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

0,3

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).

Links

Table of n, a(n) for n=0..50.

Example

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).

Mathematica

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

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.

Sequence in context: A360683 A317085 A236543 * A364674 A319079 A325329

Adjacent sequences: A360242 A360243 A360244 * A360246 A360247 A360248

Keyword

nonn

Author

Gus Wiseman, Feb 05 2023

Status

approved