cp's OEIS Frontend

A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

%I A383509 #9 May 18 2025 14:37:19
%S A383509 0,0,0,0,1,2,1,3,4,4,7,9,11,18,25,30,41,55,63,87,98,125,147,192,213,
%T A383509 271,313,389,444,551,621,767,874,1055,1209,1444,1646,1965,2244,2644,
%U A383509 2991
%N A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.
%C A383509 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 A383509 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.
%e A383509 The a(4) = 1 through a(11) = 9 partitions:
%e A383509   211  221   21111  2221    422      22221     442        222221
%e A383509        2111         22111   22211    222111    4222       322211
%e A383509                     211111  221111   2211111   222211     332111
%e A383509                             2111111  21111111  322111     422111
%e A383509                                                2221111    2222111
%e A383509                                                22111111   3221111
%e A383509                                                211111111  22211111
%e A383509                                                           221111111
%e A383509                                                           2111111111
%e A383509 Conjugates of the a(4) = 1 through a(11) = 9 partitions:
%e A383509   (3,1)  (3,2)  (5,1)  (4,3)  (5,3)      (5,4)  (6,4)      (6,5)
%e A383509          (4,1)         (5,2)  (6,2)      (6,3)  (7,3)      (7,4)
%e A383509                        (6,1)  (7,1)      (7,2)  (8,2)      (8,3)
%e A383509                               (3,3,1,1)  (8,1)  (9,1)      (9,2)
%e A383509                                                 (6,3,1)    (10,1)
%e A383509                                                 (3,3,2,2)  (6,3,2)
%e A383509                                                 (4,4,1,1)  (6,4,1)
%e A383509                                                            (7,3,1)
%e A383509                                                            (6,3,1,1)
%t A383509 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383509 conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
%t A383509 Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]=={}&]], {n,0,30}]
%Y A383509 Ranking sequences are shown in parentheses below.
%Y A383509 These partitions are ranked by (A383516).
%Y A383509 A000041 counts integer partitions, strict A000009.
%Y A383509 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383509 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383509 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383509 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383509 A351592 counts non Wilf Look-and-Say partitions (A384006).
%Y A383509 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A383509 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383509 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383509 A383519 counts section-sum Wilf partitions (A383520).
%Y A383509 Cf. A047966, A048767, A320348, A325324, A325325, A353837, A381431, A383511, A383530, A383709.
%K A383509 nonn,more
%O A383509 0,6
%A A383509 _Gus Wiseman_, May 18 2025