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A350948 Number of integer partitions of n with as many even parts as even conjugate parts.

%I A350948 #7 Mar 16 2022 16:37:48
%S A350948 1,1,0,3,1,5,3,7,6,10,10,18,19,27,31,40,47,65,75,98,115,142,170,217,
%T A350948 257,316,376,458,544,671,792,952,1129,1351,1598,1919,2259,2681,3155,
%U A350948 3739,4384,5181,6064,7129,8331,9764,11380,13308,15477,18047,20944
%N A350948 Number of integer partitions of n with as many even parts as even conjugate parts.
%e A350948 The a(0) = 1 through a(8) = 6 partitions (empty column indicated by dot):
%e A350948   ()  (1)  .  (3)    (22)  (5)      (42)    (7)        (62)
%e A350948               (21)         (41)     (321)   (61)       (332)
%e A350948               (111)        (311)    (2211)  (511)      (521)
%e A350948                            (2111)           (4111)     (4211)
%e A350948                            (11111)          (31111)    (32111)
%e A350948                                             (211111)   (221111)
%e A350948                                             (1111111)
%e A350948 For example, both (3,2,1,1,1) and its conjugate (5,2,1) have exactly 1 even part, so are counted under a(8).
%t A350948 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350948 Table[Length[Select[IntegerPartitions[n],Count[#,_?EvenQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y A350948 Comparing even to odd parts gives A045931, ranked by A325698.
%Y A350948 The odd version is A277103, even rank case A345196, ranked by A350944.
%Y A350948 Comparing even to odd conjugate parts gives A277579, ranked by A349157.
%Y A350948 Comparing product of parts to product of conjugate parts gives A325039.
%Y A350948 These partitions are ranked by A350945, the zeros of A350950.
%Y A350948 A000041 counts integer partitions, strict A000009.
%Y A350948 A103919 counts partitions by sum and alternating sum, reverse A344612.
%Y A350948 A116482 counts partitions by number of even (or even conjugate) parts.
%Y A350948 A122111 represents partition conjugation using Heinz numbers.
%Y A350948 A257991 counts odd parts, conjugate A344616.
%Y A350948 A257992 counts even parts, conjugate A350847.
%Y A350948 A351976: # even = # even conj, # odd = # odd conj, ranked by A350949.
%Y A350948 A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.
%Y A350948 A351978: # even = # odd = # even conj = # odd conj, ranked by A350947.
%Y A350948 A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.
%Y A350948 Cf. A027187, A130780, A171966, A195017, A239241, A241638, A344607, A344651, A350848, A350941, A350942, A350943.
%K A350948 nonn
%O A350948 0,4
%A A350948 _Gus Wiseman_, Mar 14 2022