A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

1, 1, 2, 3, 4, 6, 9, 11, 14, 20, 26, 33, 41, 50, 64, 81, 97, 120, 150, 176, 210, 255, 303, 362, 426, 503, 595, 703, 816, 953, 1113, 1283, 1482, 1721, 1988, 2299, 2650, 3031, 3464, 3965, 4492, 5115, 5820, 6592, 7467, 8484, 9568, 10822, 12185, 13724, 15445, 17381, 19475, 21855

Offset

0,3

Comments

Number of integer partitions of n with strictly decreasing run-sums. - Gus Wiseman, Oct 21 2022

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 71 terms from Seiichi Manyama)

Formula

a(n) <= A304405(n).

Example

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

Mathematica

Table[Length[Select[IntegerPartitions[n], Greater@@Total/@Split[#]&]], {n, 0, 30}] (* Gus Wiseman, Oct 21 2022 *)

Crossrefs

The weak version is A304405, ranked by A357875.

The weak opposite version is A304406, ranked by A357861.

The opposite version is A304430, ranked by A357864.

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

These partitions are ranked by A357862, complement A357863.

A000041 counts integer partitions, strict A000009.

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

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

Sequence in context: A022956 A331444 A039865 * A132134 A027594 A130899

Adjacent sequences: A304425 A304426 A304427 * A304429 A304430 A304431

Keyword

nonn

Author

Seiichi Manyama, May 12 2018

Status

approved