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%I #6 Mar 18 2022 00:21:40
%S 1,0,0,1,0,0,1,0,1,1,1,1,1,2,1,3,2,3,4,3,5,5,6,6,9,8,10,12,13,15,17,
%T 20,20,26,26,32,35,39,44,50,55,61,71,76,87,96,108,117,135,145,164,181,
%U 200,222,246,272,298,334,363,404,443
%N Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.
%e The a(n) strict partitions for selected n:
%e n = 3 13 15 18 20 22
%e ------------------------------------------------------------------
%e (2,1) (6,5,2) (10,5) (12,6) (12,7,1) (12,8,2)
%e (6,4,2,1) (6,4,3,2) (8,7,3) (8,5,4,3) (8,6,5,3)
%e (6,5,3,1) (8,5,3,2) (8,6,4,2) (8,7,5,2)
%e (8,6,3,1) (8,7,4,1) (12,7,2,1)
%e (8,6,3,2,1) (8,6,4,3,1)
%e (8,7,4,2,1)
%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],_?OddQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y This is the strict case of A045931, ranked by A350848 (zeros of A350941).
%Y The conjugate version is A239241, non-strict A045931 (ranked by A325698).
%Y A000041 counts integer partitions, strict A000009.
%Y A130780 counts partitions with no more even than odd parts, strict A239243.
%Y A171966 counts partitions with no more odd than even parts, strict A239240.
%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 - A277579, ranked by A349157, strict A352131.
%Y - A277103, ranked by A350944.
%Y - A277579, ranked by A350943, strict A352130.
%Y - A350948, ranked by A350945.
%Y There are three double-pairings of statistics:
%Y - A351976, ranked by A350949.
%Y - A351977, ranked by A350946, strict A352128.
%Y - A351981, ranked by A351980.
%Y The case of all four statistics equal is A351978, ranked by A350947.
%Y Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.
%K nonn
%O 0,14
%A _Gus Wiseman_, Mar 15 2022
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