cp's OEIS Frontend

A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

%I A383519 #7 May 20 2025 21:58:37
%S A383519 1,1,2,2,3,3,6,7,9,12,14,19,21,27,30,33,41,50,57,68,79,89,112,126,144,
%T A383519 172,198,220,257,298,327,383,423,477,533,621,650,760,816,920,1013
%N A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).
%C A383519 An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
%C A383519 An integer partition is Wilf iff its multiplicities are all different (ranked by A130091).
%e A383519 The a(1) = 1 through a(8) = 9 partitions:
%e A383519   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A383519        (11)  (111)  (22)    (311)    (33)      (322)      (44)
%e A383519                     (1111)  (11111)  (222)     (331)      (332)
%e A383519                                      (411)     (511)      (611)
%e A383519                                      (3111)    (4111)     (2222)
%e A383519                                      (111111)  (31111)    (5111)
%e A383519                                                (1111111)  (41111)
%e A383519                                                           (311111)
%e A383519                                                           (11111111)
%t A383519 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383519 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383519 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383519 Table[Length[Select[IntegerPartitions[n],disjointFamilies[conj[#]]!={}&&UnsameQ@@Length/@Split[#]&]],{n,0,15}]
%Y A383519 Ranking sequences are shown in parentheses below.
%Y A383519 For Look-and-Say instead of section-sum we have A098859 (A130091), conjugate (A383512).
%Y A383519 For non Wilf instead of Wilf we have A383506 (A383514).
%Y A383519 These partitions are ranked by (A383520).
%Y A383519 A000041 counts integer partitions, strict A000009.
%Y A383519 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383519 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383519 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383519 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383519 Cf. A047966, A320348, A325324, A325325, A351592, A353837, A381431, A383530, A383709, A384006.
%K A383519 nonn,more
%O A383519 0,3
%A A383519 _Gus Wiseman_, May 19 2025