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A373271
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a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.
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4
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1, 2, 3, 6, 10, 14, 24, 34, 49, 70, 103, 134, 195, 258, 347, 461, 624, 796, 1066, 1358, 1763, 2250, 2903, 3631, 4644, 5805, 7309, 9083, 11381, 13998, 17428, 21369, 26336, 32174, 39451, 47847, 58399, 70610, 85590, 103077, 124462, 149169, 179368, 214300, 256397
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OFFSET
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1,2
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COMMENTS
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The multiplicity of a part in an integer partition (or composition) is the number of times it appears in the partition, seen as a list.
The multiplicity of 3 in the partition 12 = 5+3+3+1 is 2.
For this sequence, only distinct multiplicities appearing for parts of the partition are counted, only once for a given partition.
If all multiplicities of all parts of all integer partitions of n are counted, one gets A000070 (1, 2, 4, 7, 12, 19, 30, 45, 67, 97, ...).
If all distinct multiplicities of all parts of all integer partitions are summed, one gets A373273 (1, 3, 5, 11, 18, 29, 48, 74, 107, 161, ...).
If all multiplicities of all parts of all integer partitions of n are summed, one gets A006128 (1, 3, 6, 12, 20, 35, 54, 86, 128, 192, ...).
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LINKS
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EXAMPLE
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Example for n=20:
the partition 20=4+3+3+3+3+2+1+1
has multiplicities 1, 4, 1, 2
for the parts 4,3,2,1 listed in descending order.
It has 3 different multiplicities (1, 2 and 4) and contributes 3 to a(20) = 1358.
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MATHEMATICA
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Table[Plus @@
Table[Plus @@
Map[Length[Union[Length /@ Split[#]]] &,
IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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