login
A360455
Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.
7
1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 5, 8, 10, 14, 20, 19, 26, 31, 35, 41, 55, 65, 85, 102, 118, 151, 181, 201, 236, 281, 313, 365, 424, 495, 593, 688, 825, 978, 1181, 1374, 1650, 1948, 2323, 2682, 3175, 3680, 4314, 4930, 5718, 6546, 7532, 8557, 9777, 11067, 12622
OFFSET
0,5
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) = 1 through a(11) = 8 partitions:
1 . . 22 221 3111 . 3311 333 3331 32222
211 41111 32211 33211 33221
42211 44111
322111 52211
511111 322211
332111
422111
3221111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]], {n, 0, 30}]
CROSSREFS
For mean instead of median: A114638, ranks A324570.
For parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360453.
For parts instead of distinct parts: A360456, ranks A360454.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Sequence in context: A366388 A114638 A123340 * A267486 A285229 A227425
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 10 2023
STATUS
approved