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A360690
Number of integer partitions of n with non-integer median of multiplicities.
2
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
1,5
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).
EXAMPLE
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).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]], {n, 30}]
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.
Sequence in context: A355345 A034803 A205885 * A239262 A288496 A292726
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 22 2023
STATUS
approved