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Statistic identifier: St001610

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Collection: Integer partitions

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

In particular, the value on the partition $(n)$ is the number of endofunctions on $n$ vertices up to relabelling, [[oeis:A000088]], whereas the value on the partition $(1^n)$ is the number of endofunctions [[oeis:A000312]].

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References: 

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Code:
def statistic(mu):
    h = SymmetricFunctions(QQ).h()
    A = CombinatorialSpecies()
    X = species.SingletonSpecies()
    E = species.SetSpecies()
    A.define(X*E(A))
    F = species.PermutationSpecies()(A).cycle_index_series()
    return F.coefficient(mu.size()).scalar(h(mu))


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Statistic values:

[1]                 => 1
[2]                 => 3
[1,1]               => 4
[3]                 => 7
[2,1]               => 15
[1,1,1]             => 27
[4]                 => 19
[3,1]               => 52
[2,2]               => 76
[2,1,1]             => 136
[1,1,1,1]           => 256
[5]                 => 47
[4,1]               => 175
[3,2]               => 316
[3,1,1]             => 595
[2,2,1]             => 855
[2,1,1,1]           => 1630
[1,1,1,1,1]         => 3125
[6]                 => 130
[5,1]               => 571
[4,2]               => 1270
[4,1,1]             => 2406
[3,3]               => 1614
[3,2,1]             => 4465
[3,1,1,1]           => 8598
[2,2,2]             => 6489
[2,2,1,1]           => 12468
[2,1,1,1,1]         => 24096
[1,1,1,1,1,1]       => 46656
[7]                 => 343
[6,1]               => 1838
[5,2]               => 4790
[5,1,1]             => 9216
[4,3]               => 7464
[4,2,1]             => 20955
[4,1,1,1]           => 40593
[3,3,1]             => 27084
[3,2,2]             => 39467
[3,2,1,1]           => 76563
[3,1,1,1,1]         => 148792
[2,2,2,1]           => 111685
[2,2,1,1,1]         => 217154
[2,1,1,1,1,1]       => 422709
[1,1,1,1,1,1,1]     => 823543
[8]                 => 951
[7,1]               => 5834
[6,2]               => 17590
[6,1,1]             => 34003
[5,3]               => 32213
[5,2,1]             => 91369
[5,1,1,1]           => 177819
[4,4]               => 39230
[4,3,1]             => 144428
[4,2,2]             => 211360
[4,2,1,1]           => 411731
[4,1,1,1,1]         => 803256
[3,3,2]             => 274578
[3,3,1,1]           => 535414
[3,2,2,1]           => 784072
[3,2,1,1,1]         => 1530915
[3,1,1,1,1,1]       => 2991160
[2,2,2,2]           => 1148800
[2,2,2,1,1]         => 2243520
[2,2,1,1,1,1]       => 4385024
[2,1,1,1,1,1,1]     => 8575232
[1,1,1,1,1,1,1,1]   => 16777216
[9]                 => 2615
[8,1]               => 18363
[7,2]               => 62680
[7,1,1]             => 121936
[6,3]               => 132317
[6,2,1]             => 378003
[6,1,1,1]           => 738139
[5,4]               => 189116
[5,3,1]             => 704927
[5,2,2]             => 1034264
[5,2,1,1]           => 2022314
[5,1,1,1,1]         => 3957070
[4,4,1]             => 861345
[4,3,2]             => 1648443
[4,3,1,1]           => 3225262
[4,2,2,1]           => 4736908
[4,2,1,1,1]         => 9276295
[4,1,1,1,1,1]       => 18174132
[3,3,3]             => 2150352
[3,3,2,1]           => 6182602
[3,3,1,1,1]         => 12110759
[3,2,2,2]           => 9084495
[3,2,2,1,1]         => 17799796
[3,2,1,1,1,1]       => 34890727
[3,1,1,1,1,1,1]     => 68415993
[2,2,2,2,1]         => 26167005
[2,2,2,1,1,1]       => 51304401
[2,2,1,1,1,1,1]     => 100624347
[2,1,1,1,1,1,1,1]   => 197416188
[1,1,1,1,1,1,1,1,1] => 387420489

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Created: Sep 27, 2020 at 13:38 by Martin Rubey

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Last Updated: Sep 27, 2020 at 13:38 by Martin Rubey