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Your data matches 190 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => 2
[1,2,1] => [1,1,2] => 2
[2,2] => [2,2] => 2
[3,1] => [1,3] => 3
[1,1,2,1] => [1,1,1,2] => 2
[1,2,2] => [2,1,2] => 2
[1,3,1] => [1,1,3] => 3
[2,2,1] => [1,2,2] => 2
[2,3] => [3,2] => 2
[3,2] => [2,3] => 3
[4,1] => [1,4] => 4
[1,1,1,2,1] => [1,1,1,1,2] => 2
[1,1,2,2] => [2,1,1,2] => 2
[1,1,3,1] => [1,1,1,3] => 3
[1,2,2,1] => [1,1,2,2] => 2
[1,2,3] => [3,1,2] => 2
[1,3,2] => [2,1,3] => 3
[1,4,1] => [1,1,4] => 4
[2,1,2,1] => [1,2,1,2] => 2
[2,2,2] => [2,2,2] => 2
[2,3,1] => [1,2,3] => 3
[2,4] => [4,2] => 2
[3,2,1] => [1,3,2] => 2
[3,3] => [3,3] => 3
[4,2] => [2,4] => 4
[5,1] => [1,5] => 5
[1,1,1,1,2,1] => [1,1,1,1,1,2] => 2
[1,1,1,2,2] => [2,1,1,1,2] => 2
[1,1,1,3,1] => [1,1,1,1,3] => 3
[1,1,2,2,1] => [1,1,1,2,2] => 2
[1,1,2,3] => [3,1,1,2] => 2
[1,1,3,2] => [2,1,1,3] => 3
[1,1,4,1] => [1,1,1,4] => 4
[1,2,1,2,1] => [1,1,2,1,2] => 2
[1,2,2,2] => [2,1,2,2] => 2
[1,2,3,1] => [1,1,2,3] => 3
[1,2,4] => [4,1,2] => 2
[1,3,2,1] => [1,1,3,2] => 2
[1,3,3] => [3,1,3] => 3
[1,4,2] => [2,1,4] => 4
[1,5,1] => [1,1,5] => 5
[2,1,1,2,1] => [1,2,1,1,2] => 2
[2,1,2,2] => [2,2,1,2] => 2
[2,1,3,1] => [1,2,1,3] => 3
[2,2,2,1] => [1,2,2,2] => 2
[2,2,3] => [3,2,2] => 2
[2,3,2] => [2,2,3] => 3
[2,4,1] => [1,2,4] => 4
[2,5] => [5,2] => 2
[3,1,2,1] => [1,3,1,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10,1]
=> 2
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1,1]
=> 3
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8,1]
=> 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9,1]
=> 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> [1,1]
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15,1]
=> 2
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10,1]
=> 2
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10,1,1]
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12,1]
=> 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1,1]
=> 3
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1,1,1]
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13,1]
=> 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8,1]
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8,1,1]
=> 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> 2
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10,1]
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1]
=> 4
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 5
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14,1]
=> 2
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9,1]
=> 2
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9,1,1]
=> 3
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11,1]
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 3
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1,1]
=> 4
[2,5] => ([(4,6),(5,6)],7)
=> [1,1]
=> 2
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12,1]
=> 2
Description
The length of the partition.
Matching statistic: St000081
Values
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,2,1] => ([(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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,3,1] => ([(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
[1,2,2,1] => ([(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
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,2,1] => ([(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
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> 2
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> 2
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
Description
The number of edges of a graph.
Matching statistic: St000171
Values
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,2,1] => ([(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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,3,1] => ([(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
[1,2,2,1] => ([(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
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,2,1] => ([(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
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> 2
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> 2
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000258
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000258: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000258: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => ([(1,2)],3)
=> 2
[1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 3
[1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[4,1] => [1,4] => ([(3,4)],5)
=> 4
[1,1,1,2,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 4
[5,1] => [1,5] => ([(4,5)],6)
=> 5
[1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,2,2] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,2,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,2,3] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,2,1,2,1] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,2,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,3,1] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 5
[2,1,1,2,1] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,2,2] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,3,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,3,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,4,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,1,2,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
Description
The burning number of a graph.
This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.
Matching statistic: St000271
Values
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,2,1] => ([(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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,3,1] => ([(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
[1,2,2,1] => ([(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
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,2,1] => ([(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
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> 2
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> 2
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
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 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => ([(1,2)],3)
=> 2
[1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 3
[1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[4,1] => [1,4] => ([(3,4)],5)
=> 4
[1,1,1,2,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 4
[5,1] => [1,5] => ([(4,5)],6)
=> 5
[1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,2,2] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,2,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,2,3] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,2,1,2,1] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,2,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,3,1] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 5
[2,1,1,2,1] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,2,2] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,3,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,3,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,4,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,1,2,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
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: St000312
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,2,1] => ([(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
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,3,1] => ([(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
[1,2,2,1] => ([(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
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,2,1] => ([(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
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> 2
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 2
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5] => ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> 2
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 2
Description
The number of leaves in a graph.
That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000382
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => [2,1] => 2
[1,2,1] => [1,2,1] => [2,1,1] => 2
[2,2] => [2,2] => [2,2] => 2
[3,1] => [1,3] => [3,1] => 3
[1,1,2,1] => [1,2,1,1] => [2,1,1,1] => 2
[1,2,2] => [2,2,1] => [2,1,2] => 2
[1,3,1] => [1,3,1] => [3,1,1] => 3
[2,2,1] => [1,2,2] => [2,2,1] => 2
[2,3] => [3,2] => [2,3] => 2
[3,2] => [2,3] => [3,2] => 3
[4,1] => [1,4] => [4,1] => 4
[1,1,1,2,1] => [1,2,1,1,1] => [2,1,1,1,1] => 2
[1,1,2,2] => [2,2,1,1] => [2,1,1,2] => 2
[1,1,3,1] => [1,3,1,1] => [3,1,1,1] => 3
[1,2,2,1] => [1,2,2,1] => [2,2,1,1] => 2
[1,2,3] => [3,2,1] => [2,1,3] => 2
[1,3,2] => [2,3,1] => [3,1,2] => 3
[1,4,1] => [1,4,1] => [4,1,1] => 4
[2,1,2,1] => [1,2,1,2] => [2,1,2,1] => 2
[2,2,2] => [2,2,2] => [2,2,2] => 2
[2,3,1] => [1,3,2] => [3,2,1] => 3
[2,4] => [4,2] => [2,4] => 2
[3,2,1] => [1,2,3] => [2,3,1] => 2
[3,3] => [3,3] => [3,3] => 3
[4,2] => [2,4] => [4,2] => 4
[5,1] => [1,5] => [5,1] => 5
[1,1,1,1,2,1] => [1,2,1,1,1,1] => [2,1,1,1,1,1] => 2
[1,1,1,2,2] => [2,2,1,1,1] => [2,1,1,1,2] => 2
[1,1,1,3,1] => [1,3,1,1,1] => [3,1,1,1,1] => 3
[1,1,2,2,1] => [1,2,2,1,1] => [2,2,1,1,1] => 2
[1,1,2,3] => [3,2,1,1] => [2,1,1,3] => 2
[1,1,3,2] => [2,3,1,1] => [3,1,1,2] => 3
[1,1,4,1] => [1,4,1,1] => [4,1,1,1] => 4
[1,2,1,2,1] => [1,2,1,2,1] => [2,1,2,1,1] => 2
[1,2,2,2] => [2,2,2,1] => [2,2,1,2] => 2
[1,2,3,1] => [1,3,2,1] => [3,2,1,1] => 3
[1,2,4] => [4,2,1] => [2,1,4] => 2
[1,3,2,1] => [1,2,3,1] => [2,3,1,1] => 2
[1,3,3] => [3,3,1] => [3,1,3] => 3
[1,4,2] => [2,4,1] => [4,1,2] => 4
[1,5,1] => [1,5,1] => [5,1,1] => 5
[2,1,1,2,1] => [1,2,1,1,2] => [2,1,1,2,1] => 2
[2,1,2,2] => [2,2,1,2] => [2,1,2,2] => 2
[2,1,3,1] => [1,3,1,2] => [3,1,2,1] => 3
[2,2,2,1] => [1,2,2,2] => [2,2,2,1] => 2
[2,2,3] => [3,2,2] => [2,2,3] => 2
[2,3,2] => [2,3,2] => [3,2,2] => 3
[2,4,1] => [1,4,2] => [4,2,1] => 4
[2,5] => [5,2] => [2,5] => 2
[3,1,2,1] => [1,2,1,3] => [2,1,3,1] => 2
Description
The first part of an integer composition.
Matching statistic: St000544
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => ([(1,2)],3)
=> 2
[1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 3
[1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[4,1] => [1,4] => ([(3,4)],5)
=> 4
[1,1,1,2,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 4
[5,1] => [1,5] => ([(4,5)],6)
=> 5
[1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,2,2] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,2,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,2,3] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,2,1,2,1] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,2,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,3,1] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 5
[2,1,1,2,1] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,2,2] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,3,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,3,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,4,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,1,2,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
The following 180 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000916The packing number of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001118The acyclic chromatic index of a graph. St001322The size of a minimal independent dominating set in a graph. St001479The number of bridges of a graph. St001826The maximal number of leaves on a vertex of a graph. St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St000468The Hosoya index of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St000918The 2-limited packing number of a graph. St001176The size of a partition minus its first part. St001323The independence gap of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number 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. St001957The number of Hasse diagrams with a given underlying undirected graph. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000147The largest part of an integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000297The number of leading ones in a binary word. 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. St000678The number of up steps after the last double rise of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001733The number of weak left to right maxima of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000439The position of the first down step of a Dyck path. St000475The number of parts equal to 1 in a partition. St000504The cardinality of the first block of a set partition. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000636The hull number of a graph. St000674The number of hills of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001654The monophonic hull number of a graph. St001781The interlacing number of a set partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001340The cardinality of a minimal non-edge isolating set of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001117The game chromatic index of a graph. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St001578The minimal number of edges to add or remove to make a graph a line graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001316The domatic number of a graph. St001642The Prague dimension of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000617The number of global maxima of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001060The distinguishing index of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000485The length of the longest cycle of a permutation. St000989The number of final rises of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000461The rix statistic of a permutation. St000648The number of 2-excedences of a permutation. St000776The maximal multiplicity of an eigenvalue in a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000007The number of saliances of the permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000993The multiplicity of the largest part of an integer partition. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000315The number of isolated vertices of a graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000553The number of blocks of a graph. St000654The first descent of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001672The restrained domination number of a graph. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000335The difference of lower and upper interactions. St000338The number of pixed points of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001530The depth of a Dyck path. St000469The distinguishing number of a graph. St000451The length of the longest pattern of the form k 1 2. St000028The number of stack-sorts needed to sort a permutation. St001645The pebbling number of a connected graph. St001271The competition number of a graph. St001622The number of join-irreducible elements of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000141The maximum drop size of a permutation. St000651The maximal size of a rise in a permutation. St001596The number of two-by-two squares inside a skew partition. St001964The interval resolution global dimension of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000652The maximal difference between successive positions of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000209Maximum difference of elements in cycles. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000956The maximal displacement of a permutation. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001589The nesting number of a perfect matching. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001715The number of non-records in a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001644The dimension of a graph. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000422The energy of a graph, if it is integral. St001868The number of alignments of type NE of a signed permutation. St001866The nesting alignments of a signed permutation. St001651The Frankl number of a lattice. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. 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. St001625The Möbius invariant of a lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001616The number of neutral elements in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001620The number of sublattices of a lattice.
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