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A332641
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Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
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15
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0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 14, 22, 33, 48, 69, 96, 136, 184, 248, 330, 443, 574, 756, 970, 1252, 1595, 2040, 2558, 3236, 4041, 5054, 6256, 7781, 9547, 11782, 14394, 17614, 21423, 26083, 31501, 38158, 45930, 55299, 66262, 79477, 94803, 113214
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OFFSET
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0,11
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COMMENTS
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Also partitions whose run-lengths and negated run-lengths are not both unimodal. A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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LINKS
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EXAMPLE
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The a(8) = 1 through a(13) = 14 partitions:
(3221) (4221) (5221) (4331) (4332) (5332)
(32221) (6221) (5331) (6331)
(33211) (42221) (7221) (8221)
(322211) (43221) (43321)
(332111) (44211) (44311)
(52221) (53221)
(322221) (62221)
(422211) (332221)
(3321111) (333211)
(422221)
(442111)
(522211)
(3222211)
(33211111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], !Or[LessEqual@@Length/@Split[#], GreaterEqual@@Length/@Split[#]]&]], {n, 0, 30}]
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CROSSREFS
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The complement is counted by A332745.
The Heinz numbers of these partitions are A332831.
The case of run-lengths of compositions is A332833.
Partitions whose run-lengths are weakly increasing are A100883.
Partitions whose run-lengths are weakly decreasing are A100882.
Partitions whose run-lengths are not unimodal are A332281.
Partitions whose negated run-lengths are not unimodal are A332639.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions whose negation is not unimodal are A332669.
The case of run-lengths of compositions is A332833.
Compositions that are neither increasing nor decreasing are A332834.
Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.
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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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