- FindStat

Identifier

St001609: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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
                    

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 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