 => 3
([],4)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => ([],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => ([],3)
 => 3
([],5)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(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)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(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)
 => 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)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 3
([],6)
 => ([],1)
 => ([],1)
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],2)
 => ([],2)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => ([],2)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([],3)
 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],2)
 => ([],2)
 => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([],3)
 => ([],3)
 => 3
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
The following 85 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001672The restrained domination number of a graph. St001345The Hamming dimension of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000005The bounce statistic of a Dyck path. St000144The pyramid weight of the Dyck path. St000293The number of inversions of a binary word. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000734The last entry in the first row of a standard tableau. St000784The maximum of the length and the largest part of the integer partition. St000947The major index east count of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000532The total number of rook placements on a Ferrers board. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000921The number of internal inversions of a binary word. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000806The semiperimeter of the associated bargraph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000171The degree of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001268The size of the largest ordinal summand in the poset. St000656The number of cuts of a poset. St001645The pebbling number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. 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. St000456The monochromatic index of a connected graph. St000699The toughness times the least common multiple of 1,. St001570The minimal number of edges to add to make a graph Hamiltonian. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001651The Frankl number of a lattice. St000464The Schultz index of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.