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A332640
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Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.
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16
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 17, 29, 44, 66, 92, 138, 187, 266, 359, 492, 649, 877, 1140, 1503, 1938, 2517, 3202, 4111, 5175, 6563, 8209, 10297, 12763, 15898, 19568, 24152, 29575, 36249, 44090, 53737, 65022, 78752, 94873, 114294
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OFFSET
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0,16
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COMMENTS
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A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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LINKS
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EXAMPLE
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The a(14) = 1 through a(18) = 12 partitions:
(433211) (533211) (443221) (544211) (544311)
(4332111) (633211) (733211) (553221)
(5332111) (4333211) (644211)
(43321111) (6332111) (833211)
(53321111) (4432221)
(433211111) (5333211)
(5442111)
(7332111)
(43332111)
(63321111)
(533211111)
(4332111111)
For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).
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MATHEMATICA
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unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], !unimodQ[Length/@Split[#]]&&!unimodQ[-Length/@Split[#]]&]], {n, 0, 30}]
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CROSSREFS
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Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The Heinz numbers of these partitions are A332643.
The complement is counted by A332746.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths and negated run-lengths are not both unimodal: A332641.
Compositions whose negation is not unimodal are A332669.
Run-lengths and negated run-lengths are both unimodal: A332745.
Cf. A007052, A025065, A100883, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332642, A332726, A332727.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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