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Statistic identifier: St000346

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Collection: Integer partitions

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Description: The number of coarsenings of a partition.

A partition $\mu$ coarsens a partition $\lambda$ 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]]
[4]   The number of refinements of a partition. [[St000345]]

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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_ideal([tuple(part)]))

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Statistic values:

[]                        => 1
[1]                       => 1
[2]                       => 1
[1,1]                     => 2
[3]                       => 1
[2,1]                     => 2
[1,1,1]                   => 3
[4]                       => 1
[3,1]                     => 2
[2,2]                     => 2
[2,1,1]                   => 4
[1,1,1,1]                 => 5
[5]                       => 1
[4,1]                     => 2
[3,2]                     => 2
[3,1,1]                   => 4
[2,2,1]                   => 4
[2,1,1,1]                 => 6
[1,1,1,1,1]               => 7
[6]                       => 1
[5,1]                     => 2
[4,2]                     => 2
[4,1,1]                   => 4
[3,3]                     => 2
[3,2,1]                   => 5
[3,1,1,1]                 => 7
[2,2,2]                   => 3
[2,2,1,1]                 => 8
[2,1,1,1,1]               => 10
[1,1,1,1,1,1]             => 11
[7]                       => 1
[6,1]                     => 2
[5,2]                     => 2
[5,1,1]                   => 4
[4,3]                     => 2
[4,2,1]                   => 5
[4,1,1,1]                 => 7
[3,3,1]                   => 4
[3,2,2]                   => 4
[3,2,1,1]                 => 9
[3,1,1,1,1]               => 11
[2,2,2,1]                 => 7
[2,2,1,1,1]               => 12
[2,1,1,1,1,1]             => 14
[1,1,1,1,1,1,1]           => 15
[8]                       => 1
[7,1]                     => 2
[6,2]                     => 2
[6,1,1]                   => 4
[5,3]                     => 2
[5,2,1]                   => 5
[5,1,1,1]                 => 7
[4,4]                     => 2
[4,3,1]                   => 5
[4,2,2]                   => 4
[4,2,1,1]                 => 10
[4,1,1,1,1]               => 12
[3,3,2]                   => 4
[3,3,1,1]                 => 9
[3,2,2,1]                 => 10
[3,2,1,1,1]               => 15
[3,1,1,1,1,1]             => 17
[2,2,2,2]                 => 5
[2,2,2,1,1]               => 14
[2,2,1,1,1,1]             => 19
[2,1,1,1,1,1,1]           => 21
[1,1,1,1,1,1,1,1]         => 22
[9]                       => 1
[8,1]                     => 2
[7,2]                     => 2
[7,1,1]                   => 4
[6,3]                     => 2
[6,2,1]                   => 5
[6,1,1,1]                 => 7
[5,4]                     => 2
[5,3,1]                   => 5
[5,2,2]                   => 4
[5,2,1,1]                 => 10
[5,1,1,1,1]               => 12
[4,4,1]                   => 4
[4,3,2]                   => 5
[4,3,1,1]                 => 10
[4,2,2,1]                 => 10
[4,2,1,1,1]               => 16
[4,1,1,1,1,1]             => 18
[3,3,3]                   => 3
[3,3,2,1]                 => 10
[3,3,1,1,1]               => 15
[3,2,2,2]                 => 7
[3,2,2,1,1]               => 18
[3,2,1,1,1,1]             => 23
[3,1,1,1,1,1,1]           => 25
[2,2,2,2,1]               => 12
[2,2,2,1,1,1]             => 22
[2,2,1,1,1,1,1]           => 27
[2,1,1,1,1,1,1,1]         => 29
[1,1,1,1,1,1,1,1,1]       => 30
[10]                      => 1
[9,1]                     => 2
[8,2]                     => 2
[8,1,1]                   => 4
[7,3]                     => 2
[7,2,1]                   => 5
[7,1,1,1]                 => 7
[6,4]                     => 2
[6,3,1]                   => 5
[6,2,2]                   => 4
[6,2,1,1]                 => 10
[6,1,1,1,1]               => 12
[5,5]                     => 2
[5,4,1]                   => 5
[5,3,2]                   => 5
[5,3,1,1]                 => 11
[5,2,2,1]                 => 11
[5,2,1,1,1]               => 17
[5,1,1,1,1,1]             => 19
[4,4,2]                   => 4
[4,4,1,1]                 => 9
[4,3,3]                   => 4
[4,3,2,1]                 => 13
[4,3,1,1,1]               => 18
[4,2,2,2]                 => 6
[4,2,2,1,1]               => 20
[4,2,1,1,1,1]             => 26
[4,1,1,1,1,1,1]           => 28
[3,3,3,1]                 => 7
[3,3,2,2]                 => 9
[3,3,2,1,1]               => 20
[3,3,1,1,1,1]             => 25
[3,2,2,2,1]               => 18
[3,2,2,1,1,1]             => 29
[3,2,1,1,1,1,1]           => 34
[3,1,1,1,1,1,1,1]         => 36
[2,2,2,2,2]               => 7
[2,2,2,2,1,1]             => 24
[2,2,2,1,1,1,1]           => 34
[2,2,1,1,1,1,1,1]         => 39
[2,1,1,1,1,1,1,1,1]       => 41
[1,1,1,1,1,1,1,1,1,1]     => 42
[11]                      => 1
[10,1]                    => 2
[9,2]                     => 2
[9,1,1]                   => 4
[8,3]                     => 2
[8,2,1]                   => 5
[8,1,1,1]                 => 7
[7,4]                     => 2
[7,3,1]                   => 5
[7,2,2]                   => 4
[7,2,1,1]                 => 10
[7,1,1,1,1]               => 12
[6,5]                     => 2
[6,4,1]                   => 5
[6,3,2]                   => 5
[6,3,1,1]                 => 11
[6,2,2,1]                 => 11
[6,2,1,1,1]               => 17
[6,1,1,1,1,1]             => 19
[5,5,1]                   => 4
[5,4,2]                   => 5
[5,4,1,1]                 => 10
[5,3,3]                   => 4
[5,3,2,1]                 => 13
[5,3,1,1,1]               => 19
[5,2,2,2]                 => 7
[5,2,2,1,1]               => 21
[5,2,1,1,1,1]             => 27
[5,1,1,1,1,1,1]           => 29
[4,4,3]                   => 4
[4,4,2,1]                 => 11
[4,4,1,1,1]               => 16
[4,3,3,1]                 => 10
[4,3,2,2]                 => 10
[4,3,2,1,1]               => 24
[4,3,1,1,1,1]             => 29
[4,2,2,2,1]               => 17
[4,2,2,1,1,1]             => 32
[4,2,1,1,1,1,1]           => 38
[4,1,1,1,1,1,1,1]         => 40
[3,3,3,2]                 => 7
[3,3,3,1,1]               => 16
[3,3,2,2,1]               => 21
[3,3,2,1,1,1]             => 32
[3,3,1,1,1,1,1]           => 37
[3,2,2,2,2]               => 12
[3,2,2,2,1,1]             => 32
[3,2,2,1,1,1,1]           => 43
[3,2,1,1,1,1,1,1]         => 48
[3,1,1,1,1,1,1,1,1]       => 50
[2,2,2,2,2,1]             => 19
[2,2,2,2,1,1,1]           => 38
[2,2,2,1,1,1,1,1]         => 48
[2,2,1,1,1,1,1,1,1]       => 53
[2,1,1,1,1,1,1,1,1,1]     => 55
[1,1,1,1,1,1,1,1,1,1,1]   => 56
[12]                      => 1
[11,1]                    => 2
[10,2]                    => 2
[10,1,1]                  => 4
[9,3]                     => 2
[9,2,1]                   => 5
[9,1,1,1]                 => 7
[8,4]                     => 2
[8,3,1]                   => 5
[8,2,2]                   => 4
[8,2,1,1]                 => 10
[8,1,1,1,1]               => 12
[7,5]                     => 2
[7,4,1]                   => 5
[7,3,2]                   => 5
[7,3,1,1]                 => 11
[7,2,2,1]                 => 11
[7,2,1,1,1]               => 17
[7,1,1,1,1,1]             => 19
[6,6]                     => 2
[6,5,1]                   => 5
[6,4,2]                   => 5
[6,4,1,1]                 => 11
[6,3,3]                   => 4
[6,3,2,1]                 => 14
[6,3,1,1,1]               => 20
[6,2,2,2]                 => 7
[6,2,2,1,1]               => 22
[6,2,1,1,1,1]             => 28
[6,1,1,1,1,1,1]           => 30
[5,5,2]                   => 4
[5,5,1,1]                 => 9
[5,4,3]                   => 5
[5,4,2,1]                 => 14
[5,4,1,1,1]               => 19
[5,3,3,1]                 => 11
[5,3,2,2]                 => 10
[5,3,2,1,1]               => 26
[5,3,1,1,1,1]             => 32
[5,2,2,2,1]               => 20
[5,2,2,1,1,1]             => 35
[5,2,1,1,1,1,1]           => 41
[5,1,1,1,1,1,1,1]         => 43
[4,4,4]                   => 3
[4,4,3,1]                 => 10
[4,4,2,2]                 => 8
[4,4,2,1,1]               => 23
[4,4,1,1,1,1]             => 28
[4,3,3,2]                 => 11
[4,3,3,1,1]               => 22
[4,3,2,2,1]               => 26
[4,3,2,1,1,1]             => 40
[4,3,1,1,1,1,1]           => 45
[4,2,2,2,2]               => 10
[4,2,2,2,1,1]             => 35
[4,2,2,1,1,1,1]           => 50
[4,2,1,1,1,1,1,1]         => 56
[4,1,1,1,1,1,1,1,1]       => 58
[3,3,3,3]                 => 5
[3,3,3,2,1]               => 19
[3,3,3,1,1,1]             => 29
[3,3,2,2,2]               => 16
[3,3,2,2,1,1]             => 40
[3,3,2,1,1,1,1]           => 51
[3,3,1,1,1,1,1,1]         => 56
[3,2,2,2,2,1]             => 31
[3,2,2,2,1,1,1]           => 52
[3,2,2,1,1,1,1,1]         => 63
[3,2,1,1,1,1,1,1,1]       => 68
[3,1,1,1,1,1,1,1,1,1]     => 70
[2,2,2,2,2,2]             => 11
[2,2,2,2,2,1,1]           => 39
[2,2,2,2,1,1,1,1]         => 59
[2,2,2,1,1,1,1,1,1]       => 69
[2,2,1,1,1,1,1,1,1,1]     => 74
[2,1,1,1,1,1,1,1,1,1,1]   => 76
[1,1,1,1,1,1,1,1,1,1,1,1] => 77

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Created: Dec 23, 2015 at 16:30 by Christian Stump

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Last Updated: Oct 29, 2017 at 20:59 by Martin Rubey