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A332285
Number of strict integer partitions of n whose first differences (assuming the last part is zero) are unimodal.
14
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, 93, 111, 122, 134, 152, 169, 184, 212, 232, 252, 287, 315, 342, 389, 419, 458, 512, 556, 602, 672, 727, 787, 870, 940, 1012, 1124, 1209, 1303, 1431, 1540, 1655, 1821
OFFSET
0,4
COMMENTS
First differs from A000009 at a(8) = 5, A000009(8) = 6.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.
EXAMPLE
The a(1) = 1 through a(9) = 8 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(321) (61) (71) (72)
(421) (521) (81)
(432)
(531)
(621)
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).
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n], And[UnsameQ@@#, unimodQ[Differences[Append[#, 0]]]]&]], {n, 0, 30}]
CROSSREFS
The non-strict version is A332283.
The complement is counted by A332286.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Unimodal permutations are A011782.
Partitions with unimodal run-lengths are A332280.
Sequence in context: A322077 A114094 A179255 * A324325 A318284 A241321
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 21 2020
STATUS
approved