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Your data matches 107 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000537
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000537: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000537: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The cutwidth of a graph.
This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering $\sigma$ is the maximum, among all the prefixes of $\sigma$, of the number of edges that have exactly one vertex in a prefix.
Matching statistic: St000778
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000778: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000778: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The metric dimension of a graph.
This is the length of the shortest vector of vertices, such that every vertex is uniquely determined by the vector of distances from these vertices.
Matching statistic: St000846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> 2
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000970
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1]
=> [1,0,1,0]
=> 0
{{1,2}}
=> [2,1] => [2]
=> [1,1,0,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
Matching statistic: St001270
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The bandwidth of a graph.
The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be
ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$.
We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Matching statistic: St001349
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001349: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001349: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The number of different graphs obtained from the given graph by removing an edge.
Matching statistic: St001644
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St001949
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The rigidity index of a graph.
A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter.
This statistic yields the minimal size of a base for the automorphism group of a graph.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001962The proper pathwidth of a graph. St000172The Grundy number of a graph. St000258The burning number of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001963The tree-depth of a graph. St000862The number of parts of the shifted shape of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St001128The exponens consonantiae of a partition. St001741The largest integer such that all patterns of this size are contained in the permutation. St000628The balance of a binary word. St000710The number of big deficiencies of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000485The length of the longest cycle of a permutation. St000842The breadth of a permutation. St000264The girth of a graph, which is not a tree. St000487The length of the shortest cycle of a permutation. St000990The first ascent of a permutation. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001344The neighbouring number of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000486The number of cycles of length at least 3 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. St000623The number of occurrences of the pattern 52341 in a permutation. St000629The defect of a binary word. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000929The constant term of the character polynomial of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000699The toughness times the least common multiple of 1,. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St001175The size of a partition minus the hook length of the base cell. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000259The diameter of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000891The number of distinct diagonal sums of a permutation matrix. St001520The number of strict 3-descents. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St000906The length of the shortest maximal chain in a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000845The maximal number of elements covered by an element in a poset. St001566The length of the longest arithmetic progression in a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000872The number of very big descents of a permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001096The size of the overlap set of a permutation. St001812The biclique partition number of a graph. St000458The number of permutations obtained by switching adjacencies or successions. St001642The Prague dimension of a graph. St000741The Colin de Verdière graph invariant. St001052The length of the exterior of a permutation. St001220The width of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000779The tier of a permutation. St000837The number of ascents of distance 2 of a permutation. St001516The number of cyclic bonds of a permutation. St000045The number of linear extensions of a binary tree. St001060The distinguishing index of a graph. St001569The maximal modular displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001960The number of descents of a permutation minus one if its first entry is not one. St001557The number of inversions of the second entry of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001948The number of augmented double ascents of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order.
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