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). The run-sum trajectory is obtained by repeatedly taking the run-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).
EXAMPLE
The a(0) = 0 through a(8) = 20 compositions:
. (1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (11111) (33) (1111111) (44)
(112) (222) (224)
(211) (1113) (422)
(1111) (2112) (1124)
(3111) (2114)
(11211) (2222)
(111111) (4112)
(4211)
(11114)
(21122)
(22112)
(41111)
(111122)
(112112)
(211211)
(221111)
(1111211)
(1121111)
(11111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&, #]]==1&]], {n, 0, 15}]
CROSSREFS
The lengths of trajectories of standard compositions are A353854.
These compositions are ranked by A353857.
A011782 counts compositions.
A066099 lists compositions in standard order.
A353859 counts compositions by length of run-sum trajectory.
A353860 counts collapsible compositions.
A353932 lists run-sums of standard compositions.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 17 2022
STATUS
approved