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%I #6 Mar 18 2022 00:21:49
%S 1,0,1,1,1,1,1,2,2,3,3,3,3,4,5,6,7,7,8,9,11,12,13,14,16,18,21,23,25,
%T 28,31,34,37,41,45,50,55,60,65,72,79,86,93,102,111,121,132,143,155,
%U 169,183,197,213,231,251,271,292,315,340,367,396
%N Number of strict integer partitions of n with as many odd parts as even conjugate parts.
%e The a(n) strict partitions for selected n:
%e n = 2 7 9 13 14 15 16
%e --------------------------------------------------------------------
%e (2) (6,1) (8,1) (12,1) (14) (14,1) (16)
%e (4,2,1) (4,3,2) (6,4,3) (6,5,3) (6,5,4) (8,5,3)
%e (6,2,1) (8,3,2) (10,3,1) (8,4,3) (12,3,1)
%e (10,2,1) (6,4,3,1) (10,3,2) (6,5,4,1)
%e (8,3,2,1) (12,2,1) (8,4,3,1)
%e (6,5,3,1) (10,3,2,1)
%e (6,4,3,2,1)
%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[#,_?OddQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y This is the strict case of A277579, ranked by A350943 (zeros of A350942).
%Y The conjugate version is A352131, non-strict A277579 (ranked by A349157).
%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 - A045931, ranked by A325698, strict A239241.
%Y - A045931, ranked by A350848, strict A352129.
%Y - A277103, ranked by A350944, strict new.
%Y - A350948, ranked by A350945, strict new.
%Y There are three double-pairings of statistics:
%Y - A351976, ranked by A350949, strict A010054?
%Y - A351977, ranked by A350946, strict A352128.
%Y - A351981, ranked by A351980. strict A014105?
%Y The case of all four statistics equal is A351978, ranked by A350947.
%Y Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.
%K nonn
%O 0,8
%A _Gus Wiseman_, Mar 15 2022