OFFSET
0,16
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
LINKS
MathWorld, Unimodal Sequence
EXAMPLE
The a(14) = 1 through a(18) = 12 partitions:
(433211) (533211) (443221) (544211) (544311)
(4332111) (633211) (733211) (553221)
(5332111) (4333211) (644211)
(43321111) (6332111) (833211)
(53321111) (4432221)
(433211111) (5333211)
(5442111)
(7332111)
(43332111)
(63321111)
(533211111)
(4332111111)
For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], !unimodQ[Length/@Split[#]]&&!unimodQ[-Length/@Split[#]]&]], {n, 0, 30}]
CROSSREFS
Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The Heinz numbers of these partitions are A332643.
The complement is counted by A332746.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths and negated run-lengths are not both unimodal: A332641.
Compositions whose negation is not unimodal are A332669.
Run-lengths and negated run-lengths are both unimodal: A332745.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 25 2020
STATUS
approved