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%I #6 Mar 18 2022 00:21:24
%S 1,0,0,1,0,0,0,0,1,2,1,0,2,2,2,4,2,1,6,8,7,9,13,14,15,19,21,23,32,40,
%T 41,45,66,81,80,96,124,139,160,194,221,246,303,360,390,446,546,634,
%U 703,810,971,1115,1250,1448,1685,1910
%N Number of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.
%e The a(n) partitions for selected n:
%e n = 3 9 15 18 19 20 21
%e -----------------------------------------------------------
%e 21 4221 622221 633222 633322 644321 643332
%e 4311 632211 643221 643321 653321 654321
%e 642111 643311 644221 654221 665211
%e 651111 644211 644311 654311 82222221
%e 653211 653221 82222211 83222211
%e 663111 653311 84221111 84222111
%e 654211 86111111 85221111
%e 664111 86211111
%e 87111111
%e For example, the partition (6,6,3,1,1,1) has conjugate (6,3,3,2,2,2), and has 2 even, 4 odd, 4 even conjugate, and 2 odd conjugate parts, so is counted under a(18).
%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[#],_?OddQ]&&Count[#,_?OddQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y The first condition alone is A277579, ranked by A349157.
%Y The second condition alone is A277579, ranked by A350943.
%Y These partitions are ranked by A351980.
%Y There are four statistics:
%Y - A257991 = # of odd parts, conjugate A344616.
%Y - A257992 = # of even parts, conjugate A350847.
%Y There are four other pairings of statistics:
%Y - A045931: # of even parts = # of odd parts:
%Y - conjugate also A045931
%Y - ordered A098123
%Y - strict A239241
%Y - ranked by A325698
%Y - conjugate ranked by A350848
%Y - A277103: # of odd parts = # of odd conjugate parts, ranked by A350944.
%Y - A350948: # of even parts = # of even conjugate parts, ranked by A350945.
%Y There are two other double-pairings of statistics:
%Y - A351976, ranked by A350949.
%Y - A351977, ranked by A350946.
%Y The case of all four statistics equal is A351978, ranked by A350947.
%Y Cf. A000070, A088218, A122111, A130780, A171966, A195017, A236559, A236914, A350849, A350942.
%K nonn
%O 0,10
%A _Gus Wiseman_, Mar 15 2022
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