OFFSET
0,6
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Unimodal Sequence.
EXAMPLE
The a(4) = 1 through a(9) = 17 partitions:
(211) (311) (411) (322) (422) (522)
(2111) (2211) (511) (611) (711)
(3111) (3211) (3221) (3222)
(21111) (4111) (3311) (4221)
(22111) (4211) (4311)
(31111) (5111) (5211)
(211111) (22211) (6111)
(32111) (32211)
(41111) (33111)
(221111) (42111)
(311111) (51111)
(2111111) (222111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n], !unimodQ[-Differences[Append[#, 0]]]&]], {n, 0, 30}]
CROSSREFS
The complement is counted by A332728.
The non-negated version is A332284.
The strict case is A332579.
The case of run-lengths (instead of differences) is A332639.
The Heinz numbers of these partitions are A332832.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Numbers whose negated prime signature is not unimodal are A332642.
Compositions whose negation is not unimodal are A332669.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 27 2020
STATUS
approved