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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 (list; graph; refs; listen; history; text; internal format)
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

MathWorld, 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.

Cf. A025065, A072706, A115981, A227038, A332282, A332284, A332287, A332288, A332577, A332638, A332642.

Sequence in context: A322077 A114094 A179255 * A324325 A318284 A241321

Adjacent sequences:  A332282 A332283 A332284 * A332286 A332287 A332288

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 21 2020

STATUS

approved

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Last modified June 13 00:57 EDT 2021. Contains 344980 sequences. (Running on oeis4.)