%I #6 Feb 07 2023 12:43:57

%S 0,0,0,0,1,2,3,7,9,16,22,34,44,69,88,118,163,221,280,376,473,619,800,

%T 1016,1257,1621,2038,2522,3117,3921,4767,5964,7273,8886,10838,13141,

%U 15907,19468,23424,28093,33656,40672,48273,58171,68944,81888,97596,115643

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

%F a(n) + A360250(n) = A360242(n).

%F a(n) + A360250(n) + A360243(n) = A000041(n).

%e The a(4) = 1 through a(9) = 16 partitions:

%e (211) (311) (411) (322) (422) (522)

%e (2111) (3111) (511) (611) (711)

%e (21111) (3211) (4211) (3222)

%e (4111) (5111) (4221)

%e (22111) (32111) (4311)

%e (31111) (41111) (5211)

%e (211111) (221111) (6111)

%e (311111) (32211)

%e (2111111) (33111)

%e (42111)

%e (51111)

%e (321111)

%e (411111)

%e (2211111)

%e (3111111)

%e (21111111)

%e 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 Table[Length[Select[IntegerPartitions[n],Mean[#]<Mean[Union[#]]&]],{n,0,30}]

%Y For unequal instead of less we have A360242, ranks A360246.

%Y For equal instead of less we have A360243, ranks A360247.

%Y For greater instead of less we have A360250, ranks A360252.

%Y These partitions have ranks A360253.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

%Y A058398 counts partitions by mean, also A327482.

%Y A067538 counts partitions with integer mean, strict A102627, ranks A316413.

%Y A116608 counts partitions by number of distinct parts.

%Y A240219 counts partitions with mean equal to median, ranks A359889.

%Y A359894 counts partitions with mean different from median, ranks A359890.

%Y A360071 counts partitions by number of parts and number of distinct parts.

%Y Cf. A000975, A316313, A326567/A326568, A326619/A326620, A326621, A360068, A360241, A360244, A360245.

%K nonn

%O 0,6

%A _Gus Wiseman_, Feb 06 2023