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A353858
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Number of integer compositions of n with run-sum trajectory ending in a singleton.
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7
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0, 1, 2, 2, 5, 2, 8, 2, 20, 5, 8, 2, 78, 2, 8, 8, 223, 2, 179, 2, 142, 8, 8, 2, 4808
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&, #]]==1&]], {n, 0, 15}]
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CROSSREFS
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The lengths of trajectories of standard compositions are A353854.
These compositions are ranked by A353857.
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.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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