A360687 Number of integer partitions of n whose multiplicities have integer median.

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676

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)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).
		

Crossrefs

The case of an odd number of multiplicities is A090794.

For mean instead of median we have A360069, ranks A067340.

These partitions have ranks A360553.

The complement is counted by A360690, ranks A360554.

A058398 counts partitions by mean, see also A008284, A327482.

A124010 gives prime signature, sorted A118914, mean A088529/A088530.

A325347 = partitions w/ integer median, strict A359907, complement A307683.

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

Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]