cp's OEIS Frontend

A363723 Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.

%I A363723 #7 Jun 25 2023 08:11:28
%S A363723 0,1,2,2,3,2,4,2,5,3,5,2,10,2,7,7,12,2,18,2,24,16,13,2,60,15,18,37,60,
%T A363723 2,129,2,104,80,35,104,352,2,49,168,501,2,556,2,489,763,92,2,1799,292,
%U A363723 985,649,1296,2,2233,1681,3379,1204,225,2,10661
%N A363723 Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.
%C A363723 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
%e A363723 The a(n) partitions for n = 6, 8, 12, 14, 16 (A..G = 10..16):
%e A363723   (6)       (8)         (C)             (E)               (G)
%e A363723   (33)      (44)        (66)            (77)              (88)
%e A363723   (222)     (2222)      (444)           (2222222)         (4444)
%e A363723   (111111)  (3221)      (3333)          (3222221)         (5443)
%e A363723             (11111111)  (4332)          (3322211)         (6442)
%e A363723                         (5331)          (4222211)         (7441)
%e A363723                         (222222)        (11111111111111)  (22222222)
%e A363723                         (322221)                          (32222221)
%e A363723                         (422211)                          (33222211)
%e A363723                         (111111111111)                    (42222211)
%e A363723                                                           (52222111)
%e A363723                                                           (1111111111111111)
%t A363723 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363723 Table[Length[Select[IntegerPartitions[n],{Mean[#]}==modes[#]&]],{n,30}]
%Y A363723 Partitions containing their mean are counted by A237984, ranks A327473.
%Y A363723 For median instead of mode we have A240219, ranks A359889.
%Y A363723 Partitions missing their mean are counted by A327472, ranks A327476.
%Y A363723 The case of non-constant partitions is A362562.
%Y A363723 Including median also gives A363719, ranks A363727.
%Y A363723 Allowing multiple modes gives A363724.
%Y A363723 Requiring multiple modes gives A363731.
%Y A363723 For median instead of mean we have A363740.
%Y A363723 A000041 counts partitions, strict A000009.
%Y A363723 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A363723 A359893 and A359901 count partitions by median.
%Y A363723 A362608 counts partitions with a unique mode.
%Y A363723 Cf. A325347, A326567/A326568, A363720, A363725, A363730.
%K A363723 nonn
%O A363723 0,3
%A A363723 _Gus Wiseman_, Jun 24 2023