A360686 Number 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

Offset: 1

Author

Gus Wiseman, Feb 20 2023

Keywords

• nonn

Comments

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

Examples

			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)
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).
		

Crossrefs

For all parts: A325347, strict A359907, ranks A359908, complement A307683.

For mean instead of median: A360241, ranks A326621.

These partitions have ranks A360550, complement A360551.

For multiplicities instead of distinct parts: A360687.

The complement is counted by A360689.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median.

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.

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

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

Programs

Mathematica

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