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Statistic identifier: St001627

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Collection: Integer partitions

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Description: The number of coloured connected graphs such that the multiplicities of colours are given by a partition.

In particular, the value on the partition $(n)$ is the number of unlabelled connected graphs on $n$ vertices, [[oeis:A001349]], whereas the value on the partition $(1^n)$ is the number of labelled connected graphs [[oeis:A001187]].

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References: 

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Code:
def statistic(mu):
    h = SymmetricFunctions(QQ).h()
    F = (species.SimpleGraphSpecies().cycle_index_series()-1).logarithm()
    return F.coefficient(mu.size()).scalar(h(mu))


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Statistic values:

[1]           => 1
[2]           => 1
[1,1]         => 1
[3]           => 2
[2,1]         => 3
[1,1,1]       => 4
[4]           => 6
[3,1]         => 11
[2,2]         => 16
[2,1,1]       => 23
[1,1,1,1]     => 38
[5]           => 21
[4,1]         => 58
[3,2]         => 98
[3,1,1]       => 162
[2,2,1]       => 230
[2,1,1,1]     => 402
[1,1,1,1,1]   => 728
[6]           => 112
[5,1]         => 407
[4,2]         => 879
[4,1,1]       => 1549
[3,3]         => 1087
[3,2,1]       => 2812
[3,1,1,1]     => 5204
[2,2,2]       => 4065
[2,2,1,1]     => 7490
[2,1,1,1,1]   => 14080
[1,1,1,1,1,1] => 26704

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Created: Oct 01, 2020 at 22:08 by Martin Rubey

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Last Updated: Oct 01, 2020 at 22:08 by Martin Rubey