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A332728 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.

%I A332728 #16 Feb 16 2025 08:33:59
%S A332728 1,1,2,3,4,5,7,8,10,13,14,17,22,24,28,34,37,43,53,56,64,76,83,93,111,
%T A332728 117,131,153,163,182,210,225,250,284,304,332,377,401,441,497,529,576,
%U A332728 647,687,745,830,883,955,1062,1127,1216,1339,1422,1532,1684,1779,1914
%N A332728 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.
%C A332728 First differs from A000041 at a(6) = 10, A000041(6) = 11.
%C A332728 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332728 Fausto A. C. Cariboni, <a href="/A332728/b332728.txt">Table of n, a(n) for n = 0..600</a>
%H A332728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%H A332728 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%e A332728 The a(1) = 1 through a(8) = 10 partitions:
%e A332728   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A332728        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A332728              (111)  (31)    (41)     (42)      (52)       (53)
%e A332728                     (1111)  (221)    (51)      (61)       (62)
%e A332728                             (11111)  (222)     (331)      (71)
%e A332728                                      (321)     (421)      (332)
%e A332728                                      (111111)  (2221)     (431)
%e A332728                                                (1111111)  (521)
%e A332728                                                           (2222)
%e A332728                                                           (11111111)
%t A332728 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332728 Table[Length[Select[IntegerPartitions[n],unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]
%Y A332728 The non-negated version is A332283.
%Y A332728 The non-negated complement is counted by A332284.
%Y A332728 The strict case is A332577.
%Y A332728 The case of run-lengths (instead of differences) is A332638.
%Y A332728 The complement is counted by A332744.
%Y A332728 The Heinz numbers of partitions not in this class are A332287.
%Y A332728 Unimodal compositions are A001523.
%Y A332728 Compositions whose negation is unimodal are A332578.
%Y A332728 Compositions whose run-lengths are unimodal are A332726.
%Y A332728 Cf. A007052, A332280, A332285, A332286, A332639, A332642, A332669, A332670, A332741.
%K A332728 nonn
%O A332728 0,3
%A A332728 _Gus Wiseman_, Feb 26 2020