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A363723
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Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.
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29
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0, 1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 60, 15, 18, 37, 60, 2, 129, 2, 104, 80, 35, 104, 352, 2, 49, 168, 501, 2, 556, 2, 489, 763, 92, 2, 1799, 292, 985, 649, 1296, 2, 2233, 1681, 3379, 1204, 225, 2, 10661
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OFFSET
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0,3
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COMMENTS
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A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
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LINKS
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EXAMPLE
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The a(n) partitions for n = 6, 8, 12, 14, 16 (A..G = 10..16):
(6) (8) (C) (E) (G)
(33) (44) (66) (77) (88)
(222) (2222) (444) (2222222) (4444)
(111111) (3221) (3333) (3222221) (5443)
(11111111) (4332) (3322211) (6442)
(5331) (4222211) (7441)
(222222) (11111111111111) (22222222)
(322221) (32222221)
(422211) (33222211)
(111111111111) (42222211)
(52222111)
(1111111111111111)
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MATHEMATICA
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modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}==modes[#]&]], {n, 30}]
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CROSSREFS
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Partitions containing their mean are counted by A237984, ranks A327473.
The case of non-constant partitions is A362562.
Allowing multiple modes gives A363724.
Requiring multiple modes gives A363731.
For median instead of mean we have A363740.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A362608 counts partitions with a unique mode.
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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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