reviewed

approved

proposed

reviewed

editing

proposed

For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).

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

Taking For all parts (not just distinct) gives : A325347, strict A359907, ranks A359908, complement A307683.

For mean instead of median we have : A360241, ranks A326621.

For multiplicities instead of distinct parts we have : A360687.

`A008284 counts partitions by number of parts.

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

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

Cf. `A359895, A240219, A359906, A360005, `A360071, A360244, `A360245, `A360556, A360688.

A359893 and A359901 count partitions by median, odd-length A359902, ranks A360005.

Cf. `A359895, A359906, A360005, `A360244, `A360245, `A360556, A360688.

allocated for Gus WisemanNumber of integer partitions of n whose distinct parts have integer median.

1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408

1,2

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) = 16 partitions:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (111) (22) (311) (33) (331) (44)

(31) (11111) (42) (421) (53)

(1111) (51) (511) (62)

(222) (3211) (71)

(321) (31111) (422)

(3111) (1111111) (431)

(111111) (521)

(2222)

(3221)

(3311)

(4211)

(5111)

(32111)

(311111)

(11111111)

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

Taking all parts (not just distinct) gives A325347, strict A359907, ranks A359908, complement A307683.

For mean instead of median we have A360241, ranks A326621.

These partitions have ranks A360550, complement A360551.

For multiplicities instead of distinct parts we have A360687.

The complement is counted by A360689.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median.

`A008284 counts partitions by number of parts.

A027193 counts odd-length partitions, strict A067659, ranks A026424.

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.

A359893 and A359901 count partitions by median, odd-length A359902, ranks A360005.

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

Cf. `A359895, A359906, `A360244, `A360245, `A360556, A360688.

allocated

nonn

Gus Wiseman, Feb 20 2023

approved

editing

allocating

allocated

allocated for Gus Wiseman

allocating

approved