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Statistic identifier: St001568

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Collection: Integer partitions

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Description: The smallest positive integer that does not appear twice in the partition.

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References: [1]   Ballantine, C., Merca, M. Combinatorial Proof of the Minimal Excludant Theorem [[arXiv:1908.06789]]

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Code:
def statistic(x):
    x = list(x)
    for i in range(1, x[0]+2):
        if x.count(i) < 2:
            return i

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Statistic values:

[2]                   => 1
[1,1]                 => 2
[3]                   => 1
[2,1]                 => 1
[1,1,1]               => 2
[4]                   => 1
[3,1]                 => 1
[2,2]                 => 1
[2,1,1]               => 2
[1,1,1,1]             => 2
[5]                   => 1
[4,1]                 => 1
[3,2]                 => 1
[3,1,1]               => 2
[2,2,1]               => 1
[2,1,1,1]             => 2
[1,1,1,1,1]           => 2
[6]                   => 1
[5,1]                 => 1
[4,2]                 => 1
[4,1,1]               => 2
[3,3]                 => 1
[3,2,1]               => 1
[3,1,1,1]             => 2
[2,2,2]               => 1
[2,2,1,1]             => 3
[2,1,1,1,1]           => 2
[1,1,1,1,1,1]         => 2
[7]                   => 1
[6,1]                 => 1
[5,2]                 => 1
[5,1,1]               => 2
[4,3]                 => 1
[4,2,1]               => 1
[4,1,1,1]             => 2
[3,3,1]               => 1
[3,2,2]               => 1
[3,2,1,1]             => 2
[3,1,1,1,1]           => 2
[2,2,2,1]             => 1
[2,2,1,1,1]           => 3
[2,1,1,1,1,1]         => 2
[1,1,1,1,1,1,1]       => 2
[8]                   => 1
[7,1]                 => 1
[6,2]                 => 1
[6,1,1]               => 2
[5,3]                 => 1
[5,2,1]               => 1
[5,1,1,1]             => 2
[4,4]                 => 1
[4,3,1]               => 1
[4,2,2]               => 1
[4,2,1,1]             => 2
[4,1,1,1,1]           => 2
[3,3,2]               => 1
[3,3,1,1]             => 2
[3,2,2,1]             => 1
[3,2,1,1,1]           => 2
[3,1,1,1,1,1]         => 2
[2,2,2,2]             => 1
[2,2,2,1,1]           => 3
[2,2,1,1,1,1]         => 3
[2,1,1,1,1,1,1]       => 2
[1,1,1,1,1,1,1,1]     => 2
[9]                   => 1
[8,1]                 => 1
[7,2]                 => 1
[7,1,1]               => 2
[6,3]                 => 1
[6,2,1]               => 1
[6,1,1,1]             => 2
[5,4]                 => 1
[5,3,1]               => 1
[5,2,2]               => 1
[5,2,1,1]             => 2
[5,1,1,1,1]           => 2
[4,4,1]               => 1
[4,3,2]               => 1
[4,3,1,1]             => 2
[4,2,2,1]             => 1
[4,2,1,1,1]           => 2
[4,1,1,1,1,1]         => 2
[3,3,3]               => 1
[3,3,2,1]             => 1
[3,3,1,1,1]           => 2
[3,2,2,2]             => 1
[3,2,2,1,1]           => 3
[3,2,1,1,1,1]         => 2
[3,1,1,1,1,1,1]       => 2
[2,2,2,2,1]           => 1
[2,2,2,1,1,1]         => 3
[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]   => 2
[10]                  => 1
[9,1]                 => 1
[8,2]                 => 1
[8,1,1]               => 2
[7,3]                 => 1
[7,2,1]               => 1
[7,1,1,1]             => 2
[6,4]                 => 1
[6,3,1]               => 1
[6,2,2]               => 1
[6,2,1,1]             => 2
[6,1,1,1,1]           => 2
[5,5]                 => 1
[5,4,1]               => 1
[5,3,2]               => 1
[5,3,1,1]             => 2
[5,2,2,1]             => 1
[5,2,1,1,1]           => 2
[5,1,1,1,1,1]         => 2
[4,4,2]               => 1
[4,4,1,1]             => 2
[4,3,3]               => 1
[4,3,2,1]             => 1
[4,3,1,1,1]           => 2
[4,2,2,2]             => 1
[4,2,2,1,1]           => 3
[4,2,1,1,1,1]         => 2
[4,1,1,1,1,1,1]       => 2
[3,3,3,1]             => 1
[3,3,2,2]             => 1
[3,3,2,1,1]           => 2
[3,3,1,1,1,1]         => 2
[3,2,2,2,1]           => 1
[3,2,2,1,1,1]         => 3
[3,2,1,1,1,1,1]       => 2
[3,1,1,1,1,1,1,1]     => 2
[2,2,2,2,2]           => 1
[2,2,2,2,1,1]         => 3
[2,2,2,1,1,1,1]       => 3
[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] => 2

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Created: Jul 15, 2020 at 15:01 by Martin Rubey

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Last Updated: Jul 15, 2020 at 15:01 by Martin Rubey