%I #9 Jun 22 2023 23:34:54

%S 0,0,0,0,0,0,0,2,3,5,7,16,17,34,38,50,79,115,123,198,220,291,399,536,

%T 605,815,1036,1241,1520,2059,2315,3132,3708,4491,5668,6587,7788,10259,

%U 12299,14515,17153,21558,24623,30876,35540,41476,52023,61931,70811,85545

%N Number of integer partitions of n with different mean, median, and mode.

%C If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

%e The a(7) = 2 through a(11) = 16 partitions:

%e (421) (431) (621) (532) (542)

%e (3211) (521) (3321) (541) (632)

%e (4211) (4311) (631) (641)

%e (5211) (721) (731)

%e (32211) (5311) (821)

%e (6211) (4322)

%e (322111) (4421)

%e (5321)

%e (5411)

%e (6311)

%e (7211)

%e (33221)

%e (43211)

%e (52211)

%e (332111)

%e (422111)

%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[Length[Select[IntegerPartitions[n],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,0,30}]

%Y For equal instead of unequal: A363719, ranks A363727, odd-length A363721.

%Y The case of a unique mode is A363725.

%Y These partitions have ranks A363730.

%Y For factorizations we have A363742, for equal A363741, see A359909, A359910.

%Y Just two statistics:

%Y - (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.

%Y - (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.

%Y - (mean) = (mode) gives A363723, see A363724, A363731.

%Y - (median) = (mode) gives A363740.

%Y A000041 counts partitions, strict A000009.

%Y A008284 counts partitions by length (or negative mean), strict A008289.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y A362608 counts partitions with a unique mode.

%Y Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363728.

%K nonn

%O 0,8

%A _Gus Wiseman_, Jun 21 2023