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A304405
Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.
12
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
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.
Sequence in context: A130898 A199016 A088314 * A347446 A097071 A105420
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 12 2018
STATUS
approved