The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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.
For product instead of median we have A325039, ranks A325040.
For union instead of conjugate we have A360245, complement A360244.
Median of conjugate by rank is A363219.
These partitions are ranked by A363261.
A000700 counts self-conjugate partitions, ranks A088902.
A046682 and A352487-A352490 pertain to excedance set.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.
A352491 gives n minus Heinz number of conjugate.
Sequence in context: A305841 A331530 A199624 * A093366 A204136 A168283
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 29 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 17 03:50 EDT 2024. Contains 373432 sequences. (Running on oeis4.)