cp's OEIS Frontend

A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

%I A363740 #7 Jun 27 2023 10:28:55
%S A363740 1,2,2,4,5,7,10,15,18,26,35,46,61,82,102,136,174,224,283,360,449,569,
%T A363740 708,883,1089,1352,1659,2042,2492,3039,3695,4492,5426,6555,7889,9482,
%U A363740 11360,13602,16231,19348,23005,27313,32364,38303,45227,53341,62800,73829
%N A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.
%C A363740 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 A363740 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 A363740 The a(1) = 1 through a(8) = 15 partitions:
%e A363740   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A363740        (11)  (111)  (22)    (221)    (33)      (322)      (44)
%e A363740                     (211)   (311)    (222)     (331)      (332)
%e A363740                     (1111)  (2111)   (411)     (511)      (422)
%e A363740                             (11111)  (3111)    (2221)     (611)
%e A363740                                      (21111)   (4111)     (2222)
%e A363740                                      (111111)  (22111)    (3221)
%e A363740                                                (31111)    (5111)
%e A363740                                                (211111)   (22211)
%e A363740                                                (1111111)  (32111)
%e A363740                                                           (41111)
%e A363740                                                           (221111)
%e A363740                                                           (311111)
%e A363740                                                           (2111111)
%e A363740                                                           (11111111)
%t A363740 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363740 Table[Length[Select[IntegerPartitions[n],{Median[#]}==modes[#]&]],{n,30}]
%Y A363740 For mean instead of mode we have A240219, see A359894, A359889, A359895, A359897, A359899.
%Y A363740 Including mean also gives A363719, ranks A363727.
%Y A363740 For mean instead of median we have A363723, see A363724, A363731.
%Y A363740 A000041 counts integer partitions, strict A000009.
%Y A363740 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A363740 A359893 and A359901 count partitions by median.
%Y A363740 A362608 counts partitions with a unique mode, ranks A356862.
%Y A363740 Cf. A027193, A237984, A325347, A362562, A363720, A363725, A363726.
%K A363740 nonn
%O A363740 1,2
%A A363740 _Gus Wiseman_, Jun 26 2023