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Statistic identifier: St000345

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Collection: Integer partitions

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Description: The number of refinements of a partition.

A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.

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References: [1]   Birkhoff, G. Lattice theory [[MathSciNet:0598630]]
[2]   Ziegler, Günter M. On the poset of partitions of an integer [[MathSciNet:0847552]]
[3]   Perry, J. M. Counting refinements of partitions [[MathOverflow:226656]]

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Code:
@cached_function
def PartitionPoset(n):
    return posets.IntegerPartitions(n)

def statistic(part):
    P = PartitionPoset(sum(part))
    return len(P.order_filter([tuple(part)]))

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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]                     => 3
[2,2]                     => 3
[2,1,1]                   => 2
[1,1,1,1]                 => 1
[5]                       => 7
[4,1]                     => 5
[3,2]                     => 5
[3,1,1]                   => 3
[2,2,1]                   => 3
[2,1,1,1]                 => 2
[1,1,1,1,1]               => 1
[6]                       => 11
[5,1]                     => 7
[4,2]                     => 8
[4,1,1]                   => 5
[3,3]                     => 6
[3,2,1]                   => 5
[3,1,1,1]                 => 3
[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]                     => 11
[5,2]                     => 11
[5,1,1]                   => 7
[4,3]                     => 11
[4,2,1]                   => 8
[4,1,1,1]                 => 5
[3,3,1]                   => 6
[3,2,2]                   => 7
[3,2,1,1]                 => 5
[3,1,1,1,1]               => 3
[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]                     => 15
[6,2]                     => 17
[6,1,1]                   => 11
[5,3]                     => 15
[5,2,1]                   => 11
[5,1,1,1]                 => 7
[4,4]                     => 14
[4,3,1]                   => 11
[4,2,2]                   => 11
[4,2,1,1]                 => 8
[4,1,1,1,1]               => 5
[3,3,2]                   => 9
[3,3,1,1]                 => 6
[3,2,2,1]                 => 7
[3,2,1,1,1]               => 5
[3,1,1,1,1,1]             => 3
[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]                     => 22
[7,2]                     => 23
[7,1,1]                   => 15
[6,3]                     => 23
[6,2,1]                   => 17
[6,1,1,1]                 => 11
[5,4]                     => 22
[5,3,1]                   => 15
[5,2,2]                   => 15
[5,2,1,1]                 => 11
[5,1,1,1,1]               => 7
[4,4,1]                   => 14
[4,3,2]                   => 16
[4,3,1,1]                 => 11
[4,2,2,1]                 => 11
[4,2,1,1,1]               => 8
[4,1,1,1,1,1]             => 5
[3,3,3]                   => 10
[3,3,2,1]                 => 9
[3,3,1,1,1]               => 6
[3,2,2,2]                 => 9
[3,2,2,1,1]               => 7
[3,2,1,1,1,1]             => 5
[3,1,1,1,1,1,1]           => 3
[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]                     => 30
[8,2]                     => 33
[8,1,1]                   => 22
[7,3]                     => 30
[7,2,1]                   => 23
[7,1,1,1]                 => 15
[6,4]                     => 33
[6,3,1]                   => 23
[6,2,2]                   => 23
[6,2,1,1]                 => 17
[6,1,1,1,1]               => 11
[5,5]                     => 25
[5,4,1]                   => 22
[5,3,2]                   => 22
[5,3,1,1]                 => 15
[5,2,2,1]                 => 15
[5,2,1,1,1]               => 11
[5,1,1,1,1,1]             => 7
[4,4,2]                   => 20
[4,4,1,1]                 => 14
[4,3,3]                   => 19
[4,3,2,1]                 => 16
[4,3,1,1,1]               => 11
[4,2,2,2]                 => 14
[4,2,2,1,1]               => 11
[4,2,1,1,1,1]             => 8
[4,1,1,1,1,1,1]           => 5
[3,3,3,1]                 => 10
[3,3,2,2]                 => 12
[3,3,2,1,1]               => 9
[3,3,1,1,1,1]             => 6
[3,2,2,2,1]               => 9
[3,2,2,1,1,1]             => 7
[3,2,1,1,1,1,1]           => 5
[3,1,1,1,1,1,1,1]         => 3
[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
[11]                      => 56
[10,1]                    => 42
[9,2]                     => 45
[9,1,1]                   => 30
[8,3]                     => 44
[8,2,1]                   => 33
[8,1,1,1]                 => 22
[7,4]                     => 44
[7,3,1]                   => 30
[7,2,2]                   => 31
[7,2,1,1]                 => 23
[7,1,1,1,1]               => 15
[6,5]                     => 43
[6,4,1]                   => 33
[6,3,2]                   => 33
[6,3,1,1]                 => 23
[6,2,2,1]                 => 23
[6,2,1,1,1]               => 17
[6,1,1,1,1,1]             => 11
[5,5,1]                   => 25
[5,4,2]                   => 31
[5,4,1,1]                 => 22
[5,3,3]                   => 26
[5,3,2,1]                 => 22
[5,3,1,1,1]               => 15
[5,2,2,2]                 => 19
[5,2,2,1,1]               => 15
[5,2,1,1,1,1]             => 11
[5,1,1,1,1,1,1]           => 7
[4,4,3]                   => 26
[4,4,2,1]                 => 20
[4,4,1,1,1]               => 14
[4,3,3,1]                 => 19
[4,3,2,2]                 => 21
[4,3,2,1,1]               => 16
[4,3,1,1,1,1]             => 11
[4,2,2,2,1]               => 14
[4,2,2,1,1,1]             => 11
[4,2,1,1,1,1,1]           => 8
[4,1,1,1,1,1,1,1]         => 5
[3,3,3,2]                 => 14
[3,3,3,1,1]               => 10
[3,3,2,2,1]               => 12
[3,3,2,1,1,1]             => 9
[3,3,1,1,1,1,1]           => 6
[3,2,2,2,2]               => 11
[3,2,2,2,1,1]             => 9
[3,2,2,1,1,1,1]           => 7
[3,2,1,1,1,1,1,1]         => 5
[3,1,1,1,1,1,1,1,1]       => 3
[2,2,2,2,2,1]             => 6
[2,2,2,2,1,1,1]           => 5
[2,2,2,1,1,1,1,1]         => 4
[2,2,1,1,1,1,1,1,1]       => 3
[2,1,1,1,1,1,1,1,1,1]     => 2
[1,1,1,1,1,1,1,1,1,1,1]   => 1
[12]                      => 77
[11,1]                    => 56
[10,2]                    => 62
[10,1,1]                  => 42
[9,3]                     => 58
[9,2,1]                   => 45
[9,1,1,1]                 => 30
[8,4]                     => 62
[8,3,1]                   => 44
[8,2,2]                   => 44
[8,2,1,1]                 => 33
[8,1,1,1,1]               => 22
[7,5]                     => 56
[7,4,1]                   => 44
[7,3,2]                   => 43
[7,3,1,1]                 => 30
[7,2,2,1]                 => 31
[7,2,1,1,1]               => 23
[7,1,1,1,1,1]             => 15
[6,6]                     => 53
[6,5,1]                   => 43
[6,4,2]                   => 47
[6,4,1,1]                 => 33
[6,3,3]                   => 39
[6,3,2,1]                 => 33
[6,3,1,1,1]               => 23
[6,2,2,2]                 => 29
[6,2,2,1,1]               => 23
[6,2,1,1,1,1]             => 17
[6,1,1,1,1,1,1]           => 11
[5,5,2]                   => 35
[5,5,1,1]                 => 25
[5,4,3]                   => 40
[5,4,2,1]                 => 31
[5,4,1,1,1]               => 22
[5,3,3,1]                 => 26
[5,3,2,2]                 => 29
[5,3,2,1,1]               => 22
[5,3,1,1,1,1]             => 15
[5,2,2,2,1]               => 19
[5,2,2,1,1,1]             => 15
[5,2,1,1,1,1,1]           => 11
[5,1,1,1,1,1,1,1]         => 7
[4,4,4]                   => 30
[4,4,3,1]                 => 26
[4,4,2,2]                 => 26
[4,4,2,1,1]               => 20
[4,4,1,1,1,1]             => 14
[4,3,3,2]                 => 26
[4,3,3,1,1]               => 19
[4,3,2,2,1]               => 21
[4,3,2,1,1,1]             => 16
[4,3,1,1,1,1,1]           => 11
[4,2,2,2,2]               => 17
[4,2,2,2,1,1]             => 14
[4,2,2,1,1,1,1]           => 11
[4,2,1,1,1,1,1,1]         => 8
[4,1,1,1,1,1,1,1,1]       => 5
[3,3,3,3]                 => 15
[3,3,3,2,1]               => 14
[3,3,3,1,1,1]             => 10
[3,3,2,2,2]               => 15
[3,3,2,2,1,1]             => 12
[3,3,2,1,1,1,1]           => 9
[3,3,1,1,1,1,1,1]         => 6
[3,2,2,2,2,1]             => 11
[3,2,2,2,1,1,1]           => 9
[3,2,2,1,1,1,1,1]         => 7
[3,2,1,1,1,1,1,1,1]       => 5
[3,1,1,1,1,1,1,1,1,1]     => 3
[2,2,2,2,2,2]             => 7
[2,2,2,2,2,1,1]           => 6
[2,2,2,2,1,1,1,1]         => 5
[2,2,2,1,1,1,1,1,1]       => 4
[2,2,1,1,1,1,1,1,1,1]     => 3
[2,1,1,1,1,1,1,1,1,1,1]   => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => 1

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Created: Dec 23, 2015 at 14:54 by Christian Stump

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Last Updated: Oct 29, 2017 at 20:56 by Martin Rubey