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A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

%I A332744 #16 Feb 16 2025 08:33:59
%S A332744 0,0,0,0,1,2,4,7,12,17,28,39,55,77,107,142,194,254,332,434,563,716,
%T A332744 919,1162,1464,1841,2305,2857,3555,4383,5394,6617,8099,9859,12006,
%U A332744 14551,17600,21236,25574,30688,36809,44007,52527,62574,74430,88304,104675,123799
%N A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.
%C A332744 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332744 Fausto A. C. Cariboni, <a href="/A332744/b332744.txt">Table of n, a(n) for n = 0..600</a>
%H A332744 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%H A332744 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A332744 The a(4) = 1 through a(9) = 17 partitions:
%e A332744   (211)  (311)   (411)    (322)     (422)      (522)
%e A332744          (2111)  (2211)   (511)     (611)      (711)
%e A332744                  (3111)   (3211)    (3221)     (3222)
%e A332744                  (21111)  (4111)    (3311)     (4221)
%e A332744                           (22111)   (4211)     (4311)
%e A332744                           (31111)   (5111)     (5211)
%e A332744                           (211111)  (22211)    (6111)
%e A332744                                     (32111)    (32211)
%e A332744                                     (41111)    (33111)
%e A332744                                     (221111)   (42111)
%e A332744                                     (311111)   (51111)
%e A332744                                     (2111111)  (222111)
%e A332744                                                (321111)
%e A332744                                                (411111)
%e A332744                                                (2211111)
%e A332744                                                (3111111)
%e A332744                                                (21111111)
%e A332744 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 A332744 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332744 Table[Length[Select[IntegerPartitions[n],!unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]
%Y A332744 The complement is counted by A332728.
%Y A332744 The non-negated version is A332284.
%Y A332744 The strict case is A332579.
%Y A332744 The case of run-lengths (instead of differences) is A332639.
%Y A332744 The Heinz numbers of these partitions are A332832.
%Y A332744 Unimodal compositions are A001523.
%Y A332744 Non-unimodal compositions are A115981.
%Y A332744 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332744 Partitions whose 0-appended first differences are unimodal are A332283.
%Y A332744 Compositions whose negation is unimodal are A332578.
%Y A332744 Numbers whose negated prime signature is not unimodal are A332642.
%Y A332744 Compositions whose negation is not unimodal are A332669.
%Y A332744 Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.
%K A332744 nonn
%O A332744 0,6
%A A332744 _Gus Wiseman_, Feb 27 2020