cp's OEIS Frontend

A360251 Number of integer partitions of n where the parts have lesser mean than the distinct parts.

%I A360251 #6 Feb 07 2023 12:43:57
%S A360251 0,0,0,0,1,2,3,7,9,16,22,34,44,69,88,118,163,221,280,376,473,619,800,
%T A360251 1016,1257,1621,2038,2522,3117,3921,4767,5964,7273,8886,10838,13141,
%U A360251 15907,19468,23424,28093,33656,40672,48273,58171,68944,81888,97596,115643
%N A360251 Number of integer partitions of n where the parts have lesser mean than the distinct parts.
%F A360251 a(n) + A360250(n) = A360242(n).
%F A360251 a(n) + A360250(n) + A360243(n) = A000041(n).
%e A360251 The a(4) = 1 through a(9) = 16 partitions:
%e A360251   (211)  (311)   (411)    (322)     (422)      (522)
%e A360251          (2111)  (3111)   (511)     (611)      (711)
%e A360251                  (21111)  (3211)    (4211)     (3222)
%e A360251                           (4111)    (5111)     (4221)
%e A360251                           (22111)   (32111)    (4311)
%e A360251                           (31111)   (41111)    (5211)
%e A360251                           (211111)  (221111)   (6111)
%e A360251                                     (311111)   (32211)
%e A360251                                     (2111111)  (33111)
%e A360251                                                (42111)
%e A360251                                                (51111)
%e A360251                                                (321111)
%e A360251                                                (411111)
%e A360251                                                (2211111)
%e A360251                                                (3111111)
%e A360251                                                (21111111)
%e A360251 For example, the partition y = (4,2,2,1) has mean 9/4 and distinct parts {1,2,4} with mean 7/3, so y is counted under a(9).
%t A360251 Table[Length[Select[IntegerPartitions[n],Mean[#]<Mean[Union[#]]&]],{n,0,30}]
%Y A360251 For unequal instead of less we have A360242, ranks A360246.
%Y A360251 For equal instead of less we have A360243, ranks A360247.
%Y A360251 For greater instead of less we have A360250, ranks A360252.
%Y A360251 These partitions have ranks A360253.
%Y A360251 A000041 counts integer partitions, strict A000009.
%Y A360251 A008284 counts partitions by number of parts.
%Y A360251 A058398 counts partitions by mean, also A327482.
%Y A360251 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A360251 A116608 counts partitions by number of distinct parts.
%Y A360251 A240219 counts partitions with mean equal to median, ranks A359889.
%Y A360251 A359894 counts partitions with mean different from median, ranks A359890.
%Y A360251 A360071 counts partitions by number of parts and number of distinct parts.
%Y A360251 Cf. A000975, A316313, A326567/A326568, A326619/A326620, A326621, A360068, A360241, A360244, A360245.
%K A360251 nonn
%O A360251 0,6
%A A360251 _Gus Wiseman_, Feb 06 2023