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Identifier

St000972: 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) => 5

([],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) => 5

([(0,3),(1,2),(1,3),(2,3)],4) => 10

([(0,2),(0,3),(1,2),(1,3)],4) => 12

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 13

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

([],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) => 5

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

([(1,4),(2,3),(2,4),(3,4)],5) => 10

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 20

([(1,3),(1,4),(2,3),(2,4)],5) => 12

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 24

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 13

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 20

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 26

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 34

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 35

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

([(0,1),(2,3),(2,4),(3,4)],5) => 10

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 20

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 25

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 27

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

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 34

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 26

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

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

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 40

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 38

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 43

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 47

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

([],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) => 5

([(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) => 10

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

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 40

([(2,4),(2,5),(3,4),(3,5)],6) => 12

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 24

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

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 20

([(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) => 48

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 26

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 40

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 52

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 34

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 48

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 68

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 35

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 52

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 70

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

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 97

([(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) => 10

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

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 20

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 40

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 25

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 50

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 27

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 48

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

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 54

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Description

The composition number of a graph.
This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.

References

[1] Knopfmacher, A., Mays, M. E. Graph compositions. I. Basic enumeration MathSciNet:1835361

Code

def statistic(G):
    lC = G.connected_components_subgraphs()
    res = 1
    for H in lC:
        lB, _ = H.blocks_and_cut_vertices()
        res *= prod(statistic_connected(H.subgraph(B, immutable=True)) for B in lB)
    return res

@cached_function
def statistic_connected(G):
    return len([S for S in SetPartitions(G.vertices())
                if all(G.subgraph(b).is_connected() for b in S)])

Created

Sep 04, 2017 at 09:34 by Martin Rubey

Updated

Sep 04, 2017 at 09:34 by Martin Rubey