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%I #6 Feb 26 2020 17:19:00
%S 0,0,0,0,0,0,0,0,1,1,2,4,7,10,17,25,36,51,75,102,143,192,259,346,462,
%T 599,786,1014,1309,1670,2133,2686,3402,4258,5325,6623,8226,10134,
%U 12504,15328,18779,22878,27870,33762,40916,49349,59457,71394,85679,102394
%N Number of integer partitions of n whose negated run-lengths are not unimodal.
%C A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e The a(8) = 1 through a(13) = 10 partitions:
%e (3221) (4221) (5221) (4331) (4332) (5332)
%e (32221) (6221) (5331) (6331)
%e (42221) (7221) (8221)
%e (322211) (43221) (43321)
%e (52221) (53221)
%e (322221) (62221)
%e (422211) (332221)
%e (422221)
%e (522211)
%e (3222211)
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t Table[Length[Select[IntegerPartitions[n],!unimodQ[-Length/@Split[#]]&]],{n,0,30}]
%Y The version for normal sequences is A328509.
%Y The non-negated complement is A332280.
%Y The non-negated version is A332281.
%Y The complement is counted by A332638.
%Y The case that is not unimodal either is A332640.
%Y The Heinz numbers of these partitions are A332642.
%Y The generalization to run-lengths of compositions is A332727.
%Y Unimodal compositions are A001523.
%Y Non-unimodal permutations are A059204.
%Y Non-unimodal compositions are A115981.
%Y Compositions whose negation is not unimodal are A332669.
%Y Cf. A007052, A025065, A100883, A181819, A332282, A332578, A332579, A332641, A332670, A332671, A332726, A332742, A332744.
%K nonn
%O 0,11
%A _Gus Wiseman_, Feb 25 2020
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