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A325267
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Number of integer partitions of n with omicron 2.
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2
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0, 0, 1, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887
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OFFSET
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0,5
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COMMENTS
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The Heinz numbers of these partitions are given by A304634.
The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. We define the omicron of an integer partition to be 0 if the partition is empty, 1 if it is a singleton, and otherwise the second-to-last part of its omega-sequence. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), and its omicron is 2.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(8) = 17 partitions:
(11) (21) (22) (32) (33) (43) (44)
(31) (41) (42) (52) (53)
(211) (221) (51) (61) (62)
(311) (411) (322) (71)
(2111) (2211) (331) (332)
(3111) (511) (422)
(21111) (2221) (611)
(3211) (3221)
(4111) (3311)
(22111) (4211)
(31111) (5111)
(211111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Switch[#, {}, 0, {_}, 1, _, NestWhile[Sort[Length/@Split[#]]&, #, Length[#]>1&]//First]==2&]], {n, 0, 30}]
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CROSSREFS
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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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