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