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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