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A332283 Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

%I A332283 #12 Feb 16 2025 08:33:59
%S A332283 1,1,2,3,5,7,10,13,18,24,30,38,49,59,73,90,108,129,159,184,216,258,
%T A332283 298,347,410,466,538,626,707,807,931,1043,1181,1351,1506,1691,1924,
%U A332283 2132,2382,2688,2971,3300,3704,4073,4500,5021,5510,6065,6740,7362,8078
%N A332283 Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.
%C A332283 First differs from A000041 at a(6) = 10, A000041(6) = 11.
%C A332283 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332283 Fausto A. C. Cariboni, <a href="/A332283/b332283.txt">Table of n, a(n) for n = 0..400</a>
%H A332283 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%e A332283 The a(1) = 1 through a(7) = 13 partitions:
%e A332283   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A332283        (11)  (21)   (22)    (32)     (33)      (43)
%e A332283              (111)  (31)    (41)     (42)      (52)
%e A332283                     (211)   (221)    (51)      (61)
%e A332283                     (1111)  (311)    (222)     (322)
%e A332283                             (2111)   (321)     (421)
%e A332283                             (11111)  (411)     (511)
%e A332283                                      (3111)    (2221)
%e A332283                                      (21111)   (3211)
%e A332283                                      (111111)  (4111)
%e A332283                                                (31111)
%e A332283                                                (211111)
%e A332283                                                (1111111)
%t A332283 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332283 Table[Length[Select[IntegerPartitions[n],unimodQ[Differences[Append[#,0]]]&]],{n,0,30}]
%Y A332283 Unimodal compositions are A001523.
%Y A332283 Unimodal normal sequences appear to be A007052.
%Y A332283 Partitions with unimodal run-lengths are A332280.
%Y A332283 Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y A332283 The complement is counted by A332284.
%Y A332283 The strict case is A332285.
%Y A332283 Heinz numbers of partitions not in this class are A332287.
%Y A332283 Cf. A025065, A072706, A115981, A227038, A332288, A332577, A332638, A332642.
%K A332283 nonn
%O A332283 0,3
%A A332283 _Gus Wiseman_, Feb 19 2020