%I #7 Mar 16 2022 16:37:48

%S 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 257,316,376,458,544,671,792,952,1129,1351,1598,1919,2259,2681,3155,

%U 3739,4384,5181,6064,7129,8331,9764,11380,13308,15477,18047,20944

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

%e The a(0) = 1 through a(8) = 6 partitions (empty column indicated by dot):

%e () (1) . (3) (22) (5) (42) (7) (62)

%e (21) (41) (321) (61) (332)

%e (111) (311) (2211) (511) (521)

%e (2111) (4111) (4211)

%e (11111) (31111) (32111)

%e (211111) (221111)

%e (1111111)

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

%t Table[Length[Select[IntegerPartitions[n],Count[#,_?EvenQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]

%Y Comparing even to odd parts gives A045931, ranked by A325698.

%Y The odd version is A277103, even rank case A345196, ranked by A350944.

%Y Comparing even to odd conjugate parts gives A277579, ranked by A349157.

%Y Comparing product of parts to product of conjugate parts gives A325039.

%Y These partitions are ranked by A350945, the zeros of A350950.

%Y A000041 counts integer partitions, strict A000009.

%Y A103919 counts partitions by sum and alternating sum, reverse A344612.

%Y A116482 counts partitions by number of even (or even conjugate) parts.

%Y A122111 represents partition conjugation using Heinz numbers.

%Y A257991 counts odd parts, conjugate A344616.

%Y A257992 counts even parts, conjugate A350847.

%Y A351976: # even = # even conj, # odd = # odd conj, ranked by A350949.

%Y A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.

%Y A351978: # even = # odd = # even conj = # odd conj, ranked by A350947.

%Y A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.

%Y Cf. A027187, A130780, A171966, A195017, A239241, A241638, A344607, A344651, A350848, A350941, A350942, A350943.

%K nonn

%O 0,4

%A _Gus Wiseman_, Mar 14 2022