cp's OEIS Frontend

A383511 Number of integer partitions of n that are Look-and-Say and section-sum but not Wilf.

%I A383511 #7 May 18 2025 14:37:13
%S A383511 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,1,0,3,3,0,0,5,2,1,5,6,1,10,5,12,
%T A383511 11,12,14,31,15,25,28,38
%N A383511 Number of integer partitions of n that are Look-and-Say and section-sum but not Wilf.
%C A383511 A partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
%C A383511 A partition is section-sum iff its conjugate is Look-and-Say, meaning 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 A383511 A partition is Wilf iff its multiplicities are all different (ranked by A130091).
%e A383511 The a(n) partitions for n = 12, 15, 20, 24, 28:
%e A383511   (6,3,3)  (6,6,3)    (8,8,4)    (12,6,6)         (14,7,7)
%e A383511            (6,3,3,3)  (10,5,5)   (6,6,6,3,3)      (8,8,8,4)
%e A383511                       (8,4,4,4)  (8,4,4,4,4)      (8,8,4,4,4)
%e A383511                                  (6,6,3,3,3,3)    (8,4,4,4,4,4)
%e A383511                                  (6,3,3,3,3,3,3)  (10,6,6,2,2,2)
%e A383511                                                   (11,6,6,1,1,1,1,1)
%t A383511 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383511 conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383511 Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]!={} && !UnsameQ@@Length/@Split[#]&]], {n,0,30}]
%Y A383511 Ranking sequences are shown in parentheses below.
%Y A383511 This is the non Wilf case of A383508 (A383515).
%Y A383511 These partitions are ranked by (A383518).
%Y A383511 A000041 counts integer partitions, strict A000009.
%Y A383511 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383511 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383511 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383511 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383511 A351592 counts non Wilf Look-and-Say partitions (A384006).
%Y A383511 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383511 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383511 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383511 A383519 counts section-sum Wilf partitions (A383520).
%Y A383511 Cf. A033461, A047966, A048767, A122111, A217605, A320348, A325324, A325325, A381431, A383530, A383709.
%K A383511 nonn,more
%O A383511 0,16
%A A383511 _Gus Wiseman_, May 18 2025