cp's OEIS Frontend

A383510 Number of integer partitions of n that are neither Look-and-Say nor section-sum.

%I A383510 #9 May 18 2025 14:37:23
%S A383510 0,0,0,1,0,0,3,2,5,10,14,19,33,38,55,81,107,137,201,248,349,450,596,
%T A383510 745,1000,1242,1611,2007,2567,3164,4025,4920,6166,7545,9347,11360,
%U A383510 14004,16932,20686,24949,30305,36366,43939,52521,63098,75221
%N A383510 Number of integer partitions of n that are neither Look-and-Say nor section-sum.
%C A383510 An integer 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 A383510 An integer 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 A383510 The a(3) = 1 through a(10) = 14 partitions:
%e A383510   (21)  .  .  (42)    (421)   (431)    (432)     (532)
%e A383510               (321)   (3211)  (521)    (531)     (541)
%e A383510               (2211)          (3221)   (621)     (721)
%e A383510                               (4211)   (3321)    (4321)
%e A383510                               (32111)  (4221)    (5221)
%e A383510                                        (4311)    (5311)
%e A383510                                        (5211)    (6211)
%e A383510                                        (32211)   (32221)
%e A383510                                        (42111)   (33211)
%e A383510                                        (321111)  (42211)
%e A383510                                                  (43111)
%e A383510                                                  (52111)
%e A383510                                                  (421111)
%e A383510                                                  (3211111)
%t A383510 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383510 conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
%t A383510 Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]=={} && disjointFamilies[conj[#]]=={}&]], {n,0,15}]
%Y A383510 Ranking sequences are shown in parentheses below.
%Y A383510 These partitions are ranked by (A383517).
%Y A383510 A000041 counts integer partitions, strict A000009.
%Y A383510 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A383510 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383510 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383510 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383510 A351592 counts non Wilf Look-and-Say partitions (A384006).
%Y A383510 A381431 is the section-sum transform.
%Y A383510 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A383510 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383510 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383510 A383519 counts section-sum Wilf partitions (A383520).
%Y A383510 A383530 counts partitions that are neither Wilf nor conjugate Wilf (A383531).
%Y A383510 Cf. A033461, A047966, A098859, A122111, A320348, A325324, A325325, A383511, A383709.
%K A383510 nonn
%O A383510 0,7
%A A383510 _Gus Wiseman_, May 18 2025