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%I #7 Jun 25 2023 08:11:28
%S 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,
%T 2,129,2,104,80,35,104,352,2,49,168,501,2,556,2,489,763,92,2,1799,292,
%U 985,649,1296,2,2233,1681,3379,1204,225,2,10661
%N 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.
%C 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}.
%e The a(n) partitions for n = 6, 8, 12, 14, 16 (A..G = 10..16):
%e (6) (8) (C) (E) (G)
%e (33) (44) (66) (77) (88)
%e (222) (2222) (444) (2222222) (4444)
%e (111111) (3221) (3333) (3222221) (5443)
%e (11111111) (4332) (3322211) (6442)
%e (5331) (4222211) (7441)
%e (222222) (11111111111111) (22222222)
%e (322221) (32222221)
%e (422211) (33222211)
%e (111111111111) (42222211)
%e (52222111)
%e (1111111111111111)
%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t Table[Length[Select[IntegerPartitions[n],{Mean[#]}==modes[#]&]],{n,30}]
%Y Partitions containing their mean are counted by A237984, ranks A327473.
%Y For median instead of mode we have A240219, ranks A359889.
%Y Partitions missing their mean are counted by A327472, ranks A327476.
%Y The case of non-constant partitions is A362562.
%Y Including median also gives A363719, ranks A363727.
%Y Allowing multiple modes gives A363724.
%Y Requiring multiple modes gives A363731.
%Y For median instead of mean we have A363740.
%Y A000041 counts partitions, strict A000009.
%Y A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A359893 and A359901 count partitions by median.
%Y A362608 counts partitions with a unique mode.
%Y Cf. A325347, A326567/A326568, A363720, A363725, A363730.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jun 24 2023
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