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

0,3

Comments

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

Links

Table of n, a(n) for n=0..60.

Formula

a(n) <= A304406(n).

Example

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

Mathematica

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

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.

Sequence in context: A342516 A325391 A179254 * A086609 A341140 A340285

Adjacent sequences: A304427 A304428 A304429 * A304431 A304432 A304433

Keyword

nonn

Author

Seiichi Manyama, May 12 2018

Status

approved