- FindStat

Identifier

Mp00160: Permutations —graph of inversions⟶ Graphs
St001738: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q^{2}$

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

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

$F_{4} = 2\ q^{2} + 22\ q^{3}$

$F_{5} = 2\ q^{2} + 88\ q^{3} + 30\ q^{4}$

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

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.