A360690 Number of integer partitions of n with non-integer median of multiplicities.

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Author

Gus Wiseman, Feb 22 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) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
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 not counted under a(8).
		

Crossrefs

These partitions have ranks A360554.

The complement is counted by A360687, ranks A360553.

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.

A360069 = partitions with integer mean of multiplicities, ranks A067340.

Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

Mathematica

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