A360245 Number 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

Offset: 0

Author

Gus Wiseman, Feb 05 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) = 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).
		

Crossrefs

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

These partitions have ranks A360249.

The complement is A360244, ranks 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, ranks A359889.

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

A359893 and A359901 count partitions by median.

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

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

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

Programs

Mathematica

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