cp's OEIS Frontend

A361848 Number of integer partitions of n such that (maximum) <= 2*(median).

%I A361848 #11 Mar 30 2023 20:54:44
%S A361848 1,2,3,5,6,9,12,15,19,26,31,40,49,61,75,93,112,137,165,199,238,289,
%T A361848 341,408,482,571,674,796,932,1096,1280,1495,1738,2026,2347,2724,3148,
%U A361848 3639,4191,4831,5545,6372,7298,8358,9552,10915,12439,14176,16121,18325
%N A361848 Number of integer partitions of n such that (maximum) <= 2*(median).
%C A361848 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%F A361848 a(n) = A361849(n) + A361858(n).
%F A361848 a(n) = A000041(n) - A361857(n).
%e A361848 The a(1) = 1 through a(7) = 12 partitions:
%e A361848   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A361848        (11)  (21)   (22)    (32)     (33)      (43)
%e A361848              (111)  (31)    (41)     (42)      (52)
%e A361848                     (211)   (221)    (51)      (61)
%e A361848                     (1111)  (2111)   (222)     (322)
%e A361848                             (11111)  (321)     (331)
%e A361848                                      (2211)    (421)
%e A361848                                      (21111)   (2221)
%e A361848                                      (111111)  (3211)
%e A361848                                                (22111)
%e A361848                                                (211111)
%e A361848                                                (1111111)
%e A361848 For example, the partition y = (3,2,2) has maximum 3 and median 2, and 3 <= 2*2, so y is counted under a(7).
%t A361848 Table[Length[Select[IntegerPartitions[n],Max@@#<=2*Median[#]&]],{n,30}]
%Y A361848 For length instead of median we have A237755.
%Y A361848 For minimum instead of median we have A237824.
%Y A361848 The equal case is A361849, ranks A361856.
%Y A361848 For mean instead of median we have A361851.
%Y A361848 The complement is counted by A361857, ranks A361867.
%Y A361848 The unequal case is A361858.
%Y A361848 Reversing the inequality gives A361859, ranks A361868.
%Y A361848 A000041 counts integer partitions, strict A000009.
%Y A361848 A000975 counts subsets with integer median.
%Y A361848 A325347 counts partitions with integer median, complement A307683.
%Y A361848 A359893 and A359901 count partitions by median.
%Y A361848 A360005 gives twice median of prime indices, distinct A360457.
%Y A361848 Cf. A008284, A013580, A027193, A061395, A067538, A111907, A240219, A324562, A359907, A361394, A361860.
%K A361848 nonn
%O A361848 0,2
%A A361848 _Gus Wiseman_, Mar 28 2023