edit this statistic or download as text // json
Identifier
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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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