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A332280
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Number of integer partitions of n with unimodal run-lengths.
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34
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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
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OFFSET
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0,3
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COMMENTS
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A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.
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LINKS
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EXAMPLE
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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).
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MAPLE
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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):
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MATHEMATICA
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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];
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CROSSREFS
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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.
Unimodal normal sequences are A007052.
Numbers whose unsorted prime signature is unimodal are A332288.
Cf. A007052, A025065, A072706, A100883, A115981, A227038, A317086, A328509, A329398, A332284, A332285, A332294, A332578, A332579.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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