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A373271
a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.
4
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
OFFSET
1,2
COMMENTS
Sum of the rows of A373269.
Sum of the rows of A373270.
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, ...).
LINKS
EXAMPLE
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.
MATHEMATICA
Table[Plus @@
Table[Plus @@
Map[Length[Union[Length /@ Split[#]]] &,
IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]
CROSSREFS
Sequence in context: A090035 A337611 A325724 * A111467 A215891 A254033
KEYWORD
nonn
AUTHOR
Olivier Gérard, May 29 2024
STATUS
approved