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A363720
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Number of integer partitions of n with different mean, median, and mode.
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17
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0, 0, 0, 0, 0, 0, 0, 2, 3, 5, 7, 16, 17, 34, 38, 50, 79, 115, 123, 198, 220, 291, 399, 536, 605, 815, 1036, 1241, 1520, 2059, 2315, 3132, 3708, 4491, 5668, 6587, 7788, 10259, 12299, 14515, 17153, 21558, 24623, 30876, 35540, 41476, 52023, 61931, 70811, 85545
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OFFSET
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0,8
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COMMENTS
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If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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) = 2 through a(11) = 16 partitions:
(421) (431) (621) (532) (542)
(3211) (521) (3321) (541) (632)
(4211) (4311) (631) (641)
(5211) (721) (731)
(32211) (5311) (821)
(6211) (4322)
(322111) (4421)
(5321)
(5411)
(6311)
(7211)
(33221)
(43211)
(52211)
(332111)
(422111)
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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[#]}!={Median[#]}!=modes[#]&]], {n, 0, 30}]
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CROSSREFS
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The case of a unique mode is A363725.
These partitions have ranks A363730.
Just two statistics:
A008284 counts partitions by length (or negative 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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