A361653 Number of even-length integer partitions of n with integer median.

0, 0, 1, 0, 3, 1, 5, 3, 11, 7, 17, 16, 32, 31, 52, 55, 90, 99, 144, 167, 236, 273, 371, 442, 587, 696, 901, 1078, 1379, 1651, 2074, 2489, 3102, 3707, 4571, 5467, 6692, 7982, 9696, 11543, 13949, 16563, 19891, 23572, 28185, 33299, 39640, 46737, 55418, 65164

Offset: 0

Author

Gus Wiseman, Mar 23 2023

Keywords

• nonn

Comments

The median of an even-length multiset is the average of the two middle parts.

Because any odd-length partition has integer median, the odd-length version is counted by A027193, strict case A067659.

Examples

			The a(2) = 1 through a(9) = 7 partitions:
  (11)  .  (22)    (2111)  (33)      (2221)    (44)        (3222)
           (31)            (42)      (4111)    (53)        (4221)
           (1111)          (51)      (211111)  (62)        (4311)
                           (3111)              (71)        (6111)
                           (111111)            (2222)      (321111)
                                               (3221)      (411111)
                                               (3311)      (21111111)
                                               (5111)
                                               (221111)
                                               (311111)
                                               (11111111)
For example, the partition (4,3,1,1) has length 4 and median 2, so is counted under a(9).
		

Crossrefs

The odd-length version is counted by A027193, strict A067659.

Including odd-length partitions gives A307683, complement A325347.

For mean instead of median we have A361655, any length A067538.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median, mean A051293.

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

Cf. A008284, A013580, A079309, A240219, A240850, A349156, A359897, A359908, A359912, A360005, A360952.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&IntegerQ[Median[#]]&]],{n,0,30}]