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A383507 Number of Wilf and conjugate Wilf integer partitions of n.

%I A383507 #7 May 15 2025 08:23:11
%S A383507 1,1,2,2,3,3,6,7,9,12,14,19,20,27,30,31,40,50,56,68,76,86,112,126,139,
%T A383507 170,197,216,251,297,317,378,411,466,521,607,621,745,791,892,975,1123,
%U A383507 1163,1366,1439,1635,1757,2021,2080,2464,2599,2882,3116,3572,3713
%N A383507 Number of Wilf and conjugate Wilf integer partitions of n.
%C A383507 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 A383507 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%F A383507 These partitions have Heinz numbers A130091 /\ A383512.
%e A383507 The a(1) = 1 through a(8) = 9 partitions:
%e A383507   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A383507        (11)  (111)  (22)    (311)    (33)      (322)      (44)
%e A383507                     (1111)  (11111)  (222)     (331)      (332)
%e A383507                                      (411)     (511)      (611)
%e A383507                                      (3111)    (4111)     (2222)
%e A383507                                      (111111)  (31111)    (5111)
%e A383507                                                (1111111)  (41111)
%e A383507                                                           (311111)
%e A383507                                                           (11111111)
%t A383507 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#] && UnsameQ@@DeleteCases[Differences[Append[#,0]],0]&]],{n,0,30}]
%Y A383507 A048768 gives Look-and-Say fixed points, counted by A217605.
%Y A383507 A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
%Y A383507 A239455 counts Look-and-Say partitions, complement A351293.
%Y A383507 A325349 counts partitions with distinct augmented differences, ranks A325366.
%Y A383507 A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
%Y A383507 A381431 is the section-sum transform, union A381432, complement A381433.
%Y A383507 A383534 gives 0-prepended differences by rank, see A325351.
%Y A383507 A383709 counts Wilf partitions with distinct 0-appended differences.
%Y A383507 Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.
%K A383507 nonn
%O A383507 0,3
%A A383507 _Gus Wiseman_, May 14 2025