OFFSET
0,6
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..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence.
FORMULA
a(n) + A332578(n) = 2^(n - 1) for n > 0.
EXAMPLE
The a(4) = 1 through a(6) = 11 compositions:
(121) (131) (132)
(1121) (141)
(1211) (231)
(1131)
(1212)
(1221)
(1311)
(2121)
(11121)
(11211)
(12111)
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[-#]&]], {n, 0, 10}]
CROSSREFS
The strict case is A072707.
The complement is counted by A332578.
The version for run-lengths of partitions is A332639.
The version for unsorted prime signature is A332642.
The version for 0-appended first-differences of partitions is A332744.
The case that is not unimodal either is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
A triangle for compositions with unimodal negation is A332670.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 28 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Mar 01 2020
STATUS
approved