- FindStat

Identifier

St001627: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>1
[1,1]=>1
[3]=>2
[2,1]=>3
[1,1,1]=>4
[4]=>6
[3,1]=>11
[2,2]=>16
[2,1,1]=>23
[1,1,1,1]=>38
[5]=>21
[4,1]=>58
[3,2]=>98
[3,1,1]=>162
[2,2,1]=>230
[2,1,1,1]=>402
[1,1,1,1,1]=>728
[6]=>112
[5,1]=>407
[4,2]=>879
[4,1,1]=>1549
[3,3]=>1087
[3,2,1]=>2812
[3,1,1,1]=>5204
[2,2,2]=>4065
[2,2,1,1]=>7490
[2,1,1,1,1]=>14080
[1,1,1,1,1,1]=>26704
                    

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 connected graphs such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of unlabelled connected graphs on $n$ vertices, oeis:A001349, whereas the value on the partition $(1^n)$ is the number of labelled connected graphs oeis:A001187.

Code

def statistic(mu):
    h = SymmetricFunctions(QQ).h()
    F = (species.SimpleGraphSpecies().cycle_index_series()-1).logarithm()
    return F.coefficient(mu.size()).scalar(h(mu))

Created

Oct 01, 2020 at 22:08 by Martin Rubey

Updated

Oct 01, 2020 at 22:08 by Martin Rubey