cp's OEIS Frontend

A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

%I A363728 #9 Aug 16 2023 11:01:41
%S A363728 0,0,0,0,0,0,0,1,0,1,0,4,0,3,3,7,0,12,0,18,12,9,0,50,12,14,33,54,0,
%T A363728 115,0,92,75,31,99,323,0,45,162,443,0,507,0,467,732,88,0,1551,274,833,
%U A363728 627,1228,0,2035,1556,2859,1152,221,0,9008,0,295,4835,5358
%N A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.
%C A363728 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}.
%C A363728 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e A363728 The a(8) = 1 through a(18) = 12 partitions:
%e A363728   3221  .  32221  .  4332    .  3222221  43332  5443      .  433332
%e A363728                      5331       3322211  53331  6442         443331
%e A363728                      322221     4222211  63321  7441         533322
%e A363728                      422211                     32222221     533331
%e A363728                                                 33222211     543321
%e A363728                                                 42222211     633321
%e A363728                                                 52222111     733311
%e A363728                                                              322222221
%e A363728                                                              332222211
%e A363728                                                              422222211
%e A363728                                                              432222111
%e A363728                                                              522222111
%t A363728 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363728 Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]
%Y A363728 Non-constant partitions are counted by A144300, ranks A024619.
%Y A363728 This is the non-constant case of A363719, ranks A363727.
%Y A363728 These partitions have ranks A363729.
%Y A363728 A000041 counts partitions, strict A000009.
%Y A363728 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A363728 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A363728 A362608 counts partitions with a unique mode.
%Y A363728 Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A363720, A363721.
%K A363728 nonn
%O A363728 1,12
%A A363728 _Gus Wiseman_, Jun 23 2023