cp's OEIS Frontend

A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

%I A361859 #5 Apr 03 2023 09:17:07
%S A361859 0,0,0,1,2,3,7,10,15,23,34,46,67,90,121,164,219,285,375,483,622,799,
%T A361859 1017,1284,1621,2033,2537,3158,3915,4832,5953,7303,8930,10896,13248,
%U A361859 16071,19451,23482,28272,33977,40736,48741,58201,69367,82506,97986,116139
%N A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.
%C A361859 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e A361859 The a(4) = 1 through a(9) = 15 partitions:
%e A361859   (211)  (311)   (411)    (421)     (422)      (522)
%e A361859          (2111)  (3111)   (511)     (521)      (621)
%e A361859                  (21111)  (3211)    (611)      (711)
%e A361859                           (4111)    (4211)     (4221)
%e A361859                           (22111)   (5111)     (4311)
%e A361859                           (31111)   (32111)    (5211)
%e A361859                           (211111)  (41111)    (6111)
%e A361859                                     (221111)   (33111)
%e A361859                                     (311111)   (42111)
%e A361859                                     (2111111)  (51111)
%e A361859                                                (321111)
%e A361859                                                (411111)
%e A361859                                                (2211111)
%e A361859                                                (3111111)
%e A361859                                                (21111111)
%e A361859 The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).
%t A361859 Table[Length[Select[IntegerPartitions[n],Max@@#>=2*Median[#]&]],{n,30}]
%Y A361859 For length instead of median we have A237752.
%Y A361859 For minimum instead of median we have A237821.
%Y A361859 Reversing the inequality gives A361848.
%Y A361859 The equal case is A361849, ranks A361856.
%Y A361859 The unequal case is A361857, ranks A361867.
%Y A361859 The complement is counted by A361858.
%Y A361859 These partitions have ranks A361868.
%Y A361859 For mean instead of median we have A361906.
%Y A361859 A000041 counts integer partitions, strict A000009.
%Y A361859 A000975 counts subsets with integer median.
%Y A361859 A325347 counts partitions with integer median, complement A307683.
%Y A361859 A359893 and A359901 count partitions by median.
%Y A361859 A360005 gives twice median of prime indices, distinct A360457.
%Y A361859 Cf. A008284, A027193, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.
%K A361859 nonn
%O A361859 1,5
%A A361859 _Gus Wiseman_, Apr 02 2023