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A332280
Number of integer partitions of n with unimodal run-lengths.
34
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 97, 129, 166, 215, 273, 352, 439, 557, 692, 865, 1066, 1325, 1614, 1986, 2413, 2940, 3546, 4302, 5152, 6207, 7409, 8862, 10523, 12545, 14814, 17562, 20690, 24397, 28615, 33645, 39297, 46009, 53609, 62504, 72581, 84412
OFFSET
0,3
COMMENTS
First differs from A000041 at a(10) = 41, A000041(10) = 42.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.
LINKS
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.
Sequence in context: A023030 A246580 A212187 * A035998 A137792 A039905
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 18 2020
STATUS
approved