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A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

%I A383530 #8 May 15 2025 08:23:19
%S A383530 0,0,0,1,0,0,3,2,5,12,14,19,35,38,55,83,107,137,209,252,359,462,612,
%T A383530 757,1032,1266,1649,2050,2617,3210,4111,4980,6262,7659,9479,11484,
%U A383530 14224,17132,20962,25259,30693,36744,44517,53043,63850,75955,90943,107721,128485
%N A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.
%C A383530 An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).
%H A383530 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%F A383530 These partitions have Heinz numbers A130092 /\ A383513.
%e A383530 The a(0) = 0 through a(9) = 12 partitions:
%e A383530   .  .  .  (21)  .  .  (42)    (421)   (431)    (63)
%e A383530                        (321)   (3211)  (521)    (432)
%e A383530                        (2211)          (3221)   (531)
%e A383530                                        (4211)   (621)
%e A383530                                        (32111)  (3321)
%e A383530                                                 (4221)
%e A383530                                                 (4311)
%e A383530                                                 (5211)
%e A383530                                                 (32211)
%e A383530                                                 (42111)
%e A383530                                                 (222111)
%e A383530                                                 (321111)
%t A383530 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
%t A383530 Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&!UnsameQ@@Length/@Split[conj[#]]&]], {n,0,30}]
%Y A383530 Negating both sides gives A383507, ranks A383532.
%Y A383530 These partitions are ranked by A383531.
%Y A383530 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383530 A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
%Y A383530 A239455 counts Look-and-Say partitions, complement A351293.
%Y A383530 A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
%Y A383530 A381431 is the section-sum transform, union A381432, complement A381433.
%Y A383530 A383534 gives 0-prepended differences by rank, see A325351.
%Y A383530 A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
%Y A383530 Cf. A033461, A047966, A111133, A320348, A325324, A325325, A325349, A325367, A325368, A325388, A383506.
%K A383530 nonn
%O A383530 0,7
%A A383530 _Gus Wiseman_, May 14 2025