cp's OEIS Frontend

A363720 Number of integer partitions of n with different mean, median, and mode.

%I A363720 #9 Jun 22 2023 23:34:54
%S A363720 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 A363720 605,815,1036,1241,1520,2059,2315,3132,3708,4491,5668,6587,7788,10259,
%U A363720 12299,14515,17153,21558,24623,30876,35540,41476,52023,61931,70811,85545
%N A363720 Number of integer partitions of n with different mean, median, and mode.
%C A363720 If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
%C A363720 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 A363720 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 A363720 The a(7) = 2 through a(11) = 16 partitions:
%e A363720   (421)   (431)   (621)    (532)     (542)
%e A363720   (3211)  (521)   (3321)   (541)     (632)
%e A363720           (4211)  (4311)   (631)     (641)
%e A363720                   (5211)   (721)     (731)
%e A363720                   (32211)  (5311)    (821)
%e A363720                            (6211)    (4322)
%e A363720                            (322111)  (4421)
%e A363720                                      (5321)
%e A363720                                      (5411)
%e A363720                                      (6311)
%e A363720                                      (7211)
%e A363720                                      (33221)
%e A363720                                      (43211)
%e A363720                                      (52211)
%e A363720                                      (332111)
%e A363720                                      (422111)
%t A363720 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363720 Table[Length[Select[IntegerPartitions[n],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,0,30}]
%Y A363720 For equal instead of unequal: A363719, ranks A363727, odd-length A363721.
%Y A363720 The case of a unique mode is A363725.
%Y A363720 These partitions have ranks A363730.
%Y A363720 For factorizations we have A363742, for equal A363741, see A359909, A359910.
%Y A363720 Just two statistics:
%Y A363720 - (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
%Y A363720 - (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
%Y A363720 - (mean) = (mode) gives A363723, see A363724, A363731.
%Y A363720 - (median) = (mode) gives A363740.
%Y A363720 A000041 counts partitions, strict A000009.
%Y A363720 A008284 counts partitions by length (or negative mean), strict A008289.
%Y A363720 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A363720 A362608 counts partitions with a unique mode.
%Y A363720 Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363728.
%K A363720 nonn
%O A363720 0,8
%A A363720 _Gus Wiseman_, Jun 21 2023