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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: 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
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000535
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: 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
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
Description
The rank-width of a graph.
Matching statistic: St001271
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: 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
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St001333
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: 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
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St001340
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001340: 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
St001340: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
Description
The cardinality of a minimal non-edge isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with two isolated vertices.
Matching statistic: St001393
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001393: 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
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2,2] => [3] => [1] => ([],1)
=> 0
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St000258
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000258: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000258: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2,2] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,3] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
Description
The burning number of a graph.
This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.
Matching statistic: St000918
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000918: 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
St000918: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2,2] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,3] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
Description
The 2-limited packing number of a graph.
A subset $B$ of the set of vertices of a graph is a $k$-limited packing set if its intersection with the (closed) neighbourhood of any vertex is at most $k$. The $k$-limited packing number is the largest number of vertices in a $k$-limited packing set.
Matching statistic: St001261
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: 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
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2,2] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,3] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St001185
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,1] => [2] => [1] => [1,0]
=> 0
[2] => [1] => [1] => [1,0]
=> 0
[1,1,1] => [3] => [1] => [1,0]
=> 0
[3] => [1] => [1] => [1,0]
=> 0
[1,1,1,1] => [4] => [1] => [1,0]
=> 0
[1,1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[2,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 1
[2,2] => [2] => [1] => [1,0]
=> 0
[4] => [1] => [1] => [1,0]
=> 0
[1,1,1,1,1] => [5] => [1] => [1,0]
=> 0
[1,1,1,2] => [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,3] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 1
[2,2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[3,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 1
[5] => [1] => [1] => [1,0]
=> 0
[1,1,1,1,1,1] => [6] => [1] => [1,0]
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,4] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[2,2,2] => [3] => [1] => [1,0]
=> 0
[3,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
=> 1
[3,3] => [2] => [1] => [1,0]
=> 0
[4,1,1] => [1,2] => [1,1] => [1,0,1,0]
=> 1
[6] => [1] => [1] => [1,0]
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => [1,0]
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,3] => [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,2,2,1] => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,5] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,2,2,2] => [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,3,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,3,3] => [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,2,1,2,1,2,1,1] => [2,1,1,1,1,1,2] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,1,2,1,2,2,1] => [1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,1,2,1,2,1,1,2] => [1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,1,2,1,1,2,1,1] => [1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,1,1,2,1,2,1,1] => [1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,2,1,2,1,2,1,1,1] => [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,3,1,2,1,2,1,1,1] => [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,2,1,2,1,3,1,1,1] => [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,2,1,3,1,2,1,1,1] => [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,2,1,1,2,1,2,2,1] => [1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000397The Strahler number of a rooted tree. St000442The maximal area to the right of an up step of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St000013The height of a Dyck path. St000480The number of lower covers of a partition in dominance order. St001280The number of parts of an integer partition that are at least two. St000920The logarithmic height of a Dyck path. St000769The major index of a composition regarded as a word. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001335The cardinality of a minimal cycle-isolating set of a graph. St001673The degree of asymmetry of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000379The number of Hamiltonian cycles in a graph. St000662The staircase size of the code of a permutation. St000451The length of the longest pattern of the form k 1 2. St001741The largest integer such that all patterns of this size are contained in the permutation. St000264The girth of a graph, which is not a tree. St000141The maximum drop size of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000742The number of big ascents of a permutation after prepending zero. St000864The number of circled entries of the shifted recording tableau of a permutation. St001737The number of descents of type 2 in a permutation. St000542The number of left-to-right-minima of a permutation. St001734The lettericity of a graph. St000259The diameter of a connected graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001512The minimum rank of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001093The detour number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000402Half the size of the symmetry class of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000862The number of parts of the shifted shape of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001330The hat guessing number of a graph. St001470The cyclic holeyness of a permutation. St001645The pebbling number of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St000891The number of distinct diagonal sums of a permutation matrix.
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