OFFSET
0,8
COMMENTS
If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
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}.
EXAMPLE
The a(7) = 2 through a(11) = 16 partitions:
(421) (431) (621) (532) (542)
(3211) (521) (3321) (541) (632)
(4211) (4311) (631) (641)
(5211) (721) (731)
(32211) (5311) (821)
(6211) (4322)
(322111) (4421)
(5321)
(5411)
(6311)
(7211)
(33221)
(43211)
(52211)
(332111)
(422111)
MATHEMATICA
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}!={Median[#]}!=modes[#]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 21 2023
STATUS
approved