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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)]))
-----------------------------------------------------------------------------
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
-----------------------------------------------------------------------------
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