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A332728
Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.
11
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, 117, 131, 153, 163, 182, 210, 225, 250, 284, 304, 332, 377, 401, 441, 497, 529, 576, 647, 687, 745, 830, 883, 955, 1062, 1127, 1216, 1339, 1422, 1532, 1684, 1779, 1914
OFFSET
0,3
COMMENTS
First differs from A000041 at a(6) = 10, A000041(6) = 11.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(1) = 1 through a(8) = 10 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(321) (421) (332)
(111111) (2221) (431)
(1111111) (521)
(2222)
(11111111)
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n], unimodQ[-Differences[Append[#, 0]]]&]], {n, 0, 30}]
CROSSREFS
The non-negated version is A332283.
The non-negated complement is counted by A332284.
The strict case is A332577.
The case of run-lengths (instead of differences) is A332638.
The complement is counted by A332744.
The Heinz numbers of partitions not in this class are A332287.
Unimodal compositions are A001523.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Sequence in context: A049807 A008753 A029004 * A008752 A029003 A339279
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 26 2020
STATUS
approved