A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 52, 61, 80, 97, 127, 147, 189, 220, 277, 325, 402, 469, 578, 665, 804, 933, 1121, 1282, 1537, 1754, 2081, 2374, 2793, 3179, 3739, 4232, 4923, 5587, 6477, 7305, 8445, 9519, 10949, 12323, 14110, 15825, 18099, 20229, 23005

Offset

0,3

Comments

Number of integer partitions of n with weakly decreasing run-sums, complement A357878. - Gus Wiseman, Oct 22 2022

Links

Table of n, a(n) for n=0..50.

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
  | 2+1+1                | 2, 2
  | 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
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

Mathematica

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

Crossrefs

The strict opposite version is A304430, ranked by A357864.

The strict version is A304428, ranked by A357862.

The opposite version is A304406, ranked by A357861.

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

These partitions are ranked by A357875.

A000041 counts integer partitions, strict A000009.

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

Cf. A047966, A087980, A098859, A100883, A239312, A275870, A353832, A353864.

Sequence in context: A130898 A199016 A088314 * A347446 A097071 A105420

Adjacent sequences: A304402 A304403 A304404 * A304406 A304407 A304408

Keyword

nonn

Author

Seiichi Manyama, May 12 2018

Status

approved