OFFSET
0,13
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.
EXAMPLE
The a(8) = 1 through a(18) = 7 partitions:
(431) . (541) (641) (651) (652) (752) (762) (862)
(5421) (751) (761) (861) (871)
(5431) (851) (6531) (961)
(6431) (7431) (6532)
(6521) (7521) (6541)
(7621)
(8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).
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}]
CROSSREFS
Strict partitions are A000009.
Partitions covering an initial interval are (also) A000009.
The non-strict version is A332284.
The complement is counted by A332285.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Normal partitions whose run-lengths are not unimodal are A332579.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 21 2020
STATUS
approved