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A332835
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Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.
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20
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1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031
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OFFSET
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0,3
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
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LINKS
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FORMULA
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EXAMPLE
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The a(6) = 29 compositions:
(6) (141) (213) (1113) (21111)
(51) (114) (132) (222) (12111)
(15) (33) (123) (2211) (11121)
(42) (321) (3111) (2121) (11112)
(24) (312) (1311) (1212) (111111)
(411) (231) (1131) (1122)
Missing are: (2112), (1221), (11211).
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Or[LessEqual@@Length/@Split[#], GreaterEqual@@Length/@Split[#]]&]], {n, 0, 20}]
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CROSSREFS
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The version for the compositions themselves (not run-lengths) is A329398.
Compositions with equal run-lengths are A329738.
The version for unsorted prime signature is the complement of A332831.
The complement is counted by A332833.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Neither weakly increasing nor weakly decreasing compositions are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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