Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St001738
(load all 18 compositions to match this statistic)
Mapping
St001738: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 2
([],2)
 => 2
([(0,1)],2)
 => 2
([],3)
 => 2
([(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => 2
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],5)
 => 2
([(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => 3
([(0,1),(2,4),(3,4)],5)
 => 3
([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
Description
The minimal order of a graph which is not an induced subgraph of the given graph. For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices. By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number [[St000097]].
Matching statistic: St000260
(load all 5 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 58%●distinct values known / distinct values provided: 50%
Values
([],1)
 => [1] => [1] => ([],1)
 => 0 = 2 - 2
([],2)
 => [2] => [1] => ([],1)
 => 0 = 2 - 2
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => ? = 2 - 2
([],3)
 => [3] => [1] => ([],1)
 => 0 = 2 - 2
([(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [3] => ([],3)
 => ? = 2 - 2
([],4)
 => [4] => [1] => ([],1)
 => 0 = 2 - 2
([(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(1,2)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => ? = 2 - 2
([],5)
 => [5] => [1] => ([],1)
 => 0 = 2 - 2
([(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 2 - 2
([],0)
 => ? => ? => ?
 => ? = 1 - 2
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St001545
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 49%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [1] => ([],1)
 => ? = 2 - 1
([],2)
 => [2] => [1] => ([],1)
 => ? = 2 - 1
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => ? = 2 - 1
([],3)
 => [3] => [1] => ([],1)
 => ? = 2 - 1
([(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [3] => ([],3)
 => ? = 2 - 1
([],4)
 => [4] => [1] => ([],1)
 => ? = 2 - 1
([(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => ? = 2 - 1
([],5)
 => [5] => [1] => ([],1)
 => ? = 2 - 1
([(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 2 - 1
([],0)
 => ? => ? => ?
 => ? = 1 - 1
Description
The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St000456
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 49%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [1] => ([],1)
 => ? = 2 - 2
([],2)
 => [2] => [1] => ([],1)
 => ? = 2 - 2
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => ? = 2 - 2
([],3)
 => [3] => [1] => ([],1)
 => ? = 2 - 2
([(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [3] => ([],3)
 => ? = 2 - 2
([],4)
 => [4] => [1] => ([],1)
 => ? = 2 - 2
([(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(1,2)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => ? = 2 - 2
([],5)
 => [5] => [1] => ([],1)
 => ? = 2 - 2
([(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 2 - 2
([],0)
 => ? => ? => ?
 => ? = 1 - 2
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: St000455
(load all 3 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
([],1)
 => [1] => [1] => ([],1)
 => ? = 2 - 3
([],2)
 => [2] => [2] => ([],2)
 => ? = 2 - 3
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([],3)
 => [3] => [3] => ([],3)
 => ? = 2 - 3
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 0 = 3 - 3
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 0 = 3 - 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([],4)
 => [4] => [4] => ([],4)
 => ? = 2 - 3
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0 = 3 - 3
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0 = 3 - 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0 = 3 - 3
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 3 - 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 2 - 3
([],5)
 => [5] => [5] => ([],5)
 => ? = 2 - 3
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 2 - 3
([],0)
 => ? => ? => ?
 => ? = 1 - 3
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St001875
(load all 4 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [[1],[]]
 => ([],1)
 => ? = 2
([],2)
 => [2] => [[2],[]]
 => ([],1)
 => ? = 2
([(0,1)],2)
 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? = 2
([],3)
 => [3] => [[3],[]]
 => ([],1)
 => ? = 2
([(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3
([(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? = 2
([],4)
 => [4] => [[4],[]]
 => ([],1)
 => ? = 2
([(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3
([(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? = 2
([],5)
 => [5] => [[5],[]]
 => ([],1)
 => ? = 2
([(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3
([(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3
([(1,4),(2,3)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? = 2
([],0)
 => ? => ?
 => ?
 => ? = 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001060
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [1] => ([],1)
 => ? = 2 - 1
([],2)
 => [2] => [1] => ([],1)
 => ? = 2 - 1
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => ? = 2 - 1
([],3)
 => [3] => [1] => ([],1)
 => ? = 2 - 1
([(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [3] => ([],3)
 => ? = 2 - 1
([],4)
 => [4] => [1] => ([],1)
 => ? = 2 - 1
([(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2] => ([],2)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => ? = 2 - 1
([],5)
 => [5] => [1] => ([],1)
 => ? = 2 - 1
([(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 2 - 1
([],0)
 => ? => ? => ?
 => ? = 1 - 1
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: St001630
(load all 4 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [[1],[]]
 => ([],1)
 => ? = 2 - 2
([],2)
 => [2] => [[2],[]]
 => ([],1)
 => ? = 2 - 2
([(0,1)],2)
 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],3)
 => [3] => [[3],[]]
 => ([],1)
 => ? = 2 - 2
([(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3 - 2
([(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],4)
 => [4] => [[4],[]]
 => ([],1)
 => ? = 2 - 2
([(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 2
([(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],5)
 => [5] => [[5],[]]
 => ([],1)
 => ? = 2 - 2
([(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(1,4),(2,3)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],0)
 => ? => ?
 => ?
 => ? = 1 - 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 4 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [[1],[]]
 => ([],1)
 => ? = 2 - 2
([],2)
 => [2] => [[2],[]]
 => ([],1)
 => ? = 2 - 2
([(0,1)],2)
 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],3)
 => [3] => [[3],[]]
 => ([],1)
 => ? = 2 - 2
([(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3 - 2
([(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],4)
 => [4] => [[4],[]]
 => ([],1)
 => ? = 2 - 2
([(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 2
([(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],5)
 => [5] => [[5],[]]
 => ([],1)
 => ? = 2 - 2
([(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 2
([(1,4),(2,3)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? = 2 - 2
([],0)
 => ? => ?
 => ?
 => ? = 1 - 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001876
(load all 4 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
([],1)
 => [1] => [[1],[]]
 => ([],1)
 => ? = 2 - 3
([],2)
 => [2] => [[2],[]]
 => ([],1)
 => ? = 2 - 3
([(0,1)],2)
 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? = 2 - 3
([],3)
 => [3] => [[3],[]]
 => ([],1)
 => ? = 2 - 3
([(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3 - 3
([(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? = 3 - 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? = 2 - 3
([],4)
 => [4] => [[4],[]]
 => ([],1)
 => ? = 2 - 3
([(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 3
([(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? = 3 - 3
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? = 2 - 3
([],5)
 => [5] => [[5],[]]
 => ([],1)
 => ? = 2 - 3
([(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 3
([(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? = 3 - 3
([(1,4),(2,3)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3 - 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 4 - 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? = 3 - 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0 = 3 - 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? = 3 - 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? = 2 - 3
([],0)
 => ? => ?
 => ?
 => ? = 1 - 3
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001877Number of indecomposable injective modules with projective dimension 2. St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.