|
|
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.
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
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).
|
|
LINKS
|
|
|
EXAMPLE
|
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).
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&, #, !UnsameQ@@#&]]<=1&]], {n, 0, 30}]
|
|
CROSSREFS
|
Dominated by A018818 (partitions into divisors).
The version for compositions is A353858.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|