cp's OEIS Frontend

A384392 Number of integer partitions of n whose distinct parts are maximally refined.

%I A384392 #12 Jun 10 2025 16:26:24
%S A384392 1,1,2,2,4,6,7,10,14,20,24,33,41,55,70,88,110,140,171,214,265,324,397,
%T A384392 485,588,711,861,1032,1241,1486,1773
%N A384392 Number of integer partitions of n whose distinct parts are maximally refined.
%C A384392 Given any partition, the following are equivalent:
%C A384392 1) The distinct parts are maximally refined.
%C A384392 2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
%C A384392 3) No part is a sum of distinct non-parts.
%e A384392 The a(1) = 1 through a(8) = 14 partitions:
%e A384392   (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
%e A384392        (11)  (111)  (31)    (41)     (321)     (331)      (431)
%e A384392                     (211)   (221)    (411)     (421)      (521)
%e A384392                     (1111)  (311)    (2211)    (2221)     (2222)
%e A384392                             (2111)   (3111)    (3211)     (3221)
%e A384392                             (11111)  (21111)   (4111)     (3311)
%e A384392                                      (111111)  (22111)    (4211)
%e A384392                                                (31111)    (22211)
%e A384392                                                (211111)   (32111)
%e A384392                                                (1111111)  (41111)
%e A384392                                                           (221111)
%e A384392                                                           (311111)
%e A384392                                                           (2111111)
%e A384392                                                           (11111111)
%t A384392 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A384392 Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}]
%Y A384392 The strict case is A179009, ranks A383707.
%Y A384392 For subsets instead of partitions we have A326080, complement A384350.
%Y A384392 These partitions are ranked by A384320, complement A384321.
%Y A384392 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384392 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384392 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384392 Cf. A179822, A279375, A279790, A299200, A317142, A326083, A357982, A383706, A383708, A383710, A384317, A384318, A384319, A384391.
%K A384392 nonn,more
%O A384392 0,3
%A A384392 _Gus Wiseman_, Jun 07 2025