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Statistic identifier: St001607

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Collection: Integer partitions

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Description: The number of coloured 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 graphs on $n$ vertices, [[oeis:A000088]], whereas the value on the partition $(1^n)$ is the number of labelled graphs [[oeis:A006125]].

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References: 

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


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Statistic values:

[1]             => 1
[2]             => 2
[1,1]           => 2
[3]             => 4
[2,1]           => 6
[1,1,1]         => 8
[4]             => 11
[3,1]           => 20
[2,2]           => 28
[2,1,1]         => 40
[1,1,1,1]       => 64
[5]             => 34
[4,1]           => 90
[3,2]           => 148
[3,1,1]         => 240
[2,2,1]         => 336
[2,1,1,1]       => 576
[1,1,1,1,1]     => 1024
[6]             => 156
[5,1]           => 544
[4,2]           => 1144
[4,1,1]         => 1992
[3,3]           => 1408
[3,2,1]         => 3568
[3,1,1,1]       => 6528
[2,2,2]         => 5120
[2,2,1,1]       => 9344
[2,1,1,1,1]     => 17408
[1,1,1,1,1,1]   => 32768
[7]             => 1044
[6,1]           => 5096
[5,2]           => 13128
[5,1,1]         => 24416
[4,3]           => 20364
[4,2,1]         => 55472
[4,1,1,1]       => 105536
[3,3,1]         => 71552
[3,2,2]         => 104160
[3,2,1,1]       => 199040
[3,1,1,1,1]     => 382976
[2,2,2,1]       => 290304
[2,2,1,1,1]     => 559104
[2,1,1,1,1,1]   => 1081344
[1,1,1,1,1,1,1] => 2097152

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Created: Sep 27, 2020 at 13:19 by Martin Rubey

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Last Updated: Sep 27, 2020 at 13:19 by Martin Rubey