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Identifier

St000935: Integer partitions ⟶ ℤ

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] => 4

[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] => 7

[4,1,1] => 5

[3,3] => 5

[3,2,1] => 4

[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] => 9

[5,1,1] => 7

[4,3] => 8

[4,2,1] => 7

[4,1,1,1] => 5

[3,3,1] => 5

[3,2,2] => 5

[3,2,1,1] => 4

[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] => 15

[6,1,1] => 11

[5,3] => 10

[5,2,1] => 9

[5,1,1,1] => 7

[4,4] => 11

[4,3,1] => 8

[4,2,2] => 9

[4,2,1,1] => 7

[4,1,1,1,1] => 5

[3,3,2] => 6

[3,3,1,1] => 5

[3,2,2,1] => 5

[3,2,1,1,1] => 4

[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] => 19

[7,1,1] => 15

[6,3] => 17

[6,2,1] => 15

[6,1,1,1] => 11

[5,4] => 13

[5,3,1] => 10

[5,2,2] => 11

[5,2,1,1] => 9

[5,1,1,1,1] => 7

[4,4,1] => 11

[4,3,2] => 9

[4,3,1,1] => 8

[4,2,2,1] => 9

[4,2,1,1,1] => 7

[4,1,1,1,1,1] => 5

[3,3,3] => 7

[3,3,2,1] => 6

[3,3,1,1,1] => 5

[3,2,2,2] => 6

[3,2,2,1,1] => 5

[3,2,1,1,1,1] => 4

[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] => 29

[8,1,1] => 22

>>> Load all 547 entries. <<<

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Description

The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.

Code

def refines(L,M):
    Lpartsums = set( sum(L[:i]) for i in range(1,len(L)) )
    Mpartsums = set( sum(M[:i]) for i in range(1,len(M)) )
    return Mpartsums.issubset(Lpartsums)

def statistic(M):
    return sum(1 for L in Partitions(sum(M)) if refines(L,M) )

Created

Aug 11, 2017 at 17:14 by Christian Stump

Updated

Dec 29, 2023 at 18:03 by Martin Rubey