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A363728
Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.
9
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, 115, 0, 92, 75, 31, 99, 323, 0, 45, 162, 443, 0, 507, 0, 467, 732, 88, 0, 1551, 274, 833, 627, 1228, 0, 2035, 1556, 2859, 1152, 221, 0, 9008, 0, 295, 4835, 5358
OFFSET
1,12
COMMENTS
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}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(8) = 1 through a(18) = 12 partitions:
3221 . 32221 . 4332 . 3222221 43332 5443 . 433332
5331 3322211 53331 6442 443331
322221 4222211 63321 7441 533322
422211 32222221 533331
33222211 543321
42222211 633321
52222111 733311
322222221
332222211
422222211
432222111
522222111
MATHEMATICA
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&{Mean[#]}=={Median[#]}==modes[#]&]], {n, 30}]
CROSSREFS
Non-constant partitions are counted by A144300, ranks A024619.
This is the non-constant case of A363719, ranks A363727.
These partitions have ranks A363729.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Sequence in context: A021252 A348991 A195775 * A362562 A316260 A119708
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 23 2023
STATUS
approved