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%I #14 Jan 22 2024 13:02:06
%S 0,0,0,0,0,0,1,3,8,28,74,188,468,1120,2596,5944,13324,29437,64288,
%T 138929,297442,632074,1333897,2798352,5840164,12132638,25102232,
%U 51750419,106346704,217921161,445424102,908376235,1848753273,3755839591,7617835520,15428584567,31207263000
%N Number of compositions of n whose run-lengths are not unimodal.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A composition of n is a finite sequence of positive integers summing to n.
%H Andrew Howroyd, <a href="/A332727/b332727.txt">Table of n, a(n) for n = 0..500</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F a(n) + A332726(n) = 2^(n - 1).
%e The a(6) = 1 through a(8) = 8 compositions:
%e (11211) (11311) (11411)
%e (111211) (111311)
%e (112111) (112112)
%e (113111)
%e (211211)
%e (1111211)
%e (1112111)
%e (1121111)
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,10}]
%Y Looking at the composition itself (not its run-lengths) gives A115981.
%Y The case of partitions is A332281, with complement counted by A332280.
%Y The complement is counted by A332726.
%Y Unimodal compositions are A001523.
%Y Non-unimodal normal sequences are A328509.
%Y Compositions with normal run-lengths are A329766.
%Y Numbers whose prime signature is not unimodal are A332282.
%Y Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
%Y Compositions whose negation is not unimodal are A332669.
%Y Compositions whose run-lengths are weakly increasing are A332836.
%Y Cf. A007052, A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332578, A332638, A332639, A332670, A332741, A332833.
%K nonn
%O 0,8
%A _Gus Wiseman_, Feb 29 2020
%E Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2020
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