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A356235
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Number of integer partitions of n with a neighborless singleton.
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10
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0, 1, 1, 1, 2, 3, 5, 8, 12, 16, 25, 33, 45, 62, 84, 109, 148, 192, 251, 325, 421, 536, 690, 870, 1100, 1385, 1739, 2161, 2697, 3334, 4121, 5071, 6228, 7609, 9303, 11308, 13732, 16629, 20101, 24206, 29140, 34957, 41882, 50060, 59745, 71124, 84598, 100365
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OFFSET
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0,5
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COMMENTS
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A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).
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LINKS
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EXAMPLE
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The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(31) (41) (42) (52) (53)
(311) (51) (61) (62)
(411) (331) (71)
(3111) (421) (422)
(511) (431)
(4111) (521)
(31111) (611)
(4211)
(5111)
(41111)
(311111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Min@@Length/@Split[Reverse[#], #1>=#2-1&]==1&]], {n, 0, 30}]
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CROSSREFS
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The complement is counted by A355393.
These partitions are ranked by A356237.
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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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