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A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

%I A332286 #21 Feb 16 2025 08:33:59
%S A332286 0,0,0,0,0,0,0,0,1,0,1,1,2,3,5,5,7,9,12,15,22,23,31,40,47,58,72,81,
%T A332286 100,122,144,171,206,236,280,333,381,445,522,593,694,802,914,1054,
%U A332286 1214,1376,1577,1803,2040,2324,2646,2973,3373,3817,4287,4838,5453,6096,6857
%N A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
%C A332286 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332286 Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.
%H A332286 Fausto A. C. Cariboni, <a href="/A332286/b332286.txt">Table of n, a(n) for n = 0..500</a>
%H A332286 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%e A332286 The a(8) = 1 through a(18) = 7 partitions:
%e A332286   (431)  .  (541)  (641)  (651)   (652)   (752)   (762)   (862)
%e A332286                           (5421)  (751)   (761)   (861)   (871)
%e A332286                                   (5431)  (851)   (6531)  (961)
%e A332286                                           (6431)  (7431)  (6532)
%e A332286                                           (6521)  (7521)  (6541)
%e A332286                                                           (7621)
%e A332286                                                           (8431)
%e A332286 For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).
%t A332286 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332286 Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!unimodQ[Differences[Append[#,0]]]]&]],{n,0,30}]
%Y A332286 Strict partitions are A000009.
%Y A332286 Partitions covering an initial interval are (also) A000009.
%Y A332286 The non-strict version is A332284.
%Y A332286 The complement is counted by A332285.
%Y A332286 Unimodal compositions are A001523.
%Y A332286 Non-unimodal permutations are A059204.
%Y A332286 Non-unimodal compositions are A115981.
%Y A332286 Non-unimodal normal sequences are A328509.
%Y A332286 Partitions with non-unimodal run-lengths are A332281.
%Y A332286 Normal partitions whose run-lengths are not unimodal are A332579.
%Y A332286 Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.
%K A332286 nonn
%O A332286 0,13
%A A332286 _Gus Wiseman_, Feb 21 2020