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A332639 Number of integer partitions of n whose negated run-lengths are not unimodal.

%I A332639 #7 Feb 16 2025 08:33:59
%S A332639 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 A332639 599,786,1014,1309,1670,2133,2686,3402,4258,5325,6623,8226,10134,
%U A332639 12504,15328,18779,22878,27870,33762,40916,49349,59457,71394,85679,102394
%N A332639 Number of integer partitions of n whose negated run-lengths are not unimodal.
%C A332639 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332639 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332639 The a(8) = 1 through a(13) = 10 partitions:
%e A332639   (3221)  (4221)  (5221)   (4331)    (4332)    (5332)
%e A332639                   (32221)  (6221)    (5331)    (6331)
%e A332639                            (42221)   (7221)    (8221)
%e A332639                            (322211)  (43221)   (43321)
%e A332639                                      (52221)   (53221)
%e A332639                                      (322221)  (62221)
%e A332639                                      (422211)  (332221)
%e A332639                                                (422221)
%e A332639                                                (522211)
%e A332639                                                (3222211)
%t A332639 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332639 Table[Length[Select[IntegerPartitions[n],!unimodQ[-Length/@Split[#]]&]],{n,0,30}]
%Y A332639 The version for normal sequences is A328509.
%Y A332639 The non-negated complement is A332280.
%Y A332639 The non-negated version is A332281.
%Y A332639 The complement is counted by A332638.
%Y A332639 The case that is not unimodal either is A332640.
%Y A332639 The Heinz numbers of these partitions are A332642.
%Y A332639 The generalization to run-lengths of compositions is A332727.
%Y A332639 Unimodal compositions are A001523.
%Y A332639 Non-unimodal permutations are A059204.
%Y A332639 Non-unimodal compositions are A115981.
%Y A332639 Compositions whose negation is not unimodal are A332669.
%Y A332639 Cf. A007052, A025065, A100883, A181819, A332282, A332578, A332579, A332641, A332670, A332671, A332726, A332742, A332744.
%K A332639 nonn
%O A332639 0,11
%A A332639 _Gus Wiseman_, Feb 25 2020