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A332285 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

%I A332285 #15 Feb 16 2025 08:33:59
%S A332285 1,1,1,2,2,3,4,5,5,8,9,11,13,15,17,22,25,29,34,39,42,53,58,64,75,84,
%T A332285 93,111,122,134,152,169,184,212,232,252,287,315,342,389,419,458,512,
%U A332285 556,602,672,727,787,870,940,1012,1124,1209,1303,1431,1540,1655,1821
%N A332285 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are unimodal.
%C A332285 First differs from A000009 at a(8) = 5, A000009(8) = 6.
%C A332285 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332285 Fausto A. C. Cariboni, <a href="/A332285/b332285.txt">Table of n, a(n) for n = 0..500</a>
%H A332285 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%e A332285 The a(1) = 1 through a(9) = 8 partitions:
%e A332285   (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)
%e A332285             (21)  (31)  (32)  (42)   (43)   (53)   (54)
%e A332285                         (41)  (51)   (52)   (62)   (63)
%e A332285                               (321)  (61)   (71)   (72)
%e A332285                                      (421)  (521)  (81)
%e A332285                                                    (432)
%e A332285                                                    (531)
%e A332285                                                    (621)
%e A332285 For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is not counted under a(8).
%t A332285 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332285 Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,unimodQ[Differences[Append[#,0]]]]&]],{n,0,30}]
%Y A332285 The non-strict version is A332283.
%Y A332285 The complement is counted by A332286.
%Y A332285 Unimodal compositions are A001523.
%Y A332285 Unimodal normal sequences appear to be A007052.
%Y A332285 Unimodal permutations are A011782.
%Y A332285 Partitions with unimodal run-lengths are A332280.
%Y A332285 Cf. A025065, A072706, A115981, A227038, A332282, A332284, A332287, A332288, A332577, A332638, A332642.
%K A332285 nonn
%O A332285 0,4
%A A332285 _Gus Wiseman_, Feb 21 2020