%I #11 Mar 06 2020 11:45:42
%S 1,1,2,3,5,7,11,15,21,29,40,52,70,91,118,151,195,246,310,388,484,600,
%T 743,909,1113,1359,1650,1996,2409,2895,3471,4156,4947,5885,6985,8260,
%U 9751,11503,13511,15857,18559,21705,25304,29499,34259,39785,46101,53360,61594
%N Number of integer partitions of n whose negated run-lengths are 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) = 21 partitions:
%e (8) (44) (2222)
%e (53) (332) (22211)
%e (62) (422) (32111)
%e (71) (431) (221111)
%e (521) (3311) (311111)
%e (611) (4211) (2111111)
%e (5111) (41111) (11111111)
%e Missing from this list is only (3221).
%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 non-negated version is A332280.
%Y The complement is counted by A332639.
%Y The Heinz numbers of partitions not in this class are A332642.
%Y The case of 0-appended differences (instead of run-lengths) is A332728.
%Y Unimodal compositions are A001523.
%Y Partitions whose run lengths are not unimodal are A332281.
%Y Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y Compositions whose negation is unimodal are A332578.
%Y Compositions whose run-lengths are unimodal are A332726.
%Y Cf. A007052, A100883, A115981, A181819, A332283, A332577, A332640, A332669, A332670, A332727.
%K nonn
%O 0,3
%A _Gus Wiseman_, Feb 25 2020