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A332638 Number of integer partitions of n whose negated run-lengths are unimodal.

%I A332638 #12 Feb 16 2025 08:33:59
%S A332638 1,1,2,3,5,7,11,15,21,29,40,52,70,91,118,151,195,246,310,388,484,600,
%T A332638 743,909,1113,1359,1650,1996,2409,2895,3471,4156,4947,5885,6985,8260,
%U A332638 9751,11503,13511,15857,18559,21705,25304,29499,34259,39785,46101,53360,61594
%N A332638 Number of integer partitions of n whose negated run-lengths are unimodal.
%C A332638 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332638 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332638 The a(8) = 21 partitions:
%e A332638   (8)     (44)     (2222)
%e A332638   (53)    (332)    (22211)
%e A332638   (62)    (422)    (32111)
%e A332638   (71)    (431)    (221111)
%e A332638   (521)   (3311)   (311111)
%e A332638   (611)   (4211)   (2111111)
%e A332638   (5111)  (41111)  (11111111)
%e A332638 Missing from this list is only (3221).
%t A332638 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332638 Table[Length[Select[IntegerPartitions[n],unimodQ[-Length/@Split[#]]&]],{n,0,30}]
%Y A332638 The non-negated version is A332280.
%Y A332638 The complement is counted by A332639.
%Y A332638 The Heinz numbers of partitions not in this class are A332642.
%Y A332638 The case of 0-appended differences (instead of run-lengths) is A332728.
%Y A332638 Unimodal compositions are A001523.
%Y A332638 Partitions whose run lengths are not unimodal are A332281.
%Y A332638 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332638 Compositions whose negation is unimodal are A332578.
%Y A332638 Compositions whose run-lengths are unimodal are A332726.
%Y A332638 Cf. A007052, A100883, A115981, A181819, A332283, A332577, A332640, A332669, A332670, A332727.
%K A332638 nonn
%O A332638 0,3
%A A332638 _Gus Wiseman_, Feb 25 2020