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A363719
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Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.
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16
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1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020
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OFFSET
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1,2
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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}.
Without loss of generality, we may assume there is a unique middle-part (A238478).
Includes all constant partitions.
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LINKS
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EXAMPLE
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The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
1 2 4 6 8 C E G
11 22 33 44 66 77 88
1111 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
1^16
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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, 30}]
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CROSSREFS
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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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