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

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, 203, 251, 282, 387, 375, 537, 561, 714, 888, 958, 1042, 1408, 1618, 1939, 2076, 2650, 2764, 3479, 3863, 4431, 5387, 6520, 6688, 8098, 9041, 10614, 12084, 14773, 15469

Offset: 0

Author

Gus Wiseman, Feb 06 2023

Keywords

• nonn

Examples

			The a(5) = 1 through a(12) = 5 partitions:
  (221)  .  (331)   (332)    (441)    (442)     (443)      (552)
            (2221)  (22211)  (3321)   (3331)    (551)      (4431)
                             (22221)  (222211)  (3332)     (33321)
                                                (4331)     (44211)
                                                (4421)     (2222211)
                                                (33221)
                                                (33311)
                                                (222221)
                                                (2222111)
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).
		

Crossrefs

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

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

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

These partitions have ranks A360252.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A058398 counts partitions by mean, also A327482.

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

A116608 counts partitions by number of distinct parts.

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

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

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

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

Programs

Mathematica

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

Formula

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

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