A373271 a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.

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

Alois P. Heinz, Table of n, a(n) for n = 1..200

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

Cf. A373243, A098859.

Sequence in context: A090035 A337611 A325724 * A111467 A215891 A254033

Adjacent sequences: A373268 A373269 A373270 * A373272 A373273 A373274

Keyword

nonn

Author

Olivier Gérard, May 29 2024

Status

approved