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Identifier

St000270: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

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

([],3) => 1

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

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

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

([],4) => 1

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

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

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

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

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

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

([(0,3),(1,2),(1,3),(2,3)],4) => 14

([(0,2),(0,3),(1,2),(1,3)],4) => 15

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 38

([],5) => 1

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

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

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

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

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

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

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

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

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

([(1,4),(2,3),(2,4),(3,4)],5) => 14

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 28

([(1,3),(1,4),(2,3),(2,4)],5) => 15

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 30

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 24

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 28

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 54

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 81

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

([(0,1),(2,3),(2,4),(3,4)],5) => 14

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 28

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 49

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 31

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 52

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 82

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 48

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 38

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 76

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 128

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 86

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 134

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 198

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 291

([],6) => 1

([(4,5)],6) => 2

([(3,5),(4,5)],6) => 4

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

([(1,5),(2,5),(3,5),(4,5)],6) => 16

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

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

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

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

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

([(1,5),(2,5),(3,4),(4,5)],6) => 16

([(0,1),(2,5),(3,5),(4,5)],6) => 16

([(2,5),(3,4),(3,5),(4,5)],6) => 14

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

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 28

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 56

([(2,4),(2,5),(3,4),(3,5)],6) => 15

([(0,5),(1,5),(2,4),(3,4)],6) => 16

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 30

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

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 24

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 28

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

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 60

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 56

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 54

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 60

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 108

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 81

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 162

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 189

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 270

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

([(1,5),(2,4),(3,4),(3,5)],6) => 16

([(0,1),(2,5),(3,4),(4,5)],6) => 16

([(1,2),(3,4),(3,5),(4,5)],6) => 14

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

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 28

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 28

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 56

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 49

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 98

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 31

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 60

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 52

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 62

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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 $T_G(2,1)$.

References

[1] wikipedia:Tree_(graph_theory)#forest
[2] wikipedia:Tutte polynomial#Individual_points

Code

def statistic(g):
    return g.tutte_polynomial().subs(x=2, y=1)

Created

Jul 28, 2015 at 09:17 by Martin Rubey

Updated

Jul 25, 2022 at 13:36 by Martin Rubey