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A363725
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Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.
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15
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0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 3, 8, 8, 17, 19, 28, 39, 59, 68, 106, 123, 165, 220, 301, 361, 477, 605, 745, 929, 1245, 1456, 1932, 2328, 2846, 3590, 4292, 5111, 6665, 8040, 9607, 11532, 14410, 16699, 20894, 24287, 28706, 35745, 42845, 49548, 59963, 70985
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OFFSET
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0,10
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COMMENTS
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The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {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(7) = 1 through a(13) = 17 partitions:
(3211) (4211) (3321) (5311) (4322) (4431) (4432)
(4311) (6211) (4421) (5322) (5422)
(5211) (322111) (5411) (6411) (5521)
(6311) (7311) (6322)
(7211) (8211) (6511)
(43211) (53211) (7411)
(332111) (432111) (8311)
(422111) (522111) (9211)
(54211)
(63211)
(333211)
(433111)
(442111)
(532111)
(622111)
(3322111)
(32221111)
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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], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]], {n, 0, 30}]
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CROSSREFS
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The length-4 case appears to be A325695.
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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