- FindStat

Identifier

St001610: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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