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Number of integer partitions of n where the parts have greater mean than the distinct parts.
10

%I #5 Feb 07 2023 12:44:28

%S 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 203,251,282,387,375,537,561,714,888,958,1042,1408,1618,1939,2076,

%U 2650,2764,3479,3863,4431,5387,6520,6688,8098,9041,10614,12084,14773,15469

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

%F a(n) + A360251(n) = A360242(n).

%F a(n) + A360251(n) + A360243(n) = A000041(n).

%e The a(5) = 1 through a(12) = 5 partitions:

%e (221) . (331) (332) (441) (442) (443) (552)

%e (2221) (22211) (3321) (3331) (551) (4431)

%e (22221) (222211) (3332) (33321)

%e (4331) (44211)

%e (4421) (2222211)

%e (33221)

%e (33311)

%e (222221)

%e (2222111)

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

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

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

%Y For less instead of greater we have A360251, ranks A360253.

%Y These partitions have ranks A360252.

%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 0,8

%A _Gus Wiseman_, Feb 06 2023