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Statistic identifier: St001608
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Collection: Integer partitions
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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]].
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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 = A.cycle_index_series()
return F.coefficient(mu.size()).scalar(h(mu))
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Statistic values:
[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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Created: Sep 27, 2020 at 13:05 by Martin Rubey
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Last Updated: Sep 27, 2020 at 13:05 by Martin Rubey