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Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.
6

%I #7 Jun 22 2023 23:34:48

%S 1,1,2,1,2,2,2,1,3,3,2,2,2,5,7,1,2,8,2,9,16,11,2,2,15,16,37,33,2,44,2,

%T 1,79,33,103,127,2,47,166,39,2,214,2,384,738,90,2,2,277,185,631,1077,

%U 2,1065,1560,477,1156,223,2,2863

%N Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

%C The median of an odd-length partition is the middle part.

%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(n) partitions for n = {1, 3, 9, 14, 15, 18, 20, 22} (A..M = 10..22):

%e 1 3 9 E F I K M

%e 111 333 2222222 555 666 44444 22222222222

%e 111111111 3222221 33333 222222222 54443 32222222221

%e 3322211 43332 322222221 64442 33222222211

%e 4222211 53331 332222211 65441 33322222111

%e 63321 422222211 74432 42222222211

%e 111111111111111 432222111 74441 43222222111

%e 522222111 84431 44222221111

%e 94421 52222222111

%e 53222221111

%e 62222221111

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

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

%Y All odd-length partitions are counted by A027193.

%Y For just (mean) = (median) we have A359895, also A240219, A359899, A359910.

%Y For just (mean) != (median) we have A359896, also A359894, A359900.

%Y Allowing any length gives A363719, ranks A363727, non-constant A363728.

%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 A363726 counts odd-length partitions with a unique mode.

%Y Cf. A237984, A325347, A326567/A326568, A327472, A363720, A363740, A363741.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jun 21 2023