A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

0, 0, 0, 0, 1, 3, 3, 9, 11, 19, 25, 43, 49, 82, 103, 136, 183, 258, 314, 435, 524, 687, 892, 1150, 1378, 1788, 2241, 2773, 3399, 4308, 5142, 6501, 7834, 9600, 11726, 14099, 16949, 20876, 25042, 30032, 35732, 43322, 51037, 61650, 72807, 86319, 102983, 122163

Offset

0,6

Links

Table of n, a(n) for n=0..47.

Example

The a(1) = 0 through a(9) = 19 partitions:
  .  .  .  (211)  (221)   (411)    (322)     (332)      (441)
                  (311)   (3111)   (331)     (422)      (522)
                  (2111)  (21111)  (511)     (611)      (711)
                                   (2221)    (4211)     (3222)
                                   (3211)    (5111)     (3321)
                                   (4111)    (22211)    (4221)
                                   (22111)   (32111)    (4311)
                                   (31111)   (41111)    (5211)
                                   (211111)  (221111)   (6111)
                                             (311111)   (22221)
                                             (2111111)  (32211)
                                                        (33111)
                                                        (42111)
                                                        (51111)
                                                        (321111)
                                                        (411111)
                                                        (2211111)
                                                        (3111111)
                                                        (21111111)
For example, the partition y = (32211) has mean 9/5 and distinct parts {1,2,3} with mean 2, so y is counted under a(9).

Mathematica

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

Crossrefs

The complement for multiplicities instead of distinct parts is A360068.

The complement is counted by A360243, ranks A360247.

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

These partitions have ranks A360246.

Sum of 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.

Sequence in context: A022156 A125027 A360244 * A005296 A124281 A351869

Adjacent sequences: A360239 A360240 A360241 * A360243 A360244 A360245

Keyword

nonn

Author

Gus Wiseman, Feb 04 2023

Status

approved