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Identifier
Values
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 2
[1,1,1] => 3
[4] => 2
[3,1] => 4
[2,2] => 6
[2,1,1] => 9
[1,1,1,1] => 16
[5] => 3
[4,1] => 9
[3,2] => 15
[3,1,1] => 26
[2,2,1] => 37
[2,1,1,1] => 67
[1,1,1,1,1] => 125
[6] => 6
[5,1] => 20
[4,2] => 43
[4,1,1] => 75
[3,3] => 51
[3,2,1] => 134
[3,1,1,1] => 251
[2,2,2] => 195
[2,2,1,1] => 359
[2,1,1,1,1] => 680
[1,1,1,1,1,1] => 1296
[7] => 11
[6,1] => 48
[5,2] => 116
[5,1,1] => 214
[4,3] => 175
[4,2,1] => 469
[4,1,1,1] => 888
[3,3,1] => 596
[3,2,2] => 861
[3,2,1,1] => 1636
[3,1,1,1,1] => 3135
[2,2,2,1] => 2365
[2,2,1,1,1] => 4530
[2,1,1,1,1,1] => 8716
[1,1,1,1,1,1,1] => 16807
[8] => 23
[7,1] => 115
[6,2] => 329
[6,1,1] => 612
[5,3] => 573
[5,2,1] => 1577
[5,1,1,1] => 3023
[4,4] => 698
[4,3,1] => 2445
[4,2,2] => 3559
[4,2,1,1] => 6817
[4,1,1,1,1] => 13155
[3,3,2] => 4562
[3,3,1,1] => 8786
[3,2,2,1] => 12765
[3,2,1,1,1] => 24674
[3,1,1,1,1,1] => 47787
[2,2,2,2] => 18584
[2,2,2,1,1] => 35892
[2,2,1,1,1,1] => 69552
[2,1,1,1,1,1,1] => 134960
[1,1,1,1,1,1,1,1] => 262144
[9] => 47
[8,1] => 286
[7,2] => 918
[7,1,1] => 1747
[6,3] => 1866
[6,2,1] => 5204
[6,1,1,1] => 10038
[5,4] => 2626
[5,3,1] => 9480
[5,2,2] => 13820
[5,2,1,1] => 26736
[5,1,1,1,1] => 51873
[4,4,1] => 11513
[4,3,2] => 21715
[4,3,1,1] => 42080
[4,2,2,1] => 61417
[4,2,1,1,1] => 119325
[4,1,1,1,1,1] => 232154
[3,3,3] => 28110
[3,3,2,1] => 79629
[3,3,1,1,1] => 154833
[3,2,2,2] => 116314
[3,2,2,1,1] => 226225
[3,2,1,1,1,1] => 440542
[3,1,1,1,1,1,1] => 858578
[2,2,2,2,1] => 330685
[2,2,2,1,1,1] => 644190
[2,2,1,1,1,1,1] => 1255973
[2,1,1,1,1,1,1,1] => 2450309
[1,1,1,1,1,1,1,1,1] => 4782969
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Description
The number of coloured 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 trees on $n$ vertices, oeis:A000055, whereas the value on the partition $(1^n)$ is the number of labelled trees oeis:A000272.
Code
def statistic(mu):
    h = SymmetricFunctions(QQ).h()
    A = CombinatorialSpecies()
    X = species.SingletonSpecies()
    E = species.SetSpecies()
    A.define(X*E(A))
    V = (X + species.CharacteristicSpecies(2)).cycle_index_series() - (X^2).cycle_index_series()
    F = V(A.cycle_index_series())
    return F.coefficient(mu.size()).scalar(h(mu))
Created
Sep 27, 2020 at 12:59 by Martin Rubey
Updated
Sep 27, 2020 at 12:59 by Martin Rubey