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Your data matches 87 different statistics following compositions of up to 3 maps.
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Matching statistic: St000535
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ 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%
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)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
Description
The rank-width of a graph.
Matching statistic: St001192
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001333
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ 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%
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)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 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: St001393
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ 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%
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)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 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: St000920
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001239
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001261
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ 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%
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 = 0 + 1
[1,2] => [2] => ([],2)
=> 1 = 0 + 1
[2,1] => [2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [3] => ([],3)
=> 1 = 0 + 1
[2,3,1] => [3] => ([],3)
=> 1 = 0 + 1
[3,1,2] => [3] => ([],3)
=> 1 = 0 + 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,3,4,1] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [4] => ([],4)
=> 1 = 0 + 1
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [4] => ([],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [4] => ([],4)
=> 1 = 0 + 1
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St001734
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001734: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001734: 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] => [2] => ([],2)
=> 1 = 0 + 1
[2,1] => [2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [3] => ([],3)
=> 1 = 0 + 1
[2,3,1] => [3] => ([],3)
=> 1 = 0 + 1
[3,1,2] => [3] => ([],3)
=> 1 = 0 + 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,3,4,1] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [4] => ([],4)
=> 1 = 0 + 1
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [4] => ([],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [4] => ([],4)
=> 1 = 0 + 1
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
Description
The lettericity of a graph.
Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$.
The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.
Matching statistic: St000259
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2] => ([],2)
=> ([],1)
=> 0
[2,1] => [2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => [3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [3] => ([],3)
=> ([],1)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3,4] => [4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => [4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [4] => ([],4)
=> ([],1)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1,2,4] => [4] => ([],4)
=> ([],1)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,4,1,2] => [4] => ([],4)
=> ([],1)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => [4] => ([],4)
=> ([],1)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2] => ([],2)
=> ([],1)
=> 0
[2,1] => [2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => [3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [3] => ([],3)
=> ([],1)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3,4] => [4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,3,1,4] => [4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [4] => ([],4)
=> ([],1)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1,2,4] => [4] => ([],4)
=> ([],1)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,4,1,2] => [4] => ([],4)
=> ([],1)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,1,2,3] => [4] => ([],4)
=> ([],1)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 77 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000480The number of lower covers of a partition in dominance order. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001340The cardinality of a minimal non-edge isolating set of a graph. 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. St001638The book thickness of a graph. St000058The order of a permutation. St000258The burning number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000396The register function (or Horton-Strahler number) of a binary tree. 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. St000862The number of parts of the shifted shape of a permutation. St000918The 2-limited packing number of a graph. 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. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001962The proper pathwidth of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001060The distinguishing index of a graph. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000640The rank of the largest boolean interval in a poset. St000307The number of rowmotion orbits of a poset. St000454The largest eigenvalue of a graph if it is integral. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
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