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A332284 Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

%I A332284 #13 Feb 16 2025 08:33:59
%S A332284 0,0,0,0,0,0,1,2,4,6,12,18,28,42,62,86,123,168,226,306,411,534,704,
%T A332284 908,1165,1492,1898,2384,3011,3758,4673,5799,7168,8792,10804,13192,
%U A332284 16053,19505,23633,28497,34367,41283,49470,59188,70675,84113,100048,118689,140533
%N A332284 Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
%C A332284 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332284 Fausto A. C. Cariboni, <a href="/A332284/b332284.txt">Table of n, a(n) for n = 0..400</a>
%H A332284 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%e A332284 The a(6) = 1 through a(11) = 18 partitions:
%e A332284   (2211)  (331)    (431)     (441)      (541)       (551)
%e A332284           (22111)  (3311)    (4311)     (3322)      (641)
%e A332284                    (22211)   (32211)    (3331)      (4331)
%e A332284                    (221111)  (33111)    (4411)      (4421)
%e A332284                              (222111)   (33211)     (5411)
%e A332284                              (2211111)  (42211)     (33221)
%e A332284                                         (43111)     (33311)
%e A332284                                         (222211)    (44111)
%e A332284                                         (322111)    (52211)
%e A332284                                         (331111)    (322211)
%e A332284                                         (2221111)   (332111)
%e A332284                                         (22111111)  (422111)
%e A332284                                                     (431111)
%e A332284                                                     (2222111)
%e A332284                                                     (3221111)
%e A332284                                                     (3311111)
%e A332284                                                     (22211111)
%e A332284                                                     (221111111)
%t A332284 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332284 Table[Length[Select[IntegerPartitions[n],!unimodQ[Differences[Append[#,0]]]&]],{n,30}]
%Y A332284 The complement is counted by A332283.
%Y A332284 The strict version is A332286.
%Y A332284 The Heinz numbers of these partitions are A332287.
%Y A332284 Non-unimodal permutations are A059204.
%Y A332284 Non-unimodal compositions are A115981.
%Y A332284 Non-unimodal normal sequences appear to be A328509.
%Y A332284 Partitions with non-unimodal run-lengths are A332281.
%Y A332284 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332284 Cf. A001523, A007052, A332280, A332285, A332288, A332579, A332638, A332639, A332640, A332642.
%K A332284 nonn
%O A332284 0,8
%A A332284 _Gus Wiseman_, Feb 20 2020