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Number of integer partitions of n whose conjugate has the same median.
1

%I #7 May 31 2023 10:48:53

%S 1,0,1,1,1,3,3,8,8,12,12,15,21,27,36,49,65,85,112,149,176,214,257,311,

%T 378,470,572,710,877,1080,1322,1637,1983,2416,2899,3465,4107,4891,

%U 5763,6820,8071,9542,11289,13381,15808,18710,22122,26105,30737,36156,42377

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

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e 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).

%e The a(1) = 1 through a(9) = 8 partitions:

%e (1) . (21) (22) (311) (321) (511) (332) (333)

%e (411) (4111) (422) (711)

%e (3111) (31111) (611) (4221)

%e (3311) (4311)

%e (4211) (6111)

%e (5111) (51111)

%e (41111) (411111)

%e (311111) (3111111)

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Select[IntegerPartitions[n],Median[#]==Median[conj[#]]&]],{n,30}]

%Y For mean instead of median we have A047993.

%Y For product instead of median we have A325039, ranks A325040.

%Y For union instead of conjugate we have A360245, complement A360244.

%Y Median of conjugate by rank is A363219.

%Y These partitions are ranked by A363261.

%Y A000700 counts self-conjugate partitions, ranks A088902.

%Y A046682 and A352487-A352490 pertain to excedance set.

%Y A122111 represents partition conjugation.

%Y A325347 counts partitions with integer median.

%Y A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.

%Y A352491 gives n minus Heinz number of conjugate.

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

%K nonn

%O 1,6

%A _Gus Wiseman_, May 29 2023