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A351977 Number of integer partitions of n with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

%I A351977 #5 Mar 18 2022 00:21:02
%S A351977 1,0,0,1,0,0,0,0,1,2,1,0,2,0,2,4,2,1,6,6,7,9,11,10,13,17,17,21,28,36,
%T A351977 35,41,58,71,72,90,106,121,142,178,191,216,269,320,344,400,486,564,
%U A351977 633,734,867,991,1130,1312,1509,1702,1978,2288,2582,2917,3404
%N A351977 Number of integer partitions of n with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.
%e A351977 The a(n) partitions for selected n (A..C = 10..12):
%e A351977 n = 3     9         15            18          20
%e A351977    ----------------------------------------------------------
%e A351977     (21)  (63)      (A5)          (8433)      (8543)
%e A351977           (222111)  (632211)      (8532)      (8741)
%e A351977                     (642111)      (8631)      (C611)
%e A351977                     (2222211111)  (43322211)  (43332221)
%e A351977                                   (44322111)  (44432111)
%e A351977                                   (44421111)  (84221111)
%e A351977                                               (422222111111)
%t A351977 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A351977 Table[Length[Select[IntegerPartitions[n],Count[#,_?OddQ]==Count[#,_?EvenQ]&&Count[conj[#],_?OddQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y A351977 The first condition alone is A045931, ranked by A325698, strict A239241.
%Y A351977 The second condition alone is A045931, ranked by A350848, strict A352129.
%Y A351977 These partitions are ranked by A350946.
%Y A351977 The strict case is A352128.
%Y A351977 There are four statistics:
%Y A351977 - A257991 = # of odd parts, conjugate A344616.
%Y A351977 - A257992 = # of even parts, conjugate A350847.
%Y A351977 There are four additional pairings of statistics:
%Y A351977 - A277579: # even = # odd conj, ranked by A349157, strict A352131.
%Y A351977 - A277579: # even conj = # odd, ranked by A350943, strict A352130.
%Y A351977 - A277103: # odd = # odd conj, ranked by A350944, strict A000700.
%Y A351977 - A350948: # even = # even conj, ranked by A350945.
%Y A351977 There are two additional double-pairings of statistics:
%Y A351977 - A351981, ranked by A351980.
%Y A351977 - A351976, ranked by A350949.
%Y A351977 The case of all four statistics equal is A351978, ranked by A350947.
%Y A351977 Cf. A000041, A000070, A088218, A098123, A130780, A171966, A195017, A236559, A236914, A241638, A350849.
%K A351977 nonn
%O A351977 0,10
%A A351977 _Gus Wiseman_, Mar 14 2022