- FindStat

Identifier

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001738: Graphs ⟶ ℤ

Values

[(1,2)] => [2,1] => ([(0,1)],2) => ([(0,1)],2) => 2
[(1,2),(3,4)] => [2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => 3
[(1,3),(2,4)] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 2
[(1,4),(2,3)] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 2
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2)],4) => 3
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,1)],2) => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2)],4) => 3
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) => ([(0,3),(1,2)],4) => 3
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => ([(0,1),(0,2),(1,2)],3) => 2
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(6,7)],8) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,2)],3) => 2
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => ([(0,1)],2) => 2
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) => ([(0,3),(1,2)],4) => 3
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => ([(0,2),(0,3),(1,2),(1,3),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8) => ([(0,3),(1,2)],4) => 3
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => ([(0,2),(0,3),(1,2),(1,3),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6)],8) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 3
[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(6,7)],8) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8) => ([(0,1),(0,2),(1,2)],3) => 2
[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
[(1,2),(3,7),(4,8),(5,6)] => [2,1,6,7,8,5,3,4] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

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$F_{2} = q^{2}$

$F_{4} = 2\ q^{2} + q^{3}$

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 St000097The order of the largest clique of the graph..

Map

non-nesting-exceedence permutation

Description

The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.

Map

de-duplicate

Description

The de-duplicate of a graph.
Let

$G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods

$\{N_v | v \in V\}$ of

$G$ , and has an edge

$(N_a, N_b)$ between two vertices if and only if

$(a, b)$ is an edge of

$G$ . This is well-defined, because if

$N_a = N_c$ and

$N_b = N_d$ , then

$(a, b)\in E$ if and only if

$(c, d)\in E$ .
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.

Map

graph of inversions

Description

The graph of inversions of a permutation.
For a permutation of

$\{1,\dots,n\}$ , this is the graph with vertices

$\{1,\dots,n\}$ , where

$(i,j)$ is an edge if and only if it is an inversion of the permutation.