proposed

approved

editing

proposed

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

For mean instead of median: A360242, ranked by ranks A360247, complement A360243.

These partitions are ranked by have ranks A360249.

The complement is A360244, ranked by ranks A360248.

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

A325347 = counts partitions with w/ integer median, strict A359907, ranked by ranks A359908.

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

Cf. A000975, A027193, `A067659, `A316313, ~`A326567/~`A326568, A326619/A326620, A326621, ~`A359895, `A359896, A359902, `A359906, `A360068, A360241, A360246, A360250, A360251.

allocated for Gus WisemanNumber of integer partitions of n where the parts have the same median as the distinct parts.

1, 1, 2, 3, 4, 4, 8, 6, 11, 13, 19, 19, 35, 33, 48, 66, 78, 88, 124, 138, 183, 219, 252, 306, 388, 450, 527, 643, 780, 903, 1097, 1266, 1523, 1784, 2107, 2511, 2966, 3407, 4019, 4667, 5559, 6364, 7492, 8601, 10063, 11634, 13469, 15469, 17985, 20558, 23812

0,3

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

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)

For example, the partition y = (6,4,4,4,1,1) has median 4, and the distinct parts {1,4,6} also have median 4, so y is counted under a(20).

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

For mean instead of median: A360242, ranked by A360247, complement A360243.

These partitions are ranked by A360249.

The complement is A360244, ranked by A360248.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A116608 counts partitions by number of distinct parts.

A240219 counts partitions with mean equal to median, ranked by A359889.

A325347 = partitions with integer median, strict A359907, ranked by A359908.

A359893 and A359901 count partitions by median.

A359894 = partitions with mean different from median, ranked by A359890.

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

Cf. A000975, A027193, `A067659, `A316313, ~`A326567/~`A326568, A326619/A326620, A326621, ~`A359895, `A359896, A359902, `A359906, `A360068, A360241, A360246, A360250, A360251.

allocated

nonn

Gus Wiseman, Feb 05 2023

approved

editing

allocated for Gus Wiseman

allocated

approved