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Identifier
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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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.
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))
Created
Sep 27, 2020 at 13:38 by Martin Rubey
Updated
Sep 27, 2020 at 13:38 by Martin Rubey