- FindStat

Identifier

St001607: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>2
[1,1]=>2
[3]=>4
[2,1]=>6
[1,1,1]=>8
[4]=>11
[3,1]=>20
[2,2]=>28
[2,1,1]=>40
[1,1,1,1]=>64
[5]=>34
[4,1]=>90
[3,2]=>148
[3,1,1]=>240
[2,2,1]=>336
[2,1,1,1]=>576
[1,1,1,1,1]=>1024
[6]=>156
[5,1]=>544
[4,2]=>1144
[4,1,1]=>1992
[3,3]=>1408
[3,2,1]=>3568
[3,1,1,1]=>6528
[2,2,2]=>5120
[2,2,1,1]=>9344
[2,1,1,1,1]=>17408
[1,1,1,1,1,1]=>32768
[7]=>1044
[6,1]=>5096
[5,2]=>13128
[5,1,1]=>24416
[4,3]=>20364
[4,2,1]=>55472
[4,1,1,1]=>105536
[3,3,1]=>71552
[3,2,2]=>104160
[3,2,1,1]=>199040
[3,1,1,1,1]=>382976
[2,2,2,1]=>290304
[2,2,1,1,1]=>559104
[2,1,1,1,1,1]=>1081344
[1,1,1,1,1,1,1]=>2097152
                    

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Description

The number of coloured 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 graphs on $n$ vertices, oeis:A000088, whereas the value on the partition $(1^n)$ is the number of labelled graphs oeis:A006125.

Code

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

Created

Sep 27, 2020 at 13:19 by Martin Rubey

Updated

Sep 27, 2020 at 13:19 by Martin Rubey