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

Author

Olivier Gérard, May 29 2024

Keywords

• nonn

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, ...).

Examples

			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.
		

Links

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

Crossrefs

Cf. A373243, A098859.

Programs

Mathematica

Table[Plus @@
  Table[Plus @@
    Map[Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]