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A332286
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Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
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16
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0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857
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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 and a weakly decreasing sequence.
Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.
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LINKS
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EXAMPLE
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The a(8) = 1 through a(18) = 7 partitions:
(431) . (541) (641) (651) (652) (752) (762) (862)
(5421) (751) (761) (861) (871)
(5431) (851) (6531) (961)
(6431) (7431) (6532)
(6521) (7521) (6541)
(7621)
(8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).
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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], And[UnsameQ@@#, !unimodQ[Differences[Append[#, 0]]]]&]], {n, 0, 30}]
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CROSSREFS
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Partitions covering an initial interval are (also) A000009.
The complement is counted by A332285.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Normal partitions whose run-lengths are not unimodal are A332579.
Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.
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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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