cp's OEIS Frontend

A361858 Number of integer partitions of n such that the maximum is less than twice the median.

%I A361858 #5 Apr 03 2023 09:16:53
%S A361858 1,2,3,4,5,8,8,12,15,19,22,31,34,45,55,67,78,100,115,144,170,203,238,
%T A361858 291,337,403,473,560,650,772,889,1046,1213,1414,1635,1906,2186,2533,
%U A361858 2913,3361,3847,4433,5060,5808,6628,7572,8615,9835,11158,12698,14394
%N A361858 Number of integer partitions of n such that the maximum is less than twice the median.
%C A361858 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 A361858 The a(1) = 1 through a(8) = 12 partitions:
%e A361858   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A361858        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A361858              (111)  (31)    (41)     (42)      (52)       (53)
%e A361858                     (1111)  (221)    (51)      (61)       (62)
%e A361858                             (11111)  (222)     (322)      (71)
%e A361858                                      (321)     (331)      (332)
%e A361858                                      (2211)    (2221)     (431)
%e A361858                                      (111111)  (1111111)  (2222)
%e A361858                                                           (3221)
%e A361858                                                           (3311)
%e A361858                                                           (22211)
%e A361858                                                           (11111111)
%e A361858 The partition y = (3,2,2,1) has maximum 3 and median 2, and 3 < 2*2, so y is counted under a(8).
%t A361858 Table[Length[Select[IntegerPartitions[n],Max@@#<2*Median[#]&]],{n,30}]
%Y A361858 For minimum instead of median we have A053263.
%Y A361858 For length instead of median we have A237754.
%Y A361858 Allowing equality gives A361848, strict A361850.
%Y A361858 The equal version is A361849, ranks A361856.
%Y A361858 For mean instead of median we have A361852.
%Y A361858 Reversing the inequality gives A361857, ranks A361867.
%Y A361858 The complement is counted by A361859, ranks A361868.
%Y A361858 A000041 counts integer partitions, strict A000009.
%Y A361858 A000975 counts subsets with integer median.
%Y A361858 A325347 counts partitions with integer median, complement A307683.
%Y A361858 A359893 and A359901 count partitions by median.
%Y A361858 A360005 gives twice median of prime indices, distinct A360457.
%Y A361858 Cf. A008284, A027193, A237751, A237755, A237820, A237824, A240219, A361394, A361851, A361860, A361907.
%K A361858 nonn
%O A361858 1,2
%A A361858 _Gus Wiseman_, Apr 02 2023