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%I #15 Jan 21 2024 11:07:08
%S 0,0,0,0,1,2,4,7,12,17,28,39,55,77,107,142,194,254,332,434,563,716,
%T 919,1162,1464,1841,2305,2857,3555,4383,5394,6617,8099,9859,12006,
%U 14551,17600,21236,25574,30688,36809,44007,52527,62574,74430,88304,104675,123799
%N Number of integer partitions of n whose negated first differences (assuming the last part is zero) 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 Fausto A. C. Cariboni, <a href="/A332744/b332744.txt">Table of n, a(n) for n = 0..600</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e The a(4) = 1 through a(9) = 17 partitions:
%e (211) (311) (411) (322) (422) (522)
%e (2111) (2211) (511) (611) (711)
%e (3111) (3211) (3221) (3222)
%e (21111) (4111) (3311) (4221)
%e (22111) (4211) (4311)
%e (31111) (5111) (5211)
%e (211111) (22211) (6111)
%e (32111) (32211)
%e (41111) (33111)
%e (221111) (42111)
%e (311111) (51111)
%e (2111111) (222111)
%e (321111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%e For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).
%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[-Differences[Append[#,0]]]&]],{n,0,30}]
%Y The complement is counted by A332728.
%Y The non-negated version is A332284.
%Y The strict case is A332579.
%Y The case of run-lengths (instead of differences) is A332639.
%Y The Heinz numbers of these partitions are A332832.
%Y Unimodal compositions are A001523.
%Y Non-unimodal compositions are A115981.
%Y Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y Partitions whose 0-appended first differences are unimodal are A332283.
%Y Compositions whose negation is unimodal are A332578.
%Y Numbers whose negated prime signature is not unimodal are A332642.
%Y Compositions whose negation is not unimodal are A332669.
%Y Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.
%K nonn
%O 0,6
%A _Gus Wiseman_, Feb 27 2020
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