A363728 - OEIS

A363728

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

%I #9 Aug 16 2023 11:01:41

%S 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 115,0,92,75,31,99,323,0,45,162,443,0,507,0,467,732,88,0,1551,274,833,

%U 627,1228,0,2035,1556,2859,1152,221,0,9008,0,295,4835,5358

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

%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}.

%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).

%e The a(8) = 1 through a(18) = 12 partitions:

%e 3221 . 32221 . 4332 . 3222221 43332 5443 . 433332

%e 5331 3322211 53331 6442 443331

%e 322221 4222211 63321 7441 533322

%e 422211 32222221 533331

%e 33222211 543321

%e 42222211 633321

%e 52222111 733311

%e 322222221

%e 332222211

%e 422222211

%e 432222111

%e 522222111

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

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

%Y Non-constant partitions are counted by A144300, ranks A024619.

%Y This is the non-constant case of A363719, ranks A363727.

%Y These partitions have ranks A363729.

%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, odd-length A359902.

%Y A362608 counts partitions with a unique mode.

%Y Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A363720, A363721.

%K nonn

%O 1,12

%A _Gus Wiseman_, Jun 23 2023