A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 4, 3, 2, 1, 1, 3, 7, 6, 4, 2, 1, 1, 4, 8, 8, 6, 4, 2, 1, 1, 4, 10, 12, 10, 5, 4, 2, 1, 1, 5, 12, 15, 13, 11, 6, 4, 2, 1, 1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1, 1, 6, 16, 25, 26, 21, 16, 10, 6, 4, 2, 1, 1, 6, 20, 33, 36, 34, 24, 17, 11, 6, 4, 2, 1, 1, 7, 22, 38, 46, 44, 34, 25, 17, 11, 6, 4, 2, 1, 1, 7, 25, 48, 58, 56, 50, 38, 24, 16, 11, 6, 4, 2, 1

Offset: 1

Author

Olivier Gérard, May 29 2024

Keywords

• nonn
• tabl

Examples

			Array begins:
  1,
  1, 1,
  1, 1,  1,
  1, 2,  2,  1,
  1, 2,  4,  2,  1,
  1, 3,  4,  3,  2,  1,
  1, 3,  7,  6,  4,  2,  1,
  1, 4,  8,  8,  6,  4,  2, 1,
  1, 4, 10, 12, 10,  5,  4, 2, 1,
  1, 5, 12, 15, 13, 11,  6, 4, 2, 1,
  1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1,
  ...
Example of computation:
T(9,3) = 10 because the partitions of 9 into 3 parts are
  7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
the number of different multiplicities are
  2, 1, 1, 2, 2, 1, 1,
and the sum of these multiplicities is 10.
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened (first 40 rows from Olivier Gérard)

Crossrefs

Cf. A373242, A373269, A373271.

Programs

Mathematica

Flatten@Table[
  Plus @@@
   Table[Map[Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]