%I #5 Feb 06 2023 10:06:07

%S 1,1,2,3,4,4,8,6,11,13,19,19,35,33,48,66,78,88,124,138,183,219,252,

%T 306,388,450,527,643,780,903,1097,1266,1523,1784,2107,2511,2966,3407,

%U 4019,4667,5559,6364,7492,8601,10063,11634,13469,15469,17985,20558,23812

%N Number of integer partitions of n where the parts have the same median as the distinct parts.

%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) = 11 partitions:

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

%e (11) (21) (22) (32) (33) (43) (44)

%e (111) (31) (41) (42) (52) (53)

%e (1111) (11111) (51) (61) (62)

%e (222) (421) (71)

%e (321) (1111111) (431)

%e (2211) (521)

%e (111111) (2222)

%e (3221)

%e (3311)

%e (11111111)

%e 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).

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

%Y For mean instead of median: A360242, ranks A360247, complement A360243.

%Y These partitions have ranks A360249.

%Y The complement is A360244, ranks A360248.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

%Y A116608 counts partitions by number of distinct parts.

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

%Y A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

%Y A359893 and A359901 count partitions by median.

%Y A359894 counts partitions with mean different from median, ranks A359890.

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

%Y Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360241, A360246, A360250, A360251.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 05 2023