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Identifier
Values
=>
Cc0002;cc-rep
[]=>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]=>5 [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]=>8 [4,1,1]=>5 [3,3]=>6 [3,2,1]=>5 [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]=>11 [5,1,1]=>7 [4,3]=>11 [4,2,1]=>8 [4,1,1,1]=>5 [3,3,1]=>6 [3,2,2]=>7 [3,2,1,1]=>5 [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]=>17 [6,1,1]=>11 [5,3]=>15 [5,2,1]=>11 [5,1,1,1]=>7 [4,4]=>14 [4,3,1]=>11 [4,2,2]=>11 [4,2,1,1]=>8 [4,1,1,1,1]=>5 [3,3,2]=>9 [3,3,1,1]=>6 [3,2,2,1]=>7 [3,2,1,1,1]=>5 [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]=>23 [7,1,1]=>15 [6,3]=>23 [6,2,1]=>17 [6,1,1,1]=>11 [5,4]=>22 [5,3,1]=>15 [5,2,2]=>15 [5,2,1,1]=>11 [5,1,1,1,1]=>7 [4,4,1]=>14 [4,3,2]=>16 [4,3,1,1]=>11 [4,2,2,1]=>11 [4,2,1,1,1]=>8 [4,1,1,1,1,1]=>5 [3,3,3]=>10 [3,3,2,1]=>9 [3,3,1,1,1]=>6 [3,2,2,2]=>9 [3,2,2,1,1]=>7 [3,2,1,1,1,1]=>5 [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]=>33 [8,1,1]=>22 [7,3]=>30 [7,2,1]=>23 [7,1,1,1]=>15 [6,4]=>33 [6,3,1]=>23 [6,2,2]=>23 [6,2,1,1]=>17 [6,1,1,1,1]=>11 [5,5]=>25 [5,4,1]=>22 [5,3,2]=>22 [5,3,1,1]=>15 [5,2,2,1]=>15 [5,2,1,1,1]=>11 [5,1,1,1,1,1]=>7 [4,4,2]=>20 [4,4,1,1]=>14 [4,3,3]=>19 [4,3,2,1]=>16 [4,3,1,1,1]=>11 [4,2,2,2]=>14 [4,2,2,1,1]=>11 [4,2,1,1,1,1]=>8 [4,1,1,1,1,1,1]=>5 [3,3,3,1]=>10 [3,3,2,2]=>12 [3,3,2,1,1]=>9 [3,3,1,1,1,1]=>6 [3,2,2,2,1]=>9 [3,2,2,1,1,1]=>7 [3,2,1,1,1,1,1]=>5 [3,1,1,1,1,1,1,1]=>3 [2,2,2,2,2]=>6 [2,2,2,2,1,1]=>5 [2,2,2,1,1,1,1]=>4 [2,2,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1,1]=>2 [1,1,1,1,1,1,1,1,1,1]=>1 [11]=>56 [10,1]=>42 [9,2]=>45 [9,1,1]=>30 [8,3]=>44 [8,2,1]=>33 [8,1,1,1]=>22 [7,4]=>44 [7,3,1]=>30 [7,2,2]=>31 [7,2,1,1]=>23 [7,1,1,1,1]=>15 [6,5]=>43 [6,4,1]=>33 [6,3,2]=>33 [6,3,1,1]=>23 [6,2,2,1]=>23 [6,2,1,1,1]=>17 [6,1,1,1,1,1]=>11 [5,5,1]=>25 [5,4,2]=>31 [5,4,1,1]=>22 [5,3,3]=>26 [5,3,2,1]=>22 [5,3,1,1,1]=>15 [5,2,2,2]=>19 [5,2,2,1,1]=>15 [5,2,1,1,1,1]=>11 [5,1,1,1,1,1,1]=>7 [4,4,3]=>26 [4,4,2,1]=>20 [4,4,1,1,1]=>14 [4,3,3,1]=>19 [4,3,2,2]=>21 [4,3,2,1,1]=>16 [4,3,1,1,1,1]=>11 [4,2,2,2,1]=>14 [4,2,2,1,1,1]=>11 [4,2,1,1,1,1,1]=>8 [4,1,1,1,1,1,1,1]=>5 [3,3,3,2]=>14 [3,3,3,1,1]=>10 [3,3,2,2,1]=>12 [3,3,2,1,1,1]=>9 [3,3,1,1,1,1,1]=>6 [3,2,2,2,2]=>11 [3,2,2,2,1,1]=>9 [3,2,2,1,1,1,1]=>7 [3,2,1,1,1,1,1,1]=>5 [3,1,1,1,1,1,1,1,1]=>3 [2,2,2,2,2,1]=>6 [2,2,2,2,1,1,1]=>5 [2,2,2,1,1,1,1,1]=>4 [2,2,1,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1,1,1]=>2 [1,1,1,1,1,1,1,1,1,1,1]=>1 [12]=>77 [11,1]=>56 [10,2]=>62 [10,1,1]=>42 [9,3]=>58 [9,2,1]=>45 [9,1,1,1]=>30 [8,4]=>62 [8,3,1]=>44 [8,2,2]=>44 [8,2,1,1]=>33 [8,1,1,1,1]=>22 [7,5]=>56 [7,4,1]=>44 [7,3,2]=>43 [7,3,1,1]=>30 [7,2,2,1]=>31 [7,2,1,1,1]=>23 [7,1,1,1,1,1]=>15 [6,6]=>53 [6,5,1]=>43 [6,4,2]=>47 [6,4,1,1]=>33 [6,3,3]=>39 [6,3,2,1]=>33 [6,3,1,1,1]=>23 [6,2,2,2]=>29 [6,2,2,1,1]=>23 [6,2,1,1,1,1]=>17 [6,1,1,1,1,1,1]=>11 [5,5,2]=>35 [5,5,1,1]=>25 [5,4,3]=>40 [5,4,2,1]=>31 [5,4,1,1,1]=>22 [5,3,3,1]=>26 [5,3,2,2]=>29 [5,3,2,1,1]=>22 [5,3,1,1,1,1]=>15 [5,2,2,2,1]=>19 [5,2,2,1,1,1]=>15 [5,2,1,1,1,1,1]=>11 [5,1,1,1,1,1,1,1]=>7 [4,4,4]=>30 [4,4,3,1]=>26 [4,4,2,2]=>26 [4,4,2,1,1]=>20 [4,4,1,1,1,1]=>14 [4,3,3,2]=>26 [4,3,3,1,1]=>19 [4,3,2,2,1]=>21 [4,3,2,1,1,1]=>16 [4,3,1,1,1,1,1]=>11 [4,2,2,2,2]=>17 [4,2,2,2,1,1]=>14 [4,2,2,1,1,1,1]=>11 [4,2,1,1,1,1,1,1]=>8 [4,1,1,1,1,1,1,1,1]=>5 [3,3,3,3]=>15 [3,3,3,2,1]=>14 [3,3,3,1,1,1]=>10 [3,3,2,2,2]=>15 [3,3,2,2,1,1]=>12 [3,3,2,1,1,1,1]=>9 [3,3,1,1,1,1,1,1]=>6 [3,2,2,2,2,1]=>11 [3,2,2,2,1,1,1]=>9 [3,2,2,1,1,1,1,1]=>7 [3,2,1,1,1,1,1,1,1]=>5 [3,1,1,1,1,1,1,1,1,1]=>3 [2,2,2,2,2,2]=>7 [2,2,2,2,2,1,1]=>6 [2,2,2,2,1,1,1,1]=>5 [2,2,2,1,1,1,1,1,1]=>4 [2,2,1,1,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1,1,1,1]=>2 [1,1,1,1,1,1,1,1,1,1,1,1]=>1
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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, Günter 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