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Statistic identifier: St000321

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Collection: Integer partitions

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Description: The number of integer partitions of n that are dominated by an integer partition.

A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.

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References: 

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Code:
def statistic(L):
    return len(L.dominated_partitions())

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Statistic values:

[]                    => 1
[1]                   => 1
[2]                   => 2
[1,1]                 => 1
[3]                   => 3
[2,1]                 => 2
[1,1,1]               => 1
[4]                   => 5
[3,1]                 => 4
[2,2]                 => 3
[2,1,1]               => 2
[1,1,1,1]             => 1
[5]                   => 7
[4,1]                 => 6
[3,2]                 => 5
[3,1,1]               => 4
[2,2,1]               => 3
[2,1,1,1]             => 2
[1,1,1,1,1]           => 1
[6]                   => 11
[5,1]                 => 10
[4,2]                 => 9
[4,1,1]               => 7
[3,3]                 => 7
[3,2,1]               => 6
[3,1,1,1]             => 4
[2,2,2]               => 4
[2,2,1,1]             => 3
[2,1,1,1,1]           => 2
[1,1,1,1,1,1]         => 1
[7]                   => 15
[6,1]                 => 14
[5,2]                 => 13
[5,1,1]               => 11
[4,3]                 => 11
[4,2,1]               => 10
[4,1,1,1]             => 7
[3,3,1]               => 8
[3,2,2]               => 7
[3,2,1,1]             => 6
[3,1,1,1,1]           => 4
[2,2,2,1]             => 4
[2,2,1,1,1]           => 3
[2,1,1,1,1,1]         => 2
[1,1,1,1,1,1,1]       => 1
[8]                   => 22
[7,1]                 => 21
[6,2]                 => 20
[6,1,1]               => 17
[5,3]                 => 18
[5,2,1]               => 16
[5,1,1,1]             => 12
[4,4]                 => 15
[4,3,1]               => 14
[4,2,2]               => 13
[4,2,1,1]             => 11
[4,1,1,1,1]           => 7
[3,3,2]               => 10
[3,3,1,1]             => 9
[3,2,2,1]             => 8
[3,2,1,1,1]           => 6
[3,1,1,1,1,1]         => 4
[2,2,2,2]             => 5
[2,2,2,1,1]           => 4
[2,2,1,1,1,1]         => 3
[2,1,1,1,1,1,1]       => 2
[1,1,1,1,1,1,1,1]     => 1
[9]                   => 30
[8,1]                 => 29
[7,2]                 => 28
[7,1,1]               => 25
[6,3]                 => 26
[6,2,1]               => 24
[6,1,1,1]             => 18
[5,4]                 => 23
[5,3,1]               => 22
[5,2,2]               => 20
[5,2,1,1]             => 17
[5,1,1,1,1]           => 12
[4,4,1]               => 18
[4,3,2]               => 17
[4,3,1,1]             => 15
[4,2,2,1]             => 14
[4,2,1,1,1]           => 11
[4,1,1,1,1,1]         => 7
[3,3,3]               => 12
[3,3,2,1]             => 11
[3,3,1,1,1]           => 9
[3,2,2,2]             => 9
[3,2,2,1,1]           => 8
[3,2,1,1,1,1]         => 6
[3,1,1,1,1,1,1]       => 4
[2,2,2,2,1]           => 5
[2,2,2,1,1,1]         => 4
[2,2,1,1,1,1,1]       => 3
[2,1,1,1,1,1,1,1]     => 2
[1,1,1,1,1,1,1,1,1]   => 1
[10]                  => 42
[9,1]                 => 41
[8,2]                 => 40
[8,1,1]               => 36
[7,3]                 => 38
[7,2,1]               => 35
[7,1,1,1]             => 28
[6,4]                 => 35
[6,3,1]               => 33
[6,2,2]               => 31
[6,2,1,1]             => 27
[6,1,1,1,1]           => 19
[5,5]                 => 30
[5,4,1]               => 29
[5,3,2]               => 28
[5,3,1,1]             => 25
[5,2,2,1]             => 23
[5,2,1,1,1]           => 18
[5,1,1,1,1,1]         => 12
[4,4,2]               => 23
[4,4,1,1]             => 21
[4,3,3]               => 21
[4,3,2,1]             => 20
[4,3,1,1,1]           => 16
[4,2,2,2]             => 17
[4,2,2,1,1]           => 15
[4,2,1,1,1,1]         => 11
[4,1,1,1,1,1,1]       => 7
[3,3,3,1]             => 14
[3,3,2,2]             => 13
[3,3,2,1,1]           => 12
[3,3,1,1,1,1]         => 9
[3,2,2,2,1]           => 10
[3,2,2,1,1,1]         => 8
[3,2,1,1,1,1,1]       => 6
[3,1,1,1,1,1,1,1]     => 4
[2,2,2,2,2]           => 6
[2,2,2,2,1,1]         => 5
[2,2,2,1,1,1,1]       => 4
[2,2,1,1,1,1,1,1]     => 3
[2,1,1,1,1,1,1,1,1]   => 2
[1,1,1,1,1,1,1,1,1,1] => 1

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Created: Dec 08, 2015 at 16:23 by Christian Stump

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Last Updated: Oct 29, 2017 at 20:53 by Martin Rubey