- FindStat

Identifier

St001611: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>2
[1,1]=>2
[3]=>3
[2,1]=>4
[1,1,1]=>5
[4]=>5
[3,1]=>7
[2,2]=>9
[2,1,1]=>11
[1,1,1,1]=>15
[5]=>7
[4,1]=>12
[3,2]=>16
[3,1,1]=>21
[2,2,1]=>26
[2,1,1,1]=>36
[1,1,1,1,1]=>52
[6]=>11
[5,1]=>19
[4,2]=>29
[4,1,1]=>38
[3,3]=>31
[3,2,1]=>52
[3,1,1,1]=>74
[2,2,2]=>66
[2,2,1,1]=>92
[2,1,1,1,1]=>135
[1,1,1,1,1,1]=>203
[7]=>15
[6,1]=>30
[5,2]=>47
[5,1,1]=>64
[4,3]=>57
[4,2,1]=>98
[4,1,1,1]=>141
[3,3,1]=>109
[3,2,2]=>137
[3,2,1,1]=>198
[3,1,1,1,1]=>296
[2,2,2,1]=>249
[2,2,1,1,1]=>371
[2,1,1,1,1,1]=>566
[1,1,1,1,1,1,1]=>877
[8]=>22
[7,1]=>45
[6,2]=>77
[6,1,1]=>105
[5,3]=>97
[5,2,1]=>171
[5,1,1,1]=>250
[4,4]=>109
[4,3,1]=>212
[4,2,2]=>269
[4,2,1,1]=>392
[4,1,1,1,1]=>592
[3,3,2]=>300
[3,3,1,1]=>444
[3,2,2,1]=>560
[3,2,1,1,1]=>850
[3,1,1,1,1,1]=>1315
[2,2,2,2]=>712
[2,2,2,1,1]=>1075
[2,2,1,1,1,1]=>1663
[2,1,1,1,1,1,1]=>2610
[1,1,1,1,1,1,1,1]=>4140
[9]=>30
[8,1]=>67
[7,2]=>118
[7,1,1]=>165
[6,3]=>162
[6,2,1]=>289
[6,1,1,1]=>426
[5,4]=>189
[5,3,1]=>382
[5,2,2]=>484
[5,2,1,1]=>719
[5,1,1,1,1]=>1098
[4,4,1]=>424
[4,3,2]=>606
[4,3,1,1]=>907
[4,2,2,1]=>1150
[4,2,1,1,1]=>1763
[4,1,1,1,1,1]=>2752
[3,3,3]=>686
[3,3,2,1]=>1311
[3,3,1,1,1]=>2022
[3,2,2,2]=>1668
[3,2,2,1,1]=>2569
[3,2,1,1,1,1]=>4028
[3,1,1,1,1,1,1]=>6393
[2,2,2,2,1]=>3274
[2,2,2,1,1,1]=>5133
[2,2,1,1,1,1,1]=>8155
[2,1,1,1,1,1,1,1]=>13082
[1,1,1,1,1,1,1,1,1]=>21147
                    

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Description

The number of multiset partitions such that the multiplicities of elements are given by a partition.
In particular, the value on the partition $(n)$ is the number of integer partitions of $n$, oeis:A000041, whereas the value on the partition $(1^n)$ is the number of set partitions oeis:A006110.

Code

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

Created

Sep 27, 2020 at 13:28 by Martin Rubey

Updated

Sep 27, 2020 at 13:28 by Martin Rubey