Identifier
Identifier
Values
[2] generating graphics... => 2
[1,1] generating graphics... => 1
[3] generating graphics... => 3
[2,1] generating graphics... => 2
[1,1,1] generating graphics... => 1
[4] generating graphics... => 5
[3,1] generating graphics... => 3
[2,2] generating graphics... => 3
[2,1,1] generating graphics... => 2
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 7
[4,1] generating graphics... => 5
[3,2] generating graphics... => 4
[3,1,1] generating graphics... => 3
[2,2,1] generating graphics... => 3
[2,1,1,1] generating graphics... => 2
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 11
[5,1] generating graphics... => 7
[4,2] generating graphics... => 7
[4,1,1] generating graphics... => 5
[3,3] generating graphics... => 5
[3,2,1] generating graphics... => 4
[3,1,1,1] generating graphics... => 3
[2,2,2] generating graphics... => 4
[2,2,1,1] generating graphics... => 3
[2,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 15
[6,1] generating graphics... => 11
[5,2] generating graphics... => 9
[5,1,1] generating graphics... => 7
[4,3] generating graphics... => 8
[4,2,1] generating graphics... => 7
[4,1,1,1] generating graphics... => 5
[3,3,1] generating graphics... => 5
[3,2,2] generating graphics... => 5
[3,2,1,1] generating graphics... => 4
[3,1,1,1,1] generating graphics... => 3
[2,2,2,1] generating graphics... => 4
[2,2,1,1,1] generating graphics... => 3
[2,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 22
[7,1] generating graphics... => 15
[6,2] generating graphics... => 15
[6,1,1] generating graphics... => 11
[5,3] generating graphics... => 10
[5,2,1] generating graphics... => 9
[5,1,1,1] generating graphics... => 7
[4,4] generating graphics... => 11
[4,3,1] generating graphics... => 8
[4,2,2] generating graphics... => 9
[4,2,1,1] generating graphics... => 7
[4,1,1,1,1] generating graphics... => 5
[3,3,2] generating graphics... => 6
[3,3,1,1] generating graphics... => 5
[3,2,2,1] generating graphics... => 5
[3,2,1,1,1] generating graphics... => 4
[3,1,1,1,1,1] generating graphics... => 3
[2,2,2,2] generating graphics... => 5
[2,2,2,1,1] generating graphics... => 4
[2,2,1,1,1,1] generating graphics... => 3
[2,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 30
[8,1] generating graphics... => 22
[7,2] generating graphics... => 19
[7,1,1] generating graphics... => 15
[6,3] generating graphics... => 17
[6,2,1] generating graphics... => 15
[6,1,1,1] generating graphics... => 11
[5,4] generating graphics... => 13
[5,3,1] generating graphics... => 10
[5,2,2] generating graphics... => 11
[5,2,1,1] generating graphics... => 9
[5,1,1,1,1] generating graphics... => 7
[4,4,1] generating graphics... => 11
[4,3,2] generating graphics... => 9
[4,3,1,1] generating graphics... => 8
[4,2,2,1] generating graphics... => 9
[4,2,1,1,1] generating graphics... => 7
[4,1,1,1,1,1] generating graphics... => 5
[3,3,3] generating graphics... => 7
[3,3,2,1] generating graphics... => 6
[3,3,1,1,1] generating graphics... => 5
[3,2,2,2] generating graphics... => 6
[3,2,2,1,1] generating graphics... => 5
[3,2,1,1,1,1] generating graphics... => 4
[3,1,1,1,1,1,1] generating graphics... => 3
[2,2,2,2,1] generating graphics... => 5
[2,2,2,1,1,1] generating graphics... => 4
[2,2,1,1,1,1,1] generating graphics... => 3
[2,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 42
[9,1] generating graphics... => 30
[8,2] generating graphics... => 29
[8,1,1] generating graphics... => 22
[7,3] generating graphics... => 21
[7,2,1] generating graphics... => 19
[7,1,1,1] generating graphics... => 15
[6,4] generating graphics... => 22
[6,3,1] generating graphics... => 17
[6,2,2] generating graphics... => 19
[6,2,1,1] generating graphics... => 15
[6,1,1,1,1] generating graphics... => 11
[5,5] generating graphics... => 15
[5,4,1] generating graphics... => 13
[5,3,2] generating graphics... => 11
[5,3,1,1] generating graphics... => 10
[5,2,2,1] generating graphics... => 11
[5,2,1,1,1] generating graphics... => 9
[5,1,1,1,1,1] generating graphics... => 7
[4,4,2] generating graphics... => 14
[4,4,1,1] generating graphics... => 11
[4,3,3] generating graphics... => 10
[4,3,2,1] generating graphics... => 9
[4,3,1,1,1] generating graphics... => 8
[4,2,2,2] generating graphics... => 11
[4,2,2,1,1] generating graphics... => 9
[4,2,1,1,1,1] generating graphics... => 7
[4,1,1,1,1,1,1] generating graphics... => 5
[3,3,3,1] generating graphics... => 7
[3,3,2,2] generating graphics... => 7
[3,3,2,1,1] generating graphics... => 6
[3,3,1,1,1,1] generating graphics... => 5
[3,2,2,2,1] generating graphics... => 6
[3,2,2,1,1,1] generating graphics... => 5
[3,2,1,1,1,1,1] generating graphics... => 4
[3,1,1,1,1,1,1,1] generating graphics... => 3
[2,2,2,2,2] generating graphics... => 6
[2,2,2,2,1,1] generating graphics... => 5
[2,2,2,1,1,1,1] generating graphics... => 4
[2,2,1,1,1,1,1,1] generating graphics... => 3
[2,1,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 56
[10,1] generating graphics... => 42
[9,2] generating graphics... => 38
[9,1,1] generating graphics... => 30
[8,3] generating graphics... => 32
[8,2,1] generating graphics... => 29
[8,1,1,1] generating graphics... => 22
[7,4] generating graphics... => 26
[7,3,1] generating graphics... => 21
[7,2,2] generating graphics... => 23
[7,2,1,1] generating graphics... => 19
[7,1,1,1,1] generating graphics... => 15
[6,5] generating graphics... => 25
[6,4,1] generating graphics... => 22
[6,3,2] generating graphics... => 19
[6,3,1,1] generating graphics... => 17
[6,2,2,1] generating graphics... => 19
[6,2,1,1,1] generating graphics... => 15
[6,1,1,1,1,1] generating graphics... => 11
[5,5,1] generating graphics... => 15
[5,4,2] generating graphics... => 16
[5,4,1,1] generating graphics... => 13
[5,3,3] generating graphics... => 12
[5,3,2,1] generating graphics... => 11
[5,3,1,1,1] generating graphics... => 10
[5,2,2,2] generating graphics... => 13
[5,2,2,1,1] generating graphics... => 11
[5,2,1,1,1,1] generating graphics... => 9
[5,1,1,1,1,1,1] generating graphics... => 7
[4,4,3] generating graphics... => 15
[4,4,2,1] generating graphics... => 14
[4,4,1,1,1] generating graphics... => 11
[4,3,3,1] generating graphics... => 10
[4,3,2,2] generating graphics... => 10
[4,3,2,1,1] generating graphics... => 9
[4,3,1,1,1,1] generating graphics... => 8
[4,2,2,2,1] generating graphics... => 11
[4,2,2,1,1,1] generating graphics... => 9
[4,2,1,1,1,1,1] generating graphics... => 7
[4,1,1,1,1,1,1,1] generating graphics... => 5
[3,3,3,2] generating graphics... => 8
[3,3,3,1,1] generating graphics... => 7
[3,3,2,2,1] generating graphics... => 7
[3,3,2,1,1,1] generating graphics... => 6
[3,3,1,1,1,1,1] generating graphics... => 5
[3,2,2,2,2] generating graphics... => 7
[3,2,2,2,1,1] generating graphics... => 6
[3,2,2,1,1,1,1] generating graphics... => 5
[3,2,1,1,1,1,1,1] generating graphics... => 4
[3,1,1,1,1,1,1,1,1] generating graphics... => 3
[2,2,2,2,2,1] generating graphics... => 6
[2,2,2,2,1,1,1] generating graphics... => 5
[2,2,2,1,1,1,1,1] generating graphics... => 4
[2,2,1,1,1,1,1,1,1] generating graphics... => 3
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 77
[11,1] generating graphics... => 56
[10,2] generating graphics... => 54
[10,1,1] generating graphics... => 42
[9,3] generating graphics... => 42
[9,2,1] generating graphics... => 38
[9,1,1,1] generating graphics... => 30
[8,4] generating graphics... => 41
[8,3,1] generating graphics... => 32
[8,2,2] generating graphics... => 36
[8,2,1,1] generating graphics... => 29
[8,1,1,1,1] generating graphics... => 22
[7,5] generating graphics... => 29
[7,4,1] generating graphics... => 26
[7,3,2] generating graphics... => 23
[7,3,1,1] generating graphics... => 21
[7,2,2,1] generating graphics... => 23
[7,2,1,1,1] generating graphics... => 19
[7,1,1,1,1,1] generating graphics... => 15
[6,6] generating graphics... => 33
[6,5,1] generating graphics... => 25
[6,4,2] generating graphics... => 27
[6,4,1,1] generating graphics... => 22
[6,3,3] generating graphics... => 21
[6,3,2,1] generating graphics... => 19
[6,3,1,1,1] generating graphics... => 17
[6,2,2,2] generating graphics... => 23
[6,2,2,1,1] generating graphics... => 19
[6,2,1,1,1,1] generating graphics... => 15
[6,1,1,1,1,1,1] generating graphics... => 11
[5,5,2] generating graphics... => 17
[5,5,1,1] generating graphics... => 15
[5,4,3] generating graphics... => 17
[5,4,2,1] generating graphics... => 16
[5,4,1,1,1] generating graphics... => 13
[5,3,3,1] generating graphics... => 12
[5,3,2,2] generating graphics... => 12
[5,3,2,1,1] generating graphics... => 11
[5,3,1,1,1,1] generating graphics... => 10
[5,2,2,2,1] generating graphics... => 13
[5,2,2,1,1,1] generating graphics... => 11
[5,2,1,1,1,1,1] generating graphics... => 9
[5,1,1,1,1,1,1,1] generating graphics... => 7
[4,4,4] generating graphics... => 19
[4,4,3,1] generating graphics... => 15
[4,4,2,2] generating graphics... => 17
[4,4,2,1,1] generating graphics... => 14
[4,4,1,1,1,1] generating graphics... => 11
[4,3,3,2] generating graphics... => 11
[4,3,3,1,1] generating graphics... => 10
[4,3,2,2,1] generating graphics... => 10
[4,3,2,1,1,1] generating graphics... => 9
[4,3,1,1,1,1,1] generating graphics... => 8
[4,2,2,2,2] generating graphics... => 13
[4,2,2,2,1,1] generating graphics... => 11
[4,2,2,1,1,1,1] generating graphics... => 9
[4,2,1,1,1,1,1,1] generating graphics... => 7
[4,1,1,1,1,1,1,1,1] generating graphics... => 5
[3,3,3,3] generating graphics... => 9
[3,3,3,2,1] generating graphics... => 8
[3,3,3,1,1,1] generating graphics... => 7
[3,3,2,2,2] generating graphics... => 8
[3,3,2,2,1,1] generating graphics... => 7
[3,3,2,1,1,1,1] generating graphics... => 6
[3,3,1,1,1,1,1,1] generating graphics... => 5
[3,2,2,2,2,1] generating graphics... => 7
[3,2,2,2,1,1,1] generating graphics... => 6
[3,2,2,1,1,1,1,1] generating graphics... => 5
[3,2,1,1,1,1,1,1,1] generating graphics... => 4
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 3
[2,2,2,2,2,2] generating graphics... => 7
[2,2,2,2,2,1,1] generating graphics... => 6
[2,2,2,2,1,1,1,1] generating graphics... => 5
[2,2,2,1,1,1,1,1,1] generating graphics... => 4
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 3
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
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
Aug 11, 2017 at 17:42 by Christian Stump