*****************************************************************************
*       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: St001609

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

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

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

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


-----------------------------------------------------------------------------
Statistic 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

-----------------------------------------------------------------------------
Created: Sep 27, 2020 at 12:59 by Martin Rubey

-----------------------------------------------------------------------------
Last Updated: Sep 27, 2020 at 12:59 by Martin Rubey