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A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

%I A352130 #6 Mar 18 2022 00:21:49
%S A352130 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 A352130 28,31,34,37,41,45,50,55,60,65,72,79,86,93,102,111,121,132,143,155,
%U A352130 169,183,197,213,231,251,271,292,315,340,367,396
%N A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.
%e A352130 The a(n) strict partitions for selected n:
%e A352130 n = 2    7        9        13        14         15         16
%e A352130    --------------------------------------------------------------------
%e A352130     (2)  (6,1)    (8,1)    (12,1)    (14)       (14,1)     (16)
%e A352130          (4,2,1)  (4,3,2)  (6,4,3)   (6,5,3)    (6,5,4)    (8,5,3)
%e A352130                   (6,2,1)  (8,3,2)   (10,3,1)   (8,4,3)    (12,3,1)
%e A352130                            (10,2,1)  (6,4,3,1)  (10,3,2)   (6,5,4,1)
%e A352130                                      (8,3,2,1)  (12,2,1)   (8,4,3,1)
%e A352130                                                 (6,5,3,1)  (10,3,2,1)
%e A352130                                                            (6,4,3,2,1)
%t A352130 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352130 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?OddQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y A352130 This is the strict case of A277579, ranked by A350943 (zeros of A350942).
%Y A352130 The conjugate version is A352131, non-strict A277579 (ranked by A349157).
%Y A352130 A000041 counts integer partitions, strict A000009.
%Y A352130 A130780 counts partitions with no more even than odd parts, strict A239243.
%Y A352130 A171966 counts partitions with no more odd than even parts, strict A239240.
%Y A352130 There are four statistics:
%Y A352130 - A257991 = # of odd parts, conjugate A344616.
%Y A352130 - A257992 = # of even parts, conjugate A350847.
%Y A352130 There are four other pairings of statistics:
%Y A352130 - A045931, ranked by A325698, strict A239241.
%Y A352130 - A045931, ranked by A350848, strict A352129.
%Y A352130 - A277103, ranked by A350944, strict new.
%Y A352130 - A350948, ranked by A350945, strict new.
%Y A352130 There are three double-pairings of statistics:
%Y A352130 - A351976, ranked by A350949, strict A010054?
%Y A352130 - A351977, ranked by A350946, strict A352128.
%Y A352130 - A351981, ranked by A351980. strict A014105?
%Y A352130 The case of all four statistics equal is A351978, ranked by A350947.
%Y A352130 Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.
%K A352130 nonn
%O A352130 0,8
%A A352130 _Gus Wiseman_, Mar 15 2022