A360952 Number of strict integer partitions of n with non-integer median; a(0) = 1.

1, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 6, 1, 8, 4, 11, 5, 15, 10, 20, 13, 27, 22, 36, 28, 47, 43, 63, 56, 82, 79, 107, 103, 140, 141, 180, 181, 232, 242, 299, 308, 380, 402, 483, 511, 613, 656, 772, 824, 969, 1047, 1215, 1309, 1514, 1642, 1882, 2039, 2334, 2539, 2882

Offset: 0

Author

Gus Wiseman, Mar 10 2023

Keywords

• nonn

Comments

All of these partitions have even length.

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(0) = 1 through a(15) = 11 partitions (0 = {}, A..E = 10..14):
  0  .  .  21  .  32  .  43  .  54  4321  65    6321  76    5432  87
                  41     52     63        74          85    6431  96
                         61     72        83          94    6521  A5
                                81        92          A3    8321  B4
                                          A1          B2          C3
                                          5321        C1          D2
                                                      5431        E1
                                                      7321        6432
                                                                  7431
                                                                  7521
                                                                  9321
		

Crossrefs

The non-strict version is A307683, ranks A359912.

The non-strict complement is A325347, ranks A359908.

The strict complement is counted by A359907.

For mean instead of median we have A361391, non-strict A349156.

A000041 counts partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A067538 = partitions with integer mean, complement A102627, ranks A316413.

A359893/A359901/A359902 count partitions by median.

A360005(n)/2 ranks the median statistic.

Cf. A000975, A051293, A082550, A240219, A240850, A327475, A359897.

Programs

Mathematica

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

Formula

a(n) = A000009(n) - A359907(n).