A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

1, 1, 2, 3, 4, 4, 8, 6, 11, 11, 17, 13, 28, 19, 32, 40, 48, 39, 71, 55, 103, 105, 110, 105, 197, 170, 195, 237, 319, 257, 462, 341, 515, 543, 584, 784, 1028, 761, 973, 1153, 1606, 1261, 2137, 1611, 2368, 2815, 2575, 2591, 4393, 3798, 4602, 4663, 5777, 5121

Offset: 0

Author

Gus Wiseman, Feb 04 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (321)     (1111111)  (431)
                                     (2211)               (521)
                                     (111111)             (2222)
                                                          (3221)
                                                          (3311)
                                                          (11111111)
		

Crossrefs

For multiplicities instead of distinct parts we have A360068.

The complement is counted by A360242, ranks A360246.

For median instead of mean we have A360245, complement A360244.

These partitions have ranks A360247.

Cf. A360250 and A360251, ranks A360252 and A360253.

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.

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

A360241 counts partitions whose distinct parts have integer mean.

Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156, A360069.

Programs

Mathematica

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