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A332281
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Number of integer partitions of n whose run-lengths are not unimodal.
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34
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 10, 16, 24, 33, 51, 70, 100, 137, 189, 250, 344, 450, 597, 778, 1019, 1302, 1690, 2142, 2734, 3448, 4360, 5432, 6823, 8453, 10495, 12941, 15968, 19529, 23964, 29166, 35525, 43054, 52173, 62861, 75842, 91013, 109208
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OFFSET
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0,13
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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) = 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)
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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-> combinat[numbpart](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_] := PartitionsP[n] - b[n, n, 0, True];
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CROSSREFS
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The complement is counted by A332280.
The Heinz numbers of these partitions are A332282.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Cf. A007052, A025065, A072706, A100883, A332283, A332284, A332286, A332287, A332579, A332638, A332640, A332641, A332642.
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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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