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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