- FindStat

Identifier

St001608: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>1
[1,1]=>2
[3]=>2
[2,1]=>5
[1,1,1]=>9
[4]=>4
[3,1]=>13
[2,2]=>18
[2,1,1]=>34
[1,1,1,1]=>64
[5]=>9
[4,1]=>35
[3,2]=>63
[3,1,1]=>119
[2,2,1]=>171
[2,1,1,1]=>326
[1,1,1,1,1]=>625
[6]=>20
[5,1]=>95
[4,2]=>209
[4,1,1]=>401
[3,3]=>268
[3,2,1]=>744
[3,1,1,1]=>1433
[2,2,2]=>1077
[2,2,1,1]=>2078
[2,1,1,1,1]=>4016
[1,1,1,1,1,1]=>7776
[7]=>48
[6,1]=>262
[5,2]=>683
[5,1,1]=>1316
[4,3]=>1065
[4,2,1]=>2993
[4,1,1,1]=>5799
[3,3,1]=>3868
[3,2,2]=>5637
[3,2,1,1]=>10937
[3,1,1,1,1]=>21256
[2,2,2,1]=>15955
[2,2,1,1,1]=>31022
[2,1,1,1,1,1]=>60387
[1,1,1,1,1,1,1]=>117649
[8]=>115
[7,1]=>727
[6,2]=>2189
[6,1,1]=>4247
[5,3]=>4022
[5,2,1]=>11417
[5,1,1,1]=>22224
[4,4]=>4890
[4,3,1]=>18048
[4,2,2]=>26399
[4,2,1,1]=>51463
[4,1,1,1,1]=>100407
[3,3,2]=>34316
[3,3,1,1]=>66920
[3,2,2,1]=>98005
[3,2,1,1,1]=>191361
[3,1,1,1,1,1]=>373895
[2,2,2,2]=>143568
[2,2,2,1,1]=>280440
[2,2,1,1,1,1]=>548128
[2,1,1,1,1,1,1]=>1071904
[1,1,1,1,1,1,1,1]=>2097152
[9]=>286
[8,1]=>2033
[7,2]=>6951
[7,1,1]=>13532
[6,3]=>14684
[6,2,1]=>41978
[6,1,1,1]=>81987
[5,4]=>20993
[5,3,1]=>78296
[5,2,2]=>114889
[5,2,1,1]=>224670
[5,1,1,1,1]=>439646
[4,4,1]=>95673
[4,3,2]=>183126
[4,3,1,1]=>358318
[4,2,2,1]=>526292
[4,2,1,1,1]=>1030671
[4,1,1,1,1,1]=>2019348
[3,3,3]=>238887
[3,3,2,1]=>686912
[3,3,1,1,1]=>1345583
[3,2,2,2]=>1009360
[3,2,2,1,1]=>1977724
[3,2,1,1,1,1]=>3876719
[3,1,1,1,1,1,1]=>7601777
[2,2,2,2,1]=>2907445
[2,2,2,1,1,1]=>5700489
[2,2,1,1,1,1,1]=>11180483
[2,1,1,1,1,1,1,1]=>21935132
[1,1,1,1,1,1,1,1,1]=>43046721
                    

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Description

The number of coloured rooted trees such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of unlabelled rooted trees on $n$ vertices, oeis:A000081, whereas the value on the partition $(1^n)$ is the number of labelled rooted trees oeis:A000169.

Code

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

Created

Sep 27, 2020 at 13:05 by Martin Rubey

Updated

Sep 27, 2020 at 13:05 by Martin Rubey