- FindStat

Identifier

Mp00011: Binary trees —to graph⟶ Graphs
St000270: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[.,[.,[.,[.,.]]]] => ([(0,3),(1,2),(2,3)],4) => 8

[.,[.,[[.,.],.]]] => ([(0,3),(1,2),(2,3)],4) => 8

[.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => 8

[.,[[.,[.,.]],.]] => ([(0,3),(1,2),(2,3)],4) => 8

[.,[[[.,.],.],.]] => ([(0,3),(1,2),(2,3)],4) => 8

[[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 8

[[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 8

[[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 8

[[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 8

[[.,[.,[.,.]]],.] => ([(0,3),(1,2),(2,3)],4) => 8

[[.,[[.,.],.]],.] => ([(0,3),(1,2),(2,3)],4) => 8

[[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 8

[[[.,[.,.]],.],.] => ([(0,3),(1,2),(2,3)],4) => 8

[[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => 8

[.,[.,[.,[.,[.,.]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[.,[.,[[.,.],.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[.,[.,[[.,[.,.]],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[.,[[[.,.],.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[.,[[.,.],[[.,.],.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[.,[[[.,.],.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[.,[[.,[.,[.,.]]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[[.,[[.,.],.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[.,[[[.,[.,.]],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[[[[.,.],.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[.,.],[[.,[.,.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,.],[[[.,.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[.,.]],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[.,.]],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[.,.],.],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[[.,.],.]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[[.,[.,.]],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[[.,.],.],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[.,[.,[.,.]]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[.,[[.,.],.]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[.,[[.,[.,.]],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[.,[[[.,.],.],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[[.,[.,.]],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[[.,[.,[.,.]]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[.,[[.,.],.]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 16

[[[[.,[.,.]],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[[[[[.,.],.],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16

[.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[.,[[.,.],[.,.]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[[.,.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[.,[[.,.],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[.,[[.,[.,.]],[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[.,[[[.,.],.],[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[[[.,.],[.,.]],.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[[.,.],[.,[.,[.,.]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,.],[.,[[.,.],.]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 32

[.,[[.,.],[[.,[.,.]],.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,.],[[[.,.],.],.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[.,[.,[.,.]]],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,[[.,.],.]],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 32

[.,[[[.,[.,.]],.],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[[[.,.],.],.],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[[.,[[.,.],[.,.]]],.]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32

[.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32

[.,[[[.,.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[[.,.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[[.,[.,.]],[.,.]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

[.,[[[[.,.],.],[.,.]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32

>>> Load all 625 entries. <<<

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$F_{1} = q$

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

$F_{3} = 5\ q^{4}$

$F_{4} = 14\ q^{8}$

$F_{5} = 42\ q^{16}$

$F_{6} = 132\ q^{32}$

$F_{7} = 429\ q^{64}$

Description

The number of forests contained in a graph.
That is, for a graph

$G = (V,E)$ with vertices

$V$ and edges

$E$ , the number of subsets

$E' \subseteq E$ for which the subgraph

$(V,E')$ is acyclic.
If

$T_G(x,y)$ is the Tutte polynomial [2] of

$G$ , then the number of forests contained in

$G$ is given by

$T_G(2,1)$ .

Map

to graph

Description

Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.