- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 127 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

Description

The hyper-Wiener index of a connected graph.
This is

$\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.$

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

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

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.