- FindStat

Your data matches 96 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001517
(load all 662 compositions to match this statistic)

Mapping

St001517: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of a longest pair of twins in a permutation. A pair of twins in a permutation is a pair of two disjoint subsequences which are order isomorphic.

Matching statistic: St001667
(load all 662 compositions to match this statistic)

Mapping

St001667: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal size of a pair of weak twins for a permutation. A pair of weak twins in a permutation is a pair of two disjoint subsequences of the same length with the same descent pattern. More formally, a pair of weak twins of size

$k$ for a permutation

$\pi$ of length

$n$ are two disjoint lists

$1 \leq i_1 < \dots < i_k \leq n$ and

$1 \leq j_1 < \dots < j_k \leq n$ such that

$\pi(i_a) < \pi(i_{a+1})$ if and only if

$\pi(j_a) < \pi(j_{a+1})$ for all

$1 \leq a < k$ .

Matching statistic: St000260
(load all 6 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000273
(load all 5 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St000362
(load all 4 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

Matching statistic: St000387
(load all 5 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000387: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The matching number of a graph. For a graph

$G$ , this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in

$G$ .

Matching statistic: St000916
(load all 3 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The packing number of a graph. This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint closed neighbourhoods, or, equivalently, have distance greater than two.

Matching statistic: St000985
(load all 4 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of positive eigenvalues of the adjacency matrix of the graph.

Matching statistic: St001322
(load all 5 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a minimal independent dominating set in a graph.

Matching statistic: St001339
(load all 5 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The irredundance number of a graph. A set

$S$ of vertices is irredundant, if there is no vertex in

$S$ , whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of

$S$ . The irredundance number is the smallest size of a maximal irredundant set.

The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001812The biclique partition number of a graph. St001829The common independence number of a graph. St000172The Grundy number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001116The game chromatic number of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000271The chromatic index of a graph. St000383The last part of an integer composition. St000535The rank-width of a graph. St000547The number of even non-empty partial sums of an integer partition. St000657The smallest part of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000897The number of different multiplicities of parts of an integer partition. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001316The domatic number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001569The maximal modular displacement of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001642The Prague dimension of a graph. St001716The 1-improper chromatic number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001928The number of non-overlapping descents in a permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000711The number of big exceedences of a permutation. St000918The 2-limited packing number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001271The competition number of a graph. St001350Half of the Albertson index of a graph. St001424The number of distinct squares in a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001742The difference of the maximal and the minimal degree in a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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

$eA$ . St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St000258The burning number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001568The smallest positive integer that does not appear twice in the partition. St000454The largest eigenvalue of a graph if it is integral. St001871The number of triconnected components of a graph. St000259The diameter of a connected graph. St000741The Colin de Verdière graph invariant. St001743The discrepancy of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001366The maximal multiplicity of a degree of a vertex of a graph.