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A332669
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Number of compositions of n whose negation is not unimodal.
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37
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0, 0, 0, 0, 1, 3, 11, 28, 71, 165, 372, 807, 1725, 3611, 7481, 15345, 31274, 63392, 128040, 257865, 518318, 1040277, 2085714, 4178596, 8367205, 16748151, 33515214, 67056139, 134147231, 268341515, 536746350, 1073577185, 2147266984, 4294683056, 8589563136, 17179385180
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OFFSET
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0,6
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COMMENTS
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A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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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a(n) + A332578(n) = 2^(n - 1) for n > 0.
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EXAMPLE
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The a(4) = 1 through a(6) = 11 compositions:
(121) (131) (132)
(1121) (141)
(1211) (231)
(1131)
(1212)
(1221)
(1311)
(2121)
(11121)
(11211)
(12111)
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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[Join@@Permutations/@IntegerPartitions[n], !unimodQ[-#]&]], {n, 0, 10}]
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CROSSREFS
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The complement is counted by A332578.
The version for run-lengths of partitions is A332639.
The version for unsorted prime signature is A332642.
The version for 0-appended first-differences of partitions is A332744.
The case that is not unimodal either is A332870.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
A triangle for compositions with unimodal negation is A332670.
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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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