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A351978 Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.

%I A351978 #6 Mar 18 2022 00:21:09
%S A351978 1,0,0,1,0,0,0,0,1,0,1,0,2,0,0,2,0,1,0,6,1,3,1,8,5,3,5,7,14,2,13,9,28,
%T A351978 5,22,26,44,17,30,60,59,42,41,120,84,84,66,204,143,144,131,325,268,
%U A351978 226,261,486,498,344,488,739,874
%N A351978 Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.
%e A351978 The a(n) partitions for selected n (A = 10):
%e A351978 n = 3    12     19       21       23       24         27
%e A351978    --------------------------------------------------------------
%e A351978     21   4332   633322   643332   644333   84332211   655443
%e A351978          4431   643321   654321   654332   84441111   655542
%e A351978                 644311   665211   654431   85322211   665541
%e A351978                 653221            655322   86322111   666333
%e A351978                 654211            655421   86421111   666531
%e A351978                 664111            664331              A522221111
%e A351978                                   665321              A622211111
%e A351978                                   666311
%t A351978 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A351978 Table[Length[Select[IntegerPartitions[n],Count[#,_?EvenQ]==Count[#,_?OddQ]==Count[conj[#],_?EvenQ]==Count[conj[#],_?OddQ]&]],{n,0,30}]
%Y A351978 The strict case appears to be the indicator function for A014105.
%Y A351978 These partitions are ranked by A350947.
%Y A351978 There are four statistics:
%Y A351978 - A257991 = # of odd parts, conjugate A344616.
%Y A351978 - A257992 = # of even parts, conjugate A350847.
%Y A351978 There are six pairings of statistics:
%Y A351978 - A045931: # of even parts = # of odd parts:
%Y A351978   - ordered A098123
%Y A351978   - strict A239241
%Y A351978   - ranked by A325698
%Y A351978 - A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
%Y A351978 - A277579: # even = # odd conj, ranked by A349157, strict A352131.
%Y A351978 - A277103: # odd = # odd conj, ranked by A350944, strict A000700.
%Y A351978 - A277579: # even conj = # odd, ranked by A350943, strict A352130.
%Y A351978 - A350948: # even = # even conj, ranked by A350945.
%Y A351978 There are three double-pairings of statistics:
%Y A351978 - A351976, ranked by A350949.
%Y A351978 - A351977, ranked by A350946.
%Y A351978 - A351981, ranked by A351980.
%Y A351978 A000041 counts integer partitions, strict A000009.
%Y A351978 A103919 and A116482 count partitions by sum and number of odd/even parts.
%Y A351978 A195017 = # of even parts - # of odd parts.
%Y A351978 Cf. A000070, A122111, A130780, A171966, A236559, A236914, A350849, A350941, A350942, A350950, A350951.
%K A351978 nonn
%O A351978 0,13
%A A351978 _Gus Wiseman_, Mar 15 2022