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Your data matches 219 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,5,7],[2,6],[4]]
=> [2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,5,7],[2,4],[6]]
=> [2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000479
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10 = 2 + 8
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10 = 2 + 8
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10 = 2 + 8
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 8
[[1,3,5,6,7],[2,4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,5,6,7],[2],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,5,7],[2,4,6]]
=> [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)
=> ? = 3 + 8
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,6,7],[2,5],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,3,5,7],[2,6],[4]]
=> [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)
=> ? = 3 + 8
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,5,7],[2,4],[6]]
=> [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)
=> ? = 3 + 8
[[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,5,6],[2,7],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 8
[[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 8
[[1,3,5,7],[2],[4],[6]]
=> [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)
=> ? = 3 + 8
[[1,3,5,6],[2],[4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,6],[2,5,7],[4]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,5],[2,4,6],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,6],[2,5],[4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,5],[2,6],[4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,6],[2,5],[4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,5],[2,6],[4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,5],[2,4],[6],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,5],[2],[4],[6],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3],[2,5],[4,6],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3],[2,5],[4],[6],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 10 = 2 + 8
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6,7,8],[2],[4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6,7],[2,4,8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6,7,8],[2,5],[4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6,7],[2,8],[4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6,7],[2,4],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6,7],[2],[4],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6,7],[2,5,8],[4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6],[2,4,7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6,7],[2,5],[4,8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6],[2,7],[4,8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6,7],[2,5],[4],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6],[2,7],[4],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6],[2,4],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5,6],[2],[4],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6],[2,5,7],[4,8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6],[2,5,7],[4],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5],[2,4,6],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6],[2,5],[4,7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5],[2,6],[4,7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6],[2,5],[4],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5],[2,6],[4],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5],[2,4],[6],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,5],[2],[4],[6],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3],[2,5],[4,6],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3],[2,5],[4],[6],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 10 = 2 + 8
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> 10 = 2 + 8
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => ([(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> 10 = 2 + 8
[[1,3,5,6,7,8,9],[2],[4]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> 10 = 2 + 8
[[1,3,5,6,7,8,9,10],[2],[4]]
=> [2,2,6] => ([(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> 10 = 2 + 8
Description
The Ramsey number of a graph.
This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
Matching statistic: St001667
(load all 150 compositions to match this statistic)
(load all 150 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001667: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001667: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Values
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,5,3,1] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [4,2,5,3,1] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 2
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,2,1,3,4,6] => [2,5,6,4,3,1] => 3
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [5,1,3,2,4,6] => [2,6,4,5,3,1] => 3
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,5,1,3,4,6] => [5,2,6,4,3,1] => 3
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => [4,6,2,5,3,1] => 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,2,1,3,5,4] => [1,5,6,4,2,3] => 3
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [6,1,3,2,5,4] => [1,6,4,5,2,3] => 3
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [5,3,2,1,4,6] => [2,4,5,6,3,1] => 3
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [4,5,2,1,3,6] => [3,2,5,6,4,1] => 3
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [4,1,5,3,2,6] => [3,6,2,4,5,1] => 3
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => 3
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [6,1,5,3,2,4] => [1,6,2,4,5,3] => 3
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [2,3,5,1,4,6] => [5,4,2,6,3,1] => 3
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [2,4,1,5,3,6] => [5,3,6,2,4,1] => 3
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [4,6,2,1,5,3] => [3,1,5,6,2,4] => 3
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,1,6,3,5,2] => [3,6,1,4,2,5] => 3
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [6,3,5,2,1,4] => [1,4,2,5,6,3] => 3
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [6,4,2,5,1,3] => [1,3,5,2,6,4] => 3
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => [4,6,7,5,3,2,1] => ? = 2
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [5,2,1,3,4,6,7] => [3,6,7,5,4,2,1] => ? = 3
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [6,2,1,3,4,5,7] => [2,6,7,5,4,3,1] => ? = 3
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [6,1,3,2,4,5,7] => [2,7,5,6,4,3,1] => ? = 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [6,1,2,4,3,5,7] => [2,7,6,4,5,3,1] => ? = 3
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => ? = 2
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,5,1,3,4,6,7] => [6,3,7,5,4,2,1] => ? = 3
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [2,6,1,3,4,5,7] => [6,2,7,5,4,3,1] => ? = 3
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [3,1,6,2,4,5,7] => [5,7,2,6,4,3,1] => ? = 2
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [4,1,2,6,3,5,7] => [4,7,6,2,5,3,1] => ? = 3
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [6,2,1,4,3,5,7] => [2,6,7,4,5,3,1] => ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [6,1,3,4,2,5,7] => [2,7,5,4,6,3,1] => ? = 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [6,2,1,3,5,4,7] => [2,6,7,5,3,4,1] => ? = 3
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [7,2,1,4,3,5,6] => [1,6,7,4,5,3,2] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [7,2,1,3,5,4,6] => [1,6,7,5,3,4,2] => ? = 3
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [7,2,1,3,4,6,5] => [1,6,7,5,4,2,3] => ? = 3
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [7,1,3,2,4,6,5] => [1,7,5,6,4,2,3] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [7,1,2,4,3,6,5] => [1,7,6,4,5,2,3] => ? = 3
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [5,4,2,1,3,6,7] => [3,4,6,7,5,2,1] => ? = 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [4,5,2,1,3,6,7] => [4,3,6,7,5,2,1] => ? = 3
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [6,3,2,1,4,5,7] => [2,5,6,7,4,3,1] => ? = 2
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => ? = 3
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [6,1,4,3,2,5,7] => [2,7,4,5,6,3,1] => ? = 3
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [6,5,2,1,3,4,7] => [2,3,6,7,5,4,1] => ? = 3
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [4,6,2,1,3,5,7] => [4,2,6,7,5,3,1] => ? = 3
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [4,1,6,3,2,5,7] => [4,7,2,5,6,3,1] => ? = 2
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [5,6,2,1,3,4,7] => [3,2,6,7,5,4,1] => ? = 3
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [5,1,6,3,2,4,7] => [3,7,2,5,6,4,1] => ? = 2
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [5,1,2,6,4,3,7] => [3,7,6,2,4,5,1] => ? = 3
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [7,4,2,1,3,5,6] => [1,4,6,7,5,3,2] => ? = 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [7,5,2,1,3,4,6] => [1,3,6,7,5,4,2] => 3
[[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [7,6,2,1,3,4,5] => [1,2,6,7,5,4,3] => 3
[[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [7,1,6,3,2,4,5] => [1,7,2,5,6,4,3] => ? = 2
[[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [7,1,2,6,4,3,5] => [1,7,6,2,4,5,3] => ? = 3
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [4,7,2,1,3,5,6] => [4,1,6,7,5,3,2] => ? = 2
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [5,7,2,1,3,4,6] => [3,1,6,7,5,4,2] => ? = 3
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,4,1,5,3,6,7] => [6,4,7,3,5,2,1] => ? = 3
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [2,3,6,1,4,5,7] => [6,5,2,7,4,3,1] => ? = 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,1,6,3,5,7] => [6,4,7,2,5,3,1] => ? = 3
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [3,1,4,6,2,5,7] => [5,7,4,2,6,3,1] => ? = 3
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [2,5,1,3,6,4,7] => [6,3,7,5,2,4,1] => ? = 3
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [3,1,5,2,6,4,7] => [5,7,3,6,2,4,1] => ? = 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => [6,4,7,1,5,3,2] => ? = 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,5,1,3,7,4,6] => [6,3,7,5,1,4,2] => ? = 3
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [7,4,2,1,5,3,6] => [1,4,6,7,3,5,2] => ? = 2
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [7,5,2,1,4,3,6] => [1,3,6,7,4,5,2] => 3
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [7,3,2,1,4,6,5] => [1,5,6,7,4,2,3] => ? = 2
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [7,4,2,1,3,6,5] => [1,4,6,7,5,2,3] => ? = 3
[[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [7,1,4,3,2,6,5] => [1,7,4,5,6,2,3] => ? = 3
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [7,5,2,1,3,6,4] => [1,3,6,7,5,2,4] => 3
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [7,6,2,1,4,3,5] => [1,2,6,7,4,5,3] => 3
[[1,2,5],[3,4,7],[6]]
=> [6,3,4,7,1,2,5] => [7,1,6,3,4,2,5] => [1,7,2,5,4,6,3] => ? = 2
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [7,6,2,1,3,5,4] => [1,2,6,7,5,3,4] => 3
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [7,1,6,3,2,5,4] => [1,7,2,5,6,3,4] => ? = 2
[[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [7,1,2,6,4,5,3] => [1,7,6,2,4,3,5] => ? = 3
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [6,7,2,1,4,3,5] => [2,1,6,7,4,5,3] => ? = 2
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [7,4,2,5,1,3,6] => [1,4,6,3,7,5,2] => 3
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [7,4,2,6,1,3,5] => [1,4,6,2,7,5,3] => 3
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [7,5,2,1,6,3,4] => [1,3,6,7,2,5,4] => 3
[[1,3],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3] => [7,4,2,5,6,1,3] => [1,4,6,3,2,7,5] => 3
Description
The maximal size of a pair of weak twins for a permutation.
A pair of weak twins in a permutation is a pair of two disjoint subsequences of the same length with the same descent pattern. More formally, a pair of weak twins of size $k$ for a permutation $\pi$ of length $n$ are two disjoint lists $1 \leq i_1 < \dots < i_k \leq n$ and $1 \leq j_1 < \dots < j_k \leq n$ such that $\pi(i_a) < \pi(i_{a+1})$ if and only if $\pi(j_a) < \pi(j_{a+1})$ for all $1 \leq a < k$.
Matching statistic: St000422
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3,4,6],[2,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,4,6],[3,5]]
=> [[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,4,6],[2],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,4,6],[3],[5]]
=> [[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,4],[2,5,6]]
=> [[1,2,3],[4,5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,4,6],[2,5],[3]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,6],[2,4],[5]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,6],[3,4],[5]]
=> [[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,4],[2,6],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,4],[3,6],[5]]
=> [[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,4,6],[2],[3],[5]]
=> [[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3],[2,4],[5,6]]
=> [[1,2],[3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,4],[2,6],[3],[5]]
=> [[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5,7],[2,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5,7],[3,6]]
=> [[1,3,4,6,7],[2,5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5,7],[4,6]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5,7],[2],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5,7],[3],[6]]
=> [[1,3,4,6,7],[2],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5,7],[4],[6]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3,4,6]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,7],[2,5,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,3,5],[4,6,7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,4,5],[2,6,7]]
=> [[1,2,3,4],[5,6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5],[3,6,7]]
=> [[1,3,4,7],[2,5,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5],[4,6,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,4,5,7],[2,6],[3]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3,6],[4]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,4,7],[2,6],[5]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3,4],[6]]
=> [[1,3,4,6],[2,7],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,7],[2,5],[6]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,7],[3,5],[6]]
=> [[1,3,6,7],[2,4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,7],[4,5],[6]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5],[2,7],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5],[3,7],[6]]
=> [[1,3,4,7],[2,6],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5],[4,7],[6]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,6],[2,4],[7]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2,5],[7]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,4,5,7],[2],[3],[6]]
=> [[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3],[4],[6]]
=> [[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,4,7],[2],[5],[6]]
=> [[1,2,3,6],[4],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,7],[3],[5],[6]]
=> [[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [[1,2,4,5],[3],[6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,3,5],[4],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,5,7],[4]]
=> [[1,2,5],[3,4,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,5],[3,6,7],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,3],[4,5,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,6],[3,5],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5,7]]
=> [[1,2,5],[3,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,6],[3,5],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,5],[3,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5],[7]]
=> [[1,2,3],[4,5],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,2,5],[3],[4],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5],[7]]
=> [[1,2],[3,5],[4],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000454
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,4,6],[2,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4,6],[3,5]]
=> [[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4,6],[3],[5]]
=> [[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,3],[4,5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2,5],[3]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2,4],[5]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,6],[3,4],[5]]
=> [[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4],[3,6],[5]]
=> [[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2],[3],[5]]
=> [[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2],[3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3,6]]
=> [[1,3,4,6,7],[2,5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4,6]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3],[6]]
=> [[1,3,4,6,7],[2],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4],[6]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4,6]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,3,5],[4,6,7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,5],[2,6,7]]
=> [[1,2,3,4],[5,6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,6,7]]
=> [[1,3,4,7],[2,5,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,6,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2,6],[3]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,6],[4]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,7],[2,6],[5]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4],[6]]
=> [[1,3,4,6],[2,7],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5],[6]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,7],[3,5],[6]]
=> [[1,3,6,7],[2,4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,7],[4,5],[6]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,7],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,7],[6]]
=> [[1,3,4,7],[2,6],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,7],[6]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,4],[7]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5],[7]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2],[3],[6]]
=> [[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3],[4],[6]]
=> [[1,4,6,7],[2],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,7],[2],[5],[6]]
=> [[1,2,3,6],[4],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,7],[3],[5],[6]]
=> [[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,6],[2],[4],[7]]
=> [[1,2,4,5],[3],[6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,3,5],[4],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,5,7],[4]]
=> [[1,2,5],[3,4,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,5],[3,6,7],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,3],[4,5,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,6],[3,5],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5,7]]
=> [[1,2,5],[3,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,6],[3,5],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,5],[3,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5],[7]]
=> [[1,2,3],[4,5],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,2,5],[3],[4],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4],[5],[7]]
=> [[1,2],[3,5],[4],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
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: St000259
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,6],[3,4],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4],[3,6],[5]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2,6]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3,6]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2],[6]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3],[6]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5,6]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,6,7]]
=> [[1,2,5,6],[3,4,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,6,7]]
=> [[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,6,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2,6],[3]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,6],[4]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,7],[2,6],[5]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4],[6]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5],[6]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,7],[3,5],[6]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,7],[4,5],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,7],[6]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,7],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,7],[6]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,4],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2],[3],[6]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4],[3,5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,4,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,5,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4],[3,5],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5,7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5,7]]
=> [[1,2,5],[3,4],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,4],[3,7],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,7],[3,4],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5],[7]]
=> [[1,4,7],[2,5],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5],[7]]
=> [[1,4,5],[2,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,4,7],[2],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,4],[5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,4],[5,6],[7]]
=> [[1,4],[2,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4],[5],[7]]
=> [[1,4],[2,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000456
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,6],[3,4],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2,4],[3,6],[5]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3],[2,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5,7],[2,6]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5,7],[3,6]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5,7],[4,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5,7],[2],[6]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5,7],[3],[6]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5,7],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3,4,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,7],[2,5,6]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5],[2,6,7]]
=> [[1,2,5,6],[3,4,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5],[3,6,7]]
=> [[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5],[4,6,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,4,5,7],[2,6],[3]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3,6],[4]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,4,7],[2,6],[5]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,5,7],[3,4],[6]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,7],[2,5],[6]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,7],[3,5],[6]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,7],[4,5],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4,5],[2,7],[6]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,2,4,5],[3,7],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,2,3,5],[4,7],[6]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,5,6],[2,4],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,4,6],[2,5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,4,5,7],[2],[3],[6]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4],[3,5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,4,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4],[2,5,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4],[3,5],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5,7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5,7]]
=> [[1,2,5],[3,4],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,4],[3,7],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,7],[3,4],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5],[7]]
=> [[1,4,7],[2,5],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5],[7]]
=> [[1,4,5],[2,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,4,7],[2],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,4],[5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3],[2,4],[5,6],[7]]
=> [[1,4],[2,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5],[7]]
=> [[1,4],[2,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000771
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,6],[3,4],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4],[3,6],[5]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2,6]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3,6]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2],[6]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3],[6]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5,6]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,6,7]]
=> [[1,2,5,6],[3,4,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,6,7]]
=> [[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,6,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2,6],[3]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,6],[4]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,7],[2,6],[5]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4],[6]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5],[6]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,7],[3,5],[6]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,7],[4,5],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,7],[6]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,7],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,7],[6]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,4],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2],[3],[6]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4],[3,5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,4,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,5,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4],[3,5],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5,7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5,7]]
=> [[1,2,5],[3,4],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,4],[3,7],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,7],[3,4],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5],[7]]
=> [[1,4,7],[2,5],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5],[7]]
=> [[1,4,5],[2,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,4,7],[2],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,4],[5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,4],[5,6],[7]]
=> [[1,4],[2,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4],[5],[7]]
=> [[1,4],[2,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,6],[3,4],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2,4],[3,6],[5]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2,6]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3,6]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5,7],[2],[6]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5,7],[3],[6]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5,7],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5,6]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,6,7]]
=> [[1,2,5,6],[3,4,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,6,7]]
=> [[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,6,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2,6],[3]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,6],[4]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,7],[2,6],[5]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,5,7],[3,4],[6]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,7],[2,5],[6]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,7],[3,5],[6]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,7],[4,5],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4,5],[2,7],[6]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4,5],[3,7],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,2,3,5],[4,7],[6]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,5,6],[2,4],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,4,6],[2,5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,4,5,7],[2],[3],[6]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4],[3,5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,4,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,5,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4],[3,5],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5,7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5,7]]
=> [[1,2,5],[3,4],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,4],[3,7],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,7],[3,4],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5],[7]]
=> [[1,4,7],[2,5],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5],[7]]
=> [[1,4,5],[2,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,4,7],[2],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,4],[5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,4],[5,6],[7]]
=> [[1,4],[2,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[[1,3],[2,6],[4],[5],[7]]
=> [[1,4],[2,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000260
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4],[3,5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,6],[3,4],[5]]
=> [[1,2,3],[4,5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2,4],[3,6],[5]]
=> [[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3],[2,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4,5,7],[2,6]]
=> [[1,2,4,5,6],[3,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,4,5,7],[3,6]]
=> [[1,2,3,5,6],[4,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3,5,7],[4,6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4,5,7],[2],[6]]
=> [[1,2,4,5,6],[3],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,4,5,7],[3],[6]]
=> [[1,2,3,5,6],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3,5,7],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,5,7],[3,4,6]]
=> [[1,2,3,6],[4,5,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,7],[2,5,6]]
=> [[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,7],[3,5,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,6],[2,5,7]]
=> [[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4,5],[2,6,7]]
=> [[1,2,5,6],[3,4,7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,4,5],[3,6,7]]
=> [[1,3,5,6],[2,4,7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3,5],[4,6,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,6],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,4,5,7],[2,6],[3]]
=> [[1,2,5,6],[3,7],[4]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,5,7],[3,6],[4]]
=> [[1,2,3,6],[4,7],[5]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,4,7],[2,6],[5]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,5,7],[3,4],[6]]
=> [[1,2,3,6],[4,5],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,7],[2,5],[6]]
=> [[1,2,4,5],[3,6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,4,7],[3,5],[6]]
=> [[1,2,3,5],[4,6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3,7],[4,5],[6]]
=> [[1,2,3,4],[5,6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,5,6],[2,7],[4]]
=> [[1,2,6,7],[3,4],[5]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,6],[2,7],[5]]
=> [[1,2,5,7],[3,4],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4,5],[2,7],[6]]
=> [[1,2,5,6],[3,4],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,2,4,5],[3,7],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3,5],[4,7],[6]]
=> [[1,3,4,6],[2,5],[7]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,5,6],[2,4],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4,6],[2,5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,4,5,7],[2],[3],[6]]
=> [[1,2,5,6],[3],[4],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,4,6],[2],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4],[3,5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,4,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4],[2,5,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4],[3,5],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,6],[2,4],[5,7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4],[2,6],[5,7]]
=> [[1,2,5],[3,4],[6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,4],[3,7],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,7],[3,4],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,6],[2,4],[5],[7]]
=> [[1,4,7],[2,5],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,4],[2,6],[5],[7]]
=> [[1,4,5],[2,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3,6],[2],[4],[5],[7]]
=> [[1,4,7],[2],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3],[2,6],[4,7],[5]]
=> [[1,2],[3,4],[5,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3],[2,4],[5,6],[7]]
=> [[1,4],[2,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[[1,3],[2,6],[4],[5],[7]]
=> [[1,4],[2,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 209 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000467The hyper-Wiener index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000335The difference of lower and upper interactions. St000864The number of circled entries of the shifted recording tableau of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001517The length of a longest pair of twins in a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001391The disjunction number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001578The minimal number of edges to add or remove to make a graph a line graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001742The difference of the maximal and the minimal degree in a graph. St000060The greater neighbor of the maximum. St000299The number of nonisomorphic vertex-induced subtrees. St001117The game chromatic index of a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000501The size of the first part in the decomposition of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001345The Hamming dimension of a graph. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000325The width of the tree associated to a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000741The Colin de Verdière graph invariant. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001388The number of non-attacking neighbors of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001530The depth of a Dyck path. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001812The biclique partition number of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000015The number of peaks of a Dyck path. St000051The size of the left subtree of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000080The rank of the poset. St000091The descent variation of a composition. St000143The largest repeated part of a partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000309The number of vertices with even degree. St000314The number of left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000553The number of blocks of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000822The Hadwiger number of the graph. St000831The number of indices that are either descents or recoils. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001439The number of even weak deficiencies and of odd weak exceedences. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001638The book thickness of a graph. St001649The length of a longest trail in a graph. St001783The number of odd automorphisms of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000075The orbit size of a standard tableau under promotion. St000197The number of entries equal to positive one in the alternating sign matrix. St000338The number of pixed points of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000795The mad of a permutation. St000957The number of Bruhat lower covers of a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001516The number of cyclic bonds of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001806The upper middle entry of a permutation. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St000171The degree of the graph. St000226The convexity of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000471The sum of the ascent tops of a permutation. St000625The sum of the minimal distances to a greater element. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000890The number of nonzero entries in an alternating sign matrix. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001074The number of inversions of the cyclic embedding of a permutation. St001118The acyclic chromatic index of a graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001268The size of the largest ordinal summand in the poset. St001430The number of positive entries in a signed permutation. St001497The position of the largest weak excedence of a permutation. St001519The pinnacle sum of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001725The harmonious chromatic number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001883The mutual visibility number of a graph. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000167The number of leaves of an ordered tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001746The coalition number of a graph. St000656The number of cuts of a poset. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000301The number of facets of the stable set polytope of a graph. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.
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