A304430 Number of partitions of n in which the sequence of the sum of the same summands is decreasing.

1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 10, 10, 13, 15, 18, 19, 22, 26, 33, 33, 38, 41, 50, 53, 60, 68, 77, 84, 94, 100, 116, 122, 136, 148, 172, 182, 206, 219, 246, 258, 281, 301, 341, 365, 397, 429, 466, 489, 528, 572, 623, 660, 728, 773, 849, 895, 968, 1019, 1120, 1188, 1288

Offset: 0

Author

Seiichi Manyama, May 12 2018

Keywords

• nonn

Comments

Number of integer partitions of n with strictly increasing run-sums. - Gus Wiseman, Oct 22 2022

Examples

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 2+1+1+1              | 3, 2
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 2+2+2                | 6
  | 2+1+1+1+1            | 4, 2
  | 1+1+1+1+1+1          | 6
		

Crossrefs

The weak opposite version is A304405, ranked by A357875.

The weak version is A304406, ranked by A357861.

The opposite version is A304428, ranked by A357862.

Number of rows in A354584 summing to n that are strictly decreasing.

These partitions are ranked by A357864.

A000041 counts integer partitions, strict A000009.

A304442 counts partitions with equal run-sums, distinct A353837.

Cf. A047966, A087980, A098859, A100471, A100881, A239312, A275870, A353832, A353833, A353864.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Less@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 22 2022 *)

Formula

a(n) <= A304406(n).