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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.

%I A353845 #8 May 30 2022 23:35:47
%S A353845 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,
%T A353845 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,
%U A353845 131
%N 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.
%C A353845 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).
%H A353845 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A353845 The a(1) = 1 through a(8) = 8 partitions:
%e A353845   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A353845        (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
%e A353845                     (211)            (222)                (422)
%e A353845                     (1111)           (3111)               (2222)
%e A353845                                      (111111)             (4211)
%e A353845                                                           (41111)
%e A353845                                                           (221111)
%e A353845                                                           (11111111)
%e A353845 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).
%t A353845 Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&,#,!UnsameQ@@#&]]<=1&]],{n,0,30}]
%Y A353845 Dominated by A018818 (partitions into divisors).
%Y A353845 The version for compositions is A353858.
%Y A353845 A275870 counts collapsible partitions, ranked by A300273.
%Y A353845 A304442 counts partitions with all equal run-sums, ranked by A353833.
%Y A353845 A325268 counts partitions by omicron, rank statistic A304465.
%Y A353845 A353832 represents the operation of taking run-sums of a partition.
%Y A353845 A353837 counts partitions with all distinct run-sums, ranked by A353838.
%Y A353845 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353845 A353847-A353859 pertain to composition run-sum trajectory.
%Y A353845 A353864 counts rucksack partitions, ranked by A353866.
%Y A353845 Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.
%K A353845 nonn
%O A353845 0,3
%A A353845 _Gus Wiseman_, May 26 2022