Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[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... => 5
[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... => 8
[4,1,1] generating graphics... => 5
[3,3] generating graphics... => 6
[3,2,1] generating graphics... => 5
[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... => 11
[5,1,1] generating graphics... => 7
[4,3] generating graphics... => 11
[4,2,1] generating graphics... => 8
[4,1,1,1] generating graphics... => 5
[3,3,1] generating graphics... => 6
[3,2,2] generating graphics... => 7
[3,2,1,1] generating graphics... => 5
[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... => 17
[6,1,1] generating graphics... => 11
[5,3] generating graphics... => 15
[5,2,1] generating graphics... => 11
[5,1,1,1] generating graphics... => 7
[4,4] generating graphics... => 14
[4,3,1] generating graphics... => 11
[4,2,2] generating graphics... => 11
[4,2,1,1] generating graphics... => 8
[4,1,1,1,1] generating graphics... => 5
[3,3,2] generating graphics... => 9
[3,3,1,1] generating graphics... => 6
[3,2,2,1] generating graphics... => 7
[3,2,1,1,1] generating graphics... => 5
[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... => 23
[7,1,1] generating graphics... => 15
[6,3] generating graphics... => 23
[6,2,1] generating graphics... => 17
[6,1,1,1] generating graphics... => 11
[5,4] generating graphics... => 22
[5,3,1] generating graphics... => 15
[5,2,2] generating graphics... => 15
[5,2,1,1] generating graphics... => 11
[5,1,1,1,1] generating graphics... => 7
[4,4,1] generating graphics... => 14
[4,3,2] generating graphics... => 16
[4,3,1,1] generating graphics... => 11
[4,2,2,1] generating graphics... => 11
[4,2,1,1,1] generating graphics... => 8
[4,1,1,1,1,1] generating graphics... => 5
[3,3,3] generating graphics... => 10
[3,3,2,1] generating graphics... => 9
[3,3,1,1,1] generating graphics... => 6
[3,2,2,2] generating graphics... => 9
[3,2,2,1,1] generating graphics... => 7
[3,2,1,1,1,1] generating graphics... => 5
[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... => 33
[8,1,1] generating graphics... => 22
[7,3] generating graphics... => 30
[7,2,1] generating graphics... => 23
[7,1,1,1] generating graphics... => 15
[6,4] generating graphics... => 33
[6,3,1] generating graphics... => 23
[6,2,2] generating graphics... => 23
[6,2,1,1] generating graphics... => 17
[6,1,1,1,1] generating graphics... => 11
[5,5] generating graphics... => 25
[5,4,1] generating graphics... => 22
[5,3,2] generating graphics... => 22
[5,3,1,1] generating graphics... => 15
[5,2,2,1] generating graphics... => 15
[5,2,1,1,1] generating graphics... => 11
[5,1,1,1,1,1] generating graphics... => 7
[4,4,2] generating graphics... => 20
[4,4,1,1] generating graphics... => 14
[4,3,3] generating graphics... => 19
[4,3,2,1] generating graphics... => 16
[4,3,1,1,1] generating graphics... => 11
[4,2,2,2] generating graphics... => 14
[4,2,2,1,1] generating graphics... => 11
[4,2,1,1,1,1] generating graphics... => 8
[4,1,1,1,1,1,1] generating graphics... => 5
[3,3,3,1] generating graphics... => 10
[3,3,2,2] generating graphics... => 12
[3,3,2,1,1] generating graphics... => 9
[3,3,1,1,1,1] generating graphics... => 6
[3,2,2,2,1] generating graphics... => 9
[3,2,2,1,1,1] generating graphics... => 7
[3,2,1,1,1,1,1] generating graphics... => 5
[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... => 45
[9,1,1] generating graphics... => 30
[8,3] generating graphics... => 44
[8,2,1] generating graphics... => 33
[8,1,1,1] generating graphics... => 22
[7,4] generating graphics... => 44
[7,3,1] generating graphics... => 30
[7,2,2] generating graphics... => 31
[7,2,1,1] generating graphics... => 23
[7,1,1,1,1] generating graphics... => 15
[6,5] generating graphics... => 43
[6,4,1] generating graphics... => 33
[6,3,2] generating graphics... => 33
[6,3,1,1] generating graphics... => 23
[6,2,2,1] generating graphics... => 23
[6,2,1,1,1] generating graphics... => 17
[6,1,1,1,1,1] generating graphics... => 11
[5,5,1] generating graphics... => 25
[5,4,2] generating graphics... => 31
[5,4,1,1] generating graphics... => 22
[5,3,3] generating graphics... => 26
[5,3,2,1] generating graphics... => 22
[5,3,1,1,1] generating graphics... => 15
[5,2,2,2] generating graphics... => 19
[5,2,2,1,1] generating graphics... => 15
[5,2,1,1,1,1] generating graphics... => 11
[5,1,1,1,1,1,1] generating graphics... => 7
[4,4,3] generating graphics... => 26
[4,4,2,1] generating graphics... => 20
[4,4,1,1,1] generating graphics... => 14
[4,3,3,1] generating graphics... => 19
[4,3,2,2] generating graphics... => 21
[4,3,2,1,1] generating graphics... => 16
[4,3,1,1,1,1] generating graphics... => 11
[4,2,2,2,1] generating graphics... => 14
[4,2,2,1,1,1] generating graphics... => 11
[4,2,1,1,1,1,1] generating graphics... => 8
[4,1,1,1,1,1,1,1] generating graphics... => 5
[3,3,3,2] generating graphics... => 14
[3,3,3,1,1] generating graphics... => 10
[3,3,2,2,1] generating graphics... => 12
[3,3,2,1,1,1] generating graphics... => 9
[3,3,1,1,1,1,1] generating graphics... => 6
[3,2,2,2,2] generating graphics... => 11
[3,2,2,2,1,1] generating graphics... => 9
[3,2,2,1,1,1,1] generating graphics... => 7
[3,2,1,1,1,1,1,1] generating graphics... => 5
[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... => 62
[10,1,1] generating graphics... => 42
[9,3] generating graphics... => 58
[9,2,1] generating graphics... => 45
[9,1,1,1] generating graphics... => 30
[8,4] generating graphics... => 62
[8,3,1] generating graphics... => 44
[8,2,2] generating graphics... => 44
[8,2,1,1] generating graphics... => 33
[8,1,1,1,1] generating graphics... => 22
[7,5] generating graphics... => 56
[7,4,1] generating graphics... => 44
[7,3,2] generating graphics... => 43
[7,3,1,1] generating graphics... => 30
[7,2,2,1] generating graphics... => 31
[7,2,1,1,1] generating graphics... => 23
[7,1,1,1,1,1] generating graphics... => 15
[6,6] generating graphics... => 53
[6,5,1] generating graphics... => 43
[6,4,2] generating graphics... => 47
[6,4,1,1] generating graphics... => 33
[6,3,3] generating graphics... => 39
[6,3,2,1] generating graphics... => 33
[6,3,1,1,1] generating graphics... => 23
[6,2,2,2] generating graphics... => 29
[6,2,2,1,1] generating graphics... => 23
[6,2,1,1,1,1] generating graphics... => 17
[6,1,1,1,1,1,1] generating graphics... => 11
[5,5,2] generating graphics... => 35
[5,5,1,1] generating graphics... => 25
[5,4,3] generating graphics... => 40
[5,4,2,1] generating graphics... => 31
[5,4,1,1,1] generating graphics... => 22
[5,3,3,1] generating graphics... => 26
[5,3,2,2] generating graphics... => 29
[5,3,2,1,1] generating graphics... => 22
[5,3,1,1,1,1] generating graphics... => 15
[5,2,2,2,1] generating graphics... => 19
[5,2,2,1,1,1] generating graphics... => 15
[5,2,1,1,1,1,1] generating graphics... => 11
[5,1,1,1,1,1,1,1] generating graphics... => 7
[4,4,4] generating graphics... => 30
[4,4,3,1] generating graphics... => 26
[4,4,2,2] generating graphics... => 26
[4,4,2,1,1] generating graphics... => 20
[4,4,1,1,1,1] generating graphics... => 14
[4,3,3,2] generating graphics... => 26
[4,3,3,1,1] generating graphics... => 19
[4,3,2,2,1] generating graphics... => 21
[4,3,2,1,1,1] generating graphics... => 16
[4,3,1,1,1,1,1] generating graphics... => 11
[4,2,2,2,2] generating graphics... => 17
[4,2,2,2,1,1] generating graphics... => 14
[4,2,2,1,1,1,1] generating graphics... => 11
[4,2,1,1,1,1,1,1] generating graphics... => 8
[4,1,1,1,1,1,1,1,1] generating graphics... => 5
[3,3,3,3] generating graphics... => 15
[3,3,3,2,1] generating graphics... => 14
[3,3,3,1,1,1] generating graphics... => 10
[3,3,2,2,2] generating graphics... => 15
[3,3,2,2,1,1] generating graphics... => 12
[3,3,2,1,1,1,1] generating graphics... => 9
[3,3,1,1,1,1,1,1] generating graphics... => 6
[3,2,2,2,2,1] generating graphics... => 11
[3,2,2,2,1,1,1] generating graphics... => 9
[3,2,2,1,1,1,1,1] generating graphics... => 7
[3,2,1,1,1,1,1,1,1] generating graphics... => 5
[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 refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
References
[1] Birkhoff, G. Lattice theory MathSciNet:0598630
[2] Ziegler, G√ľnter M. On the poset of partitions of an integer MathSciNet:0847552
[3] Perry, J. M. Counting refinements of partitions MathOverflow:226656
Code
@cached_function
def PartitionPoset(n):
    return posets.IntegerPartitions(n)

def statistic(part):
    P = PartitionPoset(sum(part))
    return len(P.order_filter([tuple(part)]))
Created
Dec 23, 2015 at 14:54 by Christian Stump
Updated
Oct 29, 2017 at 20:56 by Martin Rubey