A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

1, 1, 2, 2, 4, 3, 6, 5, 9, 8, 11, 11, 20, 16, 20, 21, 32, 30, 41, 38, 50, 48, 62, 64, 89, 81, 97, 100, 123, 123, 151, 154, 187, 183, 221, 221, 279, 272, 312, 316, 377, 376, 446, 460, 531, 547, 628, 641, 754, 746, 841, 856, 990, 1007, 1145, 1167, 1325, 1346, 1519, 1567, 1776

Offset

0,3

Comments

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

Links

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

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

Mathematica

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

Crossrefs

Cf. A100882.

These partitions are ranked by A357861.

The complement is A357865, ranked by A357850.

The opposite version is A304405, ranked by A357875.

The strict version is A304430, ranked by A357864.

The strict opposite version is A304428, ranked by A357862.

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

A000041 counts integer partitions, strict A000009.

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

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

Sequence in context: A062968 A239966 A341465 * A053197 A275234 A301768

Adjacent sequences: A304403 A304404 A304405 * A304407 A304408 A304409

Keyword

nonn

Author

Seiichi Manyama, May 12 2018

Status

approved