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A360456
Number of integer partitions of n for which the parts have the same median as the multiplicities.
5
1, 1, 0, 0, 1, 0, 0, 1, 2, 5, 7, 10, 14, 21, 28, 36, 51, 64, 84, 106, 132, 165, 202, 252, 311, 391, 473, 579, 713, 868, 1069, 1303, 1617, 1954, 2404, 2908, 3556, 4282, 5200, 6207, 7505, 8934, 10700, 12717, 15165, 17863, 21222, 24976, 29443, 34523, 40582, 47415
OFFSET
0,9
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) = 10 partitions:
1 . . 22 . . 2221 3311 333 4222 5222
32111 3222 33211 33221
32211 42211 52211
42111 43111 53111
321111 52111 62111
421111 322211
3211111 431111
521111
4211111
32111111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[#]&]], {n, 0, 30}]
CROSSREFS
For mean instead of median: A360068, ranks A359903.
For distinct parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360454.
For distinct parts instead of parts: A360455, ranks A360453.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Sequence in context: A094019 A161580 A024177 * A276465 A018406 A018483
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 10 2023
STATUS
approved