cp's OEIS Frontend

A360250 Number of integer partitions of n where the parts have greater mean than the distinct parts.

%I A360250 #5 Feb 07 2023 12:44:28
%S A360250 0,0,0,0,0,1,0,2,2,3,3,9,5,13,15,18,20,37,34,59,51,68,92,134,121,167,
%T A360250 203,251,282,387,375,537,561,714,888,958,1042,1408,1618,1939,2076,
%U A360250 2650,2764,3479,3863,4431,5387,6520,6688,8098,9041,10614,12084,14773,15469
%N A360250 Number of integer partitions of n where the parts have greater mean than the distinct parts.
%F A360250 a(n) + A360251(n) = A360242(n).
%F A360250 a(n) + A360251(n) + A360243(n) = A000041(n).
%e A360250 The a(5) = 1 through a(12) = 5 partitions:
%e A360250   (221)  .  (331)   (332)    (441)    (442)     (443)      (552)
%e A360250             (2221)  (22211)  (3321)   (3331)    (551)      (4431)
%e A360250                              (22221)  (222211)  (3332)     (33321)
%e A360250                                                 (4331)     (44211)
%e A360250                                                 (4421)     (2222211)
%e A360250                                                 (33221)
%e A360250                                                 (33311)
%e A360250                                                 (222221)
%e A360250                                                 (2222111)
%e A360250 For example, the partition y = (4,3,3,1) has mean 11/4 and distinct parts {1,3,4} with mean 8/5, so y is counted under a(11).
%t A360250 Table[Length[Select[IntegerPartitions[n],Mean[#]>Mean[Union[#]]&]],{n,0,30}]
%Y A360250 For unequal instead of greater we have A360242, ranks A360246.
%Y A360250 For equal instead of greater we have A360243, ranks A360247.
%Y A360250 For less instead of greater we have A360251, ranks A360253.
%Y A360250 These partitions have ranks A360252.
%Y A360250 A000041 counts integer partitions, strict A000009.
%Y A360250 A008284 counts partitions by number of parts.
%Y A360250 A058398 counts partitions by mean, also A327482.
%Y A360250 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A360250 A116608 counts partitions by number of distinct parts.
%Y A360250 A240219 counts partitions with mean equal to median, ranks A359889.
%Y A360250 A359894 counts partitions with mean different from median, ranks A359890.
%Y A360250 A360071 counts partitions by number of parts and number of distinct parts.
%Y A360250 Cf. A000975, A316313, A326567/A326568, A326619/A326620, A326621, A360068, A360241, A360244, A360245.
%K A360250 nonn
%O A360250 0,8
%A A360250 _Gus Wiseman_, Feb 06 2023