A332285 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

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

Author

Gus Wiseman, Feb 21 2020

Keywords

• nonn

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.

Examples

			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).
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Unimodal Sequence.

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.

Programs

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}]