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Statistic identifier: St001611
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Collection: Integer partitions
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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]].
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References:
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Code:
def statistic(mu):
h = SymmetricFunctions(QQ).h()
F = species.PartitionSpecies().cycle_index_series()
return F.coefficient(mu.size()).scalar(h(mu))
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Statistic values:
[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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Created: Sep 27, 2020 at 13:28 by Martin Rubey
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Last Updated: Sep 27, 2020 at 13:28 by Martin Rubey