cp's OEIS Frontend

A361857 Number of integer partitions of n such that the maximum is greater than twice the median.

%I A361857 #5 Apr 03 2023 09:16:59
%S A361857 0,0,0,0,1,2,3,7,11,16,25,37,52,74,101,138,185,248,325,428,554,713,
%T A361857 914,1167,1476,1865,2336,2922,3633,4508,5562,6854,8405,10284,12536,
%U A361857 15253,18489,22376,26994,32507,39038,46802,55963,66817,79582,94643,112315
%N A361857 Number of integer partitions of n such that the maximum is greater than twice the median.
%C A361857 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 A361857 The a(5) = 1 through a(10) = 16 partitions:
%e A361857   (311)  (411)   (511)    (521)     (522)      (622)
%e A361857          (3111)  (4111)   (611)     (621)      (721)
%e A361857                  (31111)  (4211)    (711)      (811)
%e A361857                           (5111)    (5211)     (5221)
%e A361857                           (32111)   (6111)     (5311)
%e A361857                           (41111)   (33111)    (6211)
%e A361857                           (311111)  (42111)    (7111)
%e A361857                                     (51111)    (43111)
%e A361857                                     (321111)   (52111)
%e A361857                                     (411111)   (61111)
%e A361857                                     (3111111)  (331111)
%e A361857                                                (421111)
%e A361857                                                (511111)
%e A361857                                                (3211111)
%e A361857                                                (4111111)
%e A361857                                                (31111111)
%e A361857 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 A361857 Table[Length[Select[IntegerPartitions[n],Max@@#>2*Median[#]&]],{n,30}]
%Y A361857 For length instead of median we have A237751.
%Y A361857 For minimum instead of median we have A237820.
%Y A361857 The complement is counted by A361848.
%Y A361857 The equal version is A361849, ranks A361856.
%Y A361857 Reversing the inequality gives A361858.
%Y A361857 Allowing equality gives A361859, ranks A361868.
%Y A361857 These partitions have ranks A361867.
%Y A361857 For mean instead of median we have A361907.
%Y A361857 A000041 counts integer partitions, strict A000009.
%Y A361857 A000975 counts subsets with integer median.
%Y A361857 A325347 counts partitions with integer median, complement A307683.
%Y A361857 A359893 and A359901 count partitions by median.
%Y A361857 A360005 gives twice median of prime indices, distinct A360457.
%Y A361857 Cf. A008284, A013580, A027193, A061395, A237755, A237824, A240219, A361394, A361851, A361860.
%K A361857 nonn
%O A361857 1,6
%A A361857 _Gus Wiseman_, Apr 02 2023