

A363220


Number of integer partitions of n whose conjugate has the same median.


1



1, 0, 1, 1, 1, 3, 3, 8, 8, 12, 12, 15, 21, 27, 36, 49, 65, 85, 112, 149, 176, 214, 257, 311, 378, 470, 572, 710, 877, 1080, 1322, 1637, 1983, 2416, 2899, 3465, 4107, 4891, 5763, 6820, 8071, 9542, 11289, 13381, 15808, 18710, 22122, 26105, 30737, 36156, 42377
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OFFSET

1,6


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).


LINKS



EXAMPLE

The partition y = (4,3,1,1) has median 2, and its conjugate (4,2,2,1) also has median 2, so y is counted under a(9).
The a(1) = 1 through a(9) = 8 partitions:
(1) . (21) (22) (311) (321) (511) (332) (333)
(411) (4111) (422) (711)
(3111) (31111) (611) (4221)
(3311) (4311)
(4211) (6111)
(5111) (51111)
(41111) (411111)
(311111) (3111111)


MATHEMATICA

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Median[#]==Median[conj[#]]&]], {n, 30}]


CROSSREFS

For mean instead of median we have A047993.
Median of conjugate by rank is A363219.
These partitions are ranked by A363261.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A352491 gives n minus Heinz number of conjugate.
Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.


KEYWORD

nonn


AUTHOR



STATUS

approved



