- FindStat

Your data matches 156 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000287
(load all 78 compositions to match this statistic)

Mapping

St000287: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([],5)
 => 5
([(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => 2
([(0,1),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1

Description

The number of connected components of a graph.

Matching statistic: St000010
(load all 14 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 1
([],2)
 => [1,1]
 => 2
([(0,1)],2)
 => [2]
 => 1
([],3)
 => [1,1,1]
 => 3
([(1,2)],3)
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [3]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => [1,1,1,1]
 => 4
([(2,3)],4)
 => [2,1,1]
 => 3
([(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([],5)
 => [1,1,1,1,1]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 4
([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1

Description

The length of the partition.

Matching statistic: St000286
(load all 45 compositions to match this statistic)

Mapping

Mp00111: Graphs —complement⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([(0,1)],2)
 => 2
([(0,1)],2)
 => ([],2)
 => 1
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],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
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => 1

Description

The number of connected components of the complement of a graph. The complement of a graph is the graph on the same vertex set with complementary edges.

Matching statistic: St000383
(load all 40 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1] => 1
([],2)
 => [2] => 2
([(0,1)],2)
 => [1,1] => 1
([],3)
 => [3] => 3
([(1,2)],3)
 => [1,2] => 2
([(0,2),(1,2)],3)
 => [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => 1
([],4)
 => [4] => 4
([(2,3)],4)
 => [1,3] => 3
([(1,3),(2,3)],4)
 => [1,1,2] => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 1
([(0,3),(1,2)],4)
 => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 1
([],5)
 => [5] => 5
([(3,4)],5)
 => [1,4] => 4
([(2,4),(3,4)],5)
 => [1,1,3] => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 1
([(1,4),(2,3)],5)
 => [2,3] => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 1

Description

The last part of an integer composition.

Matching statistic: St000553
(load all 25 compositions to match this statistic)

Mapping

Mp00147: Graphs —square⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 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
([(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
([],5)
 => ([],5)
 => 5
([(3,4)],5)
 => ([(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(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)
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(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)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(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
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(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)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(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)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(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)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,4),(1,3),(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)
 => 1
([(0,3),(0,4),(1,2),(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)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],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
([(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)
 => 1
([(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)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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)
 => 1

Description

The number of blocks of a graph. A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.

Matching statistic: St000093
(load all 2 compositions to match this statistic)

Mapping

Mp00147: Graphs —square⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(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
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(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
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(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
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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
([(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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([],5)
 => ([],5)
 => ([],5)
 => 5
([(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],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)
 => 2
([(0,4),(1,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => ([(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
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(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)
 => 1
([(1,4),(2,3),(2,4),(3,4)],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)
 => 2
([(0,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4)],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)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],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)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(0,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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)
 => 1
([(0,3),(0,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(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)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(0,3),(0,4),(1,2),(1,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(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)
 => 1
([(0,1),(0,4),(1,3),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 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)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000147
(load all 7 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: 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]
 => [2]
 => 2
([(0,1)],2)
 => [2]
 => [1,1]
 => 1
([],3)
 => [1,1,1]
 => [3]
 => 3
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([],4)
 => [1,1,1,1]
 => [4]
 => 4
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 3
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([],5)
 => [1,1,1,1,1]
 => [5]
 => 5
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 4
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1

Description

The largest part of an integer partition.

Matching statistic: St000271

Mapping

Mp00203: Graphs —cone⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],2)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 1
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(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)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(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)
 => ([],1)
 => 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)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 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)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 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)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([(0,2),(1,2)],3)
 => 2
([(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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 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)
 => ([],1)
 => 1

Description

The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.

Matching statistic: St000273
(load all 10 compositions to match this statistic)

Mapping

Mp00147: Graphs —square⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => ([],2)
 => 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],3)
 => ([],3)
 => 3
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],4)
 => ([],4)
 => 4
([(2,3)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(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
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(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
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(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
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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
([(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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([],5)
 => ([],5)
 => ([],5)
 => 5
([(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],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)
 => 2
([(0,4),(1,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => ([(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
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(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)
 => 1
([(1,4),(2,3),(2,4),(3,4)],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)
 => 2
([(0,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4)],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)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],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)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(0,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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)
 => 1
([(0,3),(0,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(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)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(0,3),(0,4),(1,2),(1,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(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)
 => 1
([(0,1),(0,4),(1,3),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 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)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St000288
(load all 15 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 10 => 1
([],2)
 => [1,1]
 => 110 => 2
([(0,1)],2)
 => [2]
 => 100 => 1
([],3)
 => [1,1,1]
 => 1110 => 3
([(1,2)],3)
 => [2,1]
 => 1010 => 2
([(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
([],4)
 => [1,1,1,1]
 => 11110 => 4
([(2,3)],4)
 => [2,1,1]
 => 10110 => 3
([(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
([],5)
 => [1,1,1,1,1]
 => 111110 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 4
([(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 3
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 100010 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => 100000 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 100010 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => 100000 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 100000 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

The following 146 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000469The distinguishing number of a graph. St000544The cop number of a graph. St000636The hull number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000157The number of descents of a standard tableau. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000778The metric dimension of a graph. St001323The independence gap of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001949The rigidity index of a graph. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000172The Grundy number of a graph. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000363The number of minimal vertex covers of a graph. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001530The depth of a Dyck path. St001581The achromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000090The variation of a composition. St000155The number of exceedances (also excedences) of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000806The semiperimeter of the associated bargraph. St001742The difference of the maximal and the minimal degree in a graph. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001828The Euler characteristic of a graph. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000668The least common multiple of the parts of the partition. St001812The biclique partition number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000006The dinv of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000699The toughness times the least common multiple of 1,. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001152The number of pairs with even minimum in a perfect matching. St000160The multiplicity of the smallest part of a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000264The girth of a graph, which is not a tree. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.