OFFSET
0,13
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence
EXAMPLE
The a(10) = 1 through a(15) = 10 partitions:
(33211) (332111) (44211) (44311) (55211) (44322)
(3321111) (333211) (433211) (55311)
(442111) (443111) (443211)
(33211111) (3332111) (533211)
(4421111) (552111)
(332111111) (4332111)
(4431111)
(33321111)
(44211111)
(3321111111)
MAPLE
b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
end:
a:= n-> combinat[numbpart](n)-b(n$2, 0, true):
seq(a(n), n=0..65); # Alois P. Heinz, Feb 20 2020
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], !unimodQ[Length/@Split[#]]&]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, j, t && j >= m], {j, 1, Min[If[t, Infinity, m], n/i]}] + b[n, i - 1, m, t]]];
a[n_] := PartitionsP[n] - b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
CROSSREFS
The complement is counted by A332280.
The Heinz numbers of these partitions are A332282.
The opposite version is A332639.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 19 2020
STATUS
approved