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

%I A363220 #7 May 31 2023 10:48:53
%S A363220 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 A363220 378,470,572,710,877,1080,1322,1637,1983,2416,2899,3465,4107,4891,
%U A363220 5763,6820,8071,9542,11289,13381,15808,18710,22122,26105,30737,36156,42377
%N A363220 Number of integer partitions of n whose conjugate has the same median.
%C A363220 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 A363220 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 A363220 The a(1) = 1 through a(9) = 8 partitions:
%e A363220   (1)  .  (21)  (22)  (311)  (321)   (511)    (332)     (333)
%e A363220                              (411)   (4111)   (422)     (711)
%e A363220                              (3111)  (31111)  (611)     (4221)
%e A363220                                               (3311)    (4311)
%e A363220                                               (4211)    (6111)
%e A363220                                               (5111)    (51111)
%e A363220                                               (41111)   (411111)
%e A363220                                               (311111)  (3111111)
%t A363220 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A363220 Table[Length[Select[IntegerPartitions[n],Median[#]==Median[conj[#]]&]],{n,30}]
%Y A363220 For mean instead of median we have A047993.
%Y A363220 For product instead of median we have A325039, ranks A325040.
%Y A363220 For union instead of conjugate we have A360245, complement A360244.
%Y A363220 Median of conjugate by rank is A363219.
%Y A363220 These partitions are ranked by A363261.
%Y A363220 A000700 counts self-conjugate partitions, ranks A088902.
%Y A363220 A046682 and A352487-A352490 pertain to excedance set.
%Y A363220 A122111 represents partition conjugation.
%Y A363220 A325347 counts partitions with integer median.
%Y A363220 A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.
%Y A363220 A352491 gives n minus Heinz number of conjugate.
%Y A363220 Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.
%K A363220 nonn
%O A363220 1,6
%A A363220 _Gus Wiseman_, May 29 2023