- FindStat

Identifier

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St001250: Integer partitions ⟶ ℤ

Values

([(0,1)],2) => ([(0,1)],2) => [1] => 1

([(1,2)],3) => ([(1,2)],3) => [1] => 1

([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => [1,1] => 2

([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => [3] => 0

([(2,3)],4) => ([(2,3)],4) => [1] => 1

([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => [1,1] => 2

([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => [1,1,1] => 3

([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => [1,1] => 2

([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => [1,1,1] => 3

([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => [3] => 0

([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => 1

([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => 0

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => 0

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => 0

([(3,4)],5) => ([(3,4)],5) => [1] => 1

([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => [1,1] => 2

([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => [1,1,1] => 3

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => 4

([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => [1,1] => 2

([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => [1,1,1] => 3

([(0,1),(2,4),(3,4)],5) => ([(0,1),(2,4),(3,4)],5) => [1,1,1] => 3

([(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => [3] => 0

([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => 4

([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => 1

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => 2

([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => 0

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => 2

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => 0

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => 2

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => 2

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => 1

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => 1

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => 4

([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => 1

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => 2

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => 0

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => 1

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6,1] => 1

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => 0

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6,1] => 1

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 1

([(4,5)],6) => ([(4,5)],6) => [1] => 1

([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => [1,1] => 2

([(2,5),(3,5),(4,5)],6) => ([(2,5),(3,5),(4,5)],6) => [1,1,1] => 3

([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => 4

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => 5

([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => [1,1] => 2

([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => [1,1,1] => 3

([(1,2),(3,5),(4,5)],6) => ([(1,2),(3,5),(4,5)],6) => [1,1,1] => 3

([(3,4),(3,5),(4,5)],6) => ([(3,4),(3,5),(4,5)],6) => [3] => 0

([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => 4

([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => 4

([(2,5),(3,4),(3,5),(4,5)],6) => ([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => 1

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => 5

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => 2

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => 3

([(2,4),(2,5),(3,4),(3,5)],6) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 0

([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => 4

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 2

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => 5

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 0

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => 2

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => 5

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => 3

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 2

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => 3

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => 3

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 1

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7,1] => 2

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 1

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7,1] => 2

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => 0

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => 0

([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => [1,1,1] => 3

([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => 4

([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => 4

([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => 1

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => 5

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => 2

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => 2

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => 3

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => 0

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => 1

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => 1

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => 3

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 1

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => 2

([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => 1

([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => 3

([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [7,1] => 2

([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => 3

([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 1

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [7,1] => 2

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search for individual values

searching the database for the individual values of this statistic

Description

The number of parts of a partition that are not congruent 0 modulo 3.

Map

Ore closure

Description

The Ore closure of a graph.
The Ore closure of a connected graph $G$ has the same vertices as $G$, and the smallest set of edges containing the edges of $G$ such that for any two vertices $u$ and $v$ whose sum of degrees is at least the number of vertices, then $(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.

Map

to edge-partition of biconnected components

Description

Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.