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A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.

%I A332579 #11 Feb 16 2025 08:33:59
%S A332579 0,0,0,0,0,0,0,0,0,0,1,1,1,2,3,4,7,8,10,14,19,22,30,36,43,56,69,80,
%T A332579 101,121,141,172,202,234,282,332,384,452,527,602,706,815,929,1077,
%U A332579 1236,1403,1615,1842,2082,2379,2702,3044,3458,3908,4388,4963,5589,6252
%N A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.
%C A332579 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332579 Also the number of strict integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.
%H A332579 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332579 The a(10) = 1 through a(16) = 7 partitions:
%e A332579   33211  332111  3321111  333211    433211     443211      443221
%e A332579                           33211111  3332111    4332111     3333211
%e A332579                                     332111111  33321111    4432111
%e A332579                                                3321111111  33322111
%e A332579                                                            43321111
%e A332579                                                            333211111
%e A332579                                                            33211111111
%t A332579 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A332579 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332579 Table[Length[Select[IntegerPartitions[n],normQ[#]&&!unimodQ[Length/@Split[#]]&]],{n,0,30}]
%Y A332579 The complement is counted by A332577.
%Y A332579 Not requiring the partition to cover an initial interval gives A332281.
%Y A332579 The opposite version is A332286.
%Y A332579 A version for compositions is A332743.
%Y A332579 Partitions covering an initial interval of positive integers are A000009.
%Y A332579 Unimodal compositions are A001523.
%Y A332579 Non-unimodal permutations are A059204.
%Y A332579 Non-unimodal compositions are A115981.
%Y A332579 Non-unimodal normal sequences are A328509.
%Y A332579 Numbers whose prime signature is not unimodal are A332282.
%Y A332579 Partitions whose 0-appended first differences are unimodal are A332283.
%Y A332579 Compositions whose negated run-lengths are not unimodal are A332727.
%Y A332579 Cf. A007052, A100883, A107429, A227038, A332280, A332284, A332638, A332639, A332640, A332671, A332672, A332728.
%K A332579 nonn
%O A332579 0,14
%A A332579 _Gus Wiseman_, Feb 25 2020