A353845 Number of integer partitions of n such that if you repeatedly take the multiset of run-sums (or condensation), you eventually reach an empty set or singleton.

1, 1, 2, 2, 4, 2, 5, 2, 8, 3, 5, 2, 15, 2, 5, 4, 18, 2, 13, 2, 14, 4, 5, 2, 62, 3, 5, 5, 14, 2, 18, 2, 48, 4, 5, 4, 71, 2, 5, 4, 54, 2, 18, 2, 14, 10, 5, 2, 374, 3, 9, 4, 14, 2, 37, 4, 54, 4, 5, 2, 131

Offset: 0

Author

Gus Wiseman, May 26 2022

Keywords

• nonn

Comments

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Examples

			The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (3111)               (2222)
                                     (111111)             (4211)
                                                          (41111)
                                                          (221111)
                                                          (11111111)
For example, the partition (3,2,2,2,1,1,1) has trajectory: (1,1,1,2,2,2,3) -> (3,3,6) -> (6,6) -> (12), so is counted under a(12).
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

Dominated by A018818 (partitions into divisors).

The version for compositions is A353858.

A275870 counts collapsible partitions, ranked by A300273.

A304442 counts partitions with all equal run-sums, ranked by A353833.

A325268 counts partitions by omicron, rank statistic A304465.

A353832 represents the operation of taking run-sums of a partition.

A353837 counts partitions with all distinct run-sums, ranked by A353838.

A353840-A353846 pertain to partition run-sum trajectory.

A353847-A353859 pertain to composition run-sum trajectory.

A353864 counts rucksack partitions, ranked by A353866.

Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&,#,!UnsameQ@@#&]]<=1&]],{n,0,30}]