OFFSET
0,8
COMMENTS
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
A composition of n is a finite sequence of positive integers summing to n.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.
FORMULA
a(n) + A332726(n) = 2^(n - 1).
EXAMPLE
The a(6) = 1 through a(8) = 8 compositions:
(11211) (11311) (11411)
(111211) (111311)
(112111) (112112)
(113111)
(211211)
(1111211)
(1112111)
(1121111)
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !unimodQ[Length/@Split[#]]&]], {n, 0, 10}]
CROSSREFS
Looking at the composition itself (not its run-lengths) gives A115981.
The complement is counted by A332726.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negation is not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 29 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020
STATUS
approved