cp's OEIS Frontend

A383506 Number of non Wilf section-sum partitions of n.

%I A383506 #10 May 18 2025 14:37:28
%S A383506 0,0,0,0,1,2,1,3,4,4,7,9,12,18,25,32,42,55,64,87,101,128,147,192,218,
%T A383506 273,314,394,450,552,631,772,886,1066,1221,1458,1677,1980,2269,2672,
%U A383506 3029
%N A383506 Number of non Wilf section-sum partitions of n.
%C A383506 An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
%C A383506 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.
%e A383506 The a(4) = 1 through a(12) = 12 partitions (A=10, B=11):
%e A383506   (31)  (32)  (51)  (43)  (53)    (54)  (64)    (65)    (75)
%e A383506         (41)        (52)  (62)    (63)  (73)    (74)    (84)
%e A383506                     (61)  (71)    (72)  (82)    (83)    (93)
%e A383506                           (3311)  (81)  (91)    (92)    (A2)
%e A383506                                         (631)   (A1)    (B1)
%e A383506                                         (3322)  (632)   (732)
%e A383506                                         (4411)  (641)   (831)
%e A383506                                                 (731)   (5511)
%e A383506                                                 (6311)  (6411)
%e A383506                                                         (7311)
%e A383506                                                         (63111)
%e A383506                                                         (333111)
%t A383506 disjointDiffs[y_]:=Select[Tuples[IntegerPartitions /@ Differences[Prepend[Sort[y],0]]], UnsameQ@@Join@@#&];
%t A383506 Table[Length[Select[IntegerPartitions[n], disjointDiffs[#]!={} && !UnsameQ@@Length/@Split[#]&]],{n,0,15}]
%Y A383506 Ranking sequences are shown in parentheses below.
%Y A383506 For Look-and-Say instead of section-sum we have A351592 (A384006).
%Y A383506 The Look-and-Say case is A383511 (A383518).
%Y A383506 These partitions are ranked by (A383514).
%Y A383506 For Wilf instead of non Wilf we have A383519 (A383520).
%Y A383506 A000041 counts integer partitions, strict A000009.
%Y A383506 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383506 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383506 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383506 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383506 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A383506 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383506 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383506 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383506 Cf. A047966, A048767, A122111, A320348, A325324, A325325, A353837, A381431, A383530, A383709.
%K A383506 nonn,more
%O A383506 0,6
%A A383506 _Gus Wiseman_, May 18 2025