proposed

approved

editing

proposed

The complement is counted by A360243, ranked by ranks A360247.

These partitions are ranked by have ranks A360246.

Sum of A360250 and A360251, ranked by ranks A360252 and A360253.

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

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

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).

Cf. `A051293, A067340, A240219, `A316313, A326567/A326568, A326619/A326620, `A326621, `~A327475, A349156, `A360069.

These partitions are ranked by A360246.

These partitions are ranked by A360246.

For The complement for multiplicities instead of distinct parts we have new, complement is A360068.

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

A067538 counts = partitions with integer mean, strict A102627, ranked by A316413.

For median instead of mean we have A360244.

allocated for Gus WisemanNumber of integer partitions 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

0,6

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)

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

These partitions are ranked by A360246.

For multiplicities instead of distinct parts we have new, complement A360068.

The complement is counted by A360243, ranked by A360247.

Sum of A360250 and A360251, ranked by 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, ranked by 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, `~A327475, A349156, `A360069.

allocated

nonn

Gus Wiseman, Feb 04 2023

approved

editing

allocated for Gus Wiseman

allocated

approved