%I #9 Feb 22 2023 08:08:12

%S 1,2,2,4,3,8,7,16,17,31,35,60,67,99,121,170,200,270,328,436,522,674,

%T 828,1061,1292,1626,1983,2507,3035,3772,4582,5661,6801,8358,10059,

%U 12231,14627,17702,21069,25423,30147,36100,42725,50936,60081,71388,84007,99408

%N Number of integer partitions of n whose distinct parts have integer median.

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

%e The a(1) = 1 through a(8) = 16 partitions:

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

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

%e (31) (11111) (42) (421) (53)

%e (1111) (51) (511) (62)

%e (222) (3211) (71)

%e (321) (31111) (422)

%e (3111) (1111111) (431)

%e (111111) (521)

%e (2222)

%e (3221)

%e (3311)

%e (4211)

%e (5111)

%e (32111)

%e (311111)

%e (11111111)

%e 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).

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

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

%Y For mean instead of median: A360241, ranks A326621.

%Y These partitions have ranks A360550, complement A360551.

%Y For multiplicities instead of distinct parts: A360687.

%Y The complement is counted by A360689.

%Y A000041 counts integer partitions, strict A000009.

%Y A000975 counts subsets with integer median.

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

%Y A067538 counts partitions with integer mean, strict A102627, ranks A316413.

%Y A116608 counts partitions by number of distinct parts.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 20 2023