- FindStat

Identifier

St000346: Integer partitions ⟶ ℤ

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

>>> Load all 272 entries. <<<

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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 $\lambda$.

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

Code

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

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

Created

Dec 23, 2015 at 16:30 by Christian Stump

Updated

Oct 29, 2017 at 20:59 by Martin Rubey