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

1,2

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=1..47.

Example

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).

Mathematica

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

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.

Sequence in context: A262511 A075177 A062096 * A176008 A143827 A100797

Adjacent sequences: A360684 A360685 A360686 * A360688 A360689 A360690

Keyword

nonn

Author

Gus Wiseman, Feb 20 2023

Status

approved