cp's OEIS Frontend

A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

%I A363725 #9 Jun 23 2023 23:29:30
%S A363725 0,0,0,0,0,0,0,1,1,3,3,8,8,17,19,28,39,59,68,106,123,165,220,301,361,
%T A363725 477,605,745,929,1245,1456,1932,2328,2846,3590,4292,5111,6665,8040,
%U A363725 9607,11532,14410,16699,20894,24287,28706,35745,42845,49548,59963,70985
%N A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.
%C A363725 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%C A363725 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 A363725 The a(7) = 1 through a(13) = 17 partitions:
%e A363725   (3211)  (4211)  (3321)  (5311)    (4322)    (4431)    (4432)
%e A363725                   (4311)  (6211)    (4421)    (5322)    (5422)
%e A363725                   (5211)  (322111)  (5411)    (6411)    (5521)
%e A363725                                     (6311)    (7311)    (6322)
%e A363725                                     (7211)    (8211)    (6511)
%e A363725                                     (43211)   (53211)   (7411)
%e A363725                                     (332111)  (432111)  (8311)
%e A363725                                     (422111)  (522111)  (9211)
%e A363725                                                         (54211)
%e A363725                                                         (63211)
%e A363725                                                         (333211)
%e A363725                                                         (433111)
%e A363725                                                         (442111)
%e A363725                                                         (532111)
%e A363725                                                         (622111)
%e A363725                                                         (3322111)
%e A363725                                                         (32221111)
%t A363725 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363725 Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]],{n,0,30}]
%Y A363725 The length-4 case appears to be A325695.
%Y A363725 For equal instead of unequal we have A363719, ranks A363727.
%Y A363725 Allowing multiple modes gives A363720, ranks A363730.
%Y A363725 A000041 counts partitions, strict A000009.
%Y A363725 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A363725 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A363725 A362608 counts partitions with a unique mode.
%Y A363725 Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A359896, A359900, A363723, A363728.
%K A363725 nonn
%O A363725 0,10
%A A363725 _Gus Wiseman_, Jun 22 2023