cp's OEIS Frontend

A384318 Number of strict integer partitions of n that are not maximally refined.

%I A384318 #14 Jun 11 2025 23:41:02
%S A384318 0,0,0,1,1,1,3,4,4,5,9,10,13,15,17,26,29,36,43,49,57,74,84,101,118,
%T A384318 136,158,181,219,248,291
%N A384318 Number of strict integer partitions of n that are not maximally refined.
%C A384318 This is the number of strict integer partitions of n containing at least one sum of distinct non-parts.
%C A384318 Conjecture: Also the number of strict integer partitions of n such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each part.
%F A384318 a(n) = A000009(n) - A179009(n).
%e A384318 For y = (5,4,2) we have 4 = 3+1 so y is counted under a(11).
%e A384318 On the other hand, no part of z = (6,4,1) is a subset-sum of the non-parts {2,3,5}, so z is not counted under a(11).
%e A384318 The a(3) = 1 through a(11) = 10 strict partitions:
%e A384318   (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
%e A384318                  (4,2)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
%e A384318                  (5,1)  (5,2)  (6,2)  (6,3)  (7,3)    (7,4)
%e A384318                         (6,1)  (7,1)  (7,2)  (8,2)    (8,3)
%e A384318                                       (8,1)  (9,1)    (9,2)
%e A384318                                              (5,3,2)  (10,1)
%e A384318                                              (5,4,1)  (5,4,2)
%e A384318                                              (6,3,1)  (6,3,2)
%e A384318                                              (7,2,1)  (7,3,1)
%e A384318                                                       (8,2,1)
%t A384318 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A384318 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,30}]
%Y A384318 The strict complement is A179009, ranks A383707.
%Y A384318 The non-strict version for at least one choice is A383708, for none A383710.
%Y A384318 The non-strict version is A384317, ranks A384321, complement A384392, ranks A384320.
%Y A384318 These partitions are ranked by A384322.
%Y A384318 For subsets instead of partitions we have A384350, complement A326080.
%Y A384318 Cf. A357982, A383706 (disjoint), A384319, A384323 (non-strict).
%Y A384318 Cf. A048767, A098859, A179822, A239455, A279375, A317142, A351293, A382525, A383533, A383711, A384391.
%K A384318 nonn,more
%O A384318 0,7
%A A384318 _Gus Wiseman_, May 28 2025