A373269 T(n,k) is the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 1

Offset: 1

Author

Olivier Gérard, May 29 2024

Keywords

• nonn
• tabf

Comments

The regular array for partitions of n of length k is A373270.

Row sums are A373271.

Examples

			Array begins:
  1,
  1,1,
  1,1,1,
  1,1,1,2,1,
  1,1,1,2,2,2,1,
  1,1,1,2,1,1,2,1,1,2,1,
  1,1,1,2,1,1,2,2,2,2,2,2,2,2,1
  ...
T(10,34) is the first term with value 3. It corresponds to partition 3+2+2+1+1+1 of 10, which has three different multiplicities.
		

Links

• Olivier Gérard, Table of n, a(n) for n = 1..215307

Programs

Mathematica

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