*****************************************************************************
*       www.FindStat.org - The Combinatorial Statistic Finder               *
*                                                                           *
*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
*                                                                           *
*    This information is distributed in the hope that it will be useful,    *
*    but WITHOUT ANY WARRANTY; without even the implied warranty of         *
*    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.                   *
*****************************************************************************

-----------------------------------------------------------------------------
Statistic identifier: St001608

-----------------------------------------------------------------------------
Collection: Integer partitions

-----------------------------------------------------------------------------
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]].

-----------------------------------------------------------------------------
References: 

-----------------------------------------------------------------------------
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))


-----------------------------------------------------------------------------
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

-----------------------------------------------------------------------------
Created: Sep 27, 2020 at 13:05 by Martin Rubey

-----------------------------------------------------------------------------
Last Updated: Sep 27, 2020 at 13:05 by Martin Rubey