A363220 Number of integer partitions of n whose conjugate has the same median.

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

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

Table of n, a(n) for n=1..51.

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.

Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.

Sequence in context: A305841 A331530 A199624 * A093366 A204136 A168283

Adjacent sequences: A363217 A363218 A363219 * A363221 A363222 A363223

Keyword

nonn

Author

Gus Wiseman, May 29 2023

Status

approved