OFFSET
0,3
COMMENTS
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) = 41 partitions (A = 10) are:
(A) (61111) (4321) (3211111)
(91) (55) (43111) (31111111)
(82) (541) (4222) (22222)
(811) (532) (42211) (222211)
(73) (5311) (421111) (2221111)
(721) (5221) (4111111) (22111111)
(7111) (52111) (3331) (211111111)
(64) (511111) (3322) (1111111111)
(631) (442) (331111)
(622) (4411) (32221)
(6211) (433) (322111)
Missing from this list is only (33211).
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-> 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_] := 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 A332281.
Heinz numbers of these partitions are the complement of A332282.
Taking 0-appended first-differences instead of run-lengths gives A332283.
The normal case is A332577.
The opposite version is A332638.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Numbers whose unsorted prime signature is unimodal are A332288.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 18 2020
STATUS
approved