OFFSET
0,11
COMMENTS
Also partitions whose run-lengths and negated run-lengths are not both unimodal. 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(8) = 1 through a(13) = 14 partitions:
(3221) (4221) (5221) (4331) (4332) (5332)
(32221) (6221) (5331) (6331)
(33211) (42221) (7221) (8221)
(322211) (43221) (43321)
(332111) (44211) (44311)
(52221) (53221)
(322221) (62221)
(422211) (332221)
(3321111) (333211)
(422221)
(442111)
(522211)
(3222211)
(33211111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !Or[LessEqual@@Length/@Split[#], GreaterEqual@@Length/@Split[#]]&]], {n, 0, 30}]
CROSSREFS
The complement is counted by A332745.
The Heinz numbers of these partitions are A332831.
The case of run-lengths of compositions is A332833.
Partitions whose run-lengths are weakly increasing are A100883.
Partitions whose run-lengths are weakly decreasing are A100882.
Partitions whose run-lengths are not unimodal are A332281.
Partitions whose negated run-lengths are not unimodal are A332639.
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.
Compositions whose negation is not unimodal are A332669.
The case of run-lengths of compositions is A332833.
Compositions that are neither increasing nor decreasing are A332834.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 26 2020
STATUS
approved