cp's OEIS Frontend

A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

%I A381990 #17 Mar 29 2025 13:49:45
%S A381990 0,0,1,1,2,2,5,6,9,13,17,23,33,42,58,76,97,127,168,208,267,343,431,
%T A381990 536,676,836,1045,1283,1582,1949,2395,2895,3549,4298,5216,6281,7569,
%U A381990 9104,10953,13078,15652,18627,22207,26325,31278,37002,43708,51597,60807,71533,84031
%N A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.
%e A381990 The partition y = (3,3,3,2,2,1,1,1,1) has only one multiset partition into a set of sets, namely {{1},{3},{1,2},{1,3},{1,2,3}}, but this does not have distinct sums, so y is counted under a(17).
%e A381990 The a(2) = 1 through a(8) = 9 partitions:
%e A381990   (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
%e A381990                (1111)  (11111)  (222)     (4111)     (2222)
%e A381990                                 (3111)    (22111)    (5111)
%e A381990                                 (21111)   (31111)    (22211)
%e A381990                                 (111111)  (211111)   (41111)
%e A381990                                           (1111111)  (221111)
%e A381990                                                      (311111)
%e A381990                                                      (2111111)
%e A381990                                                      (11111111)
%t A381990 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381990 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A381990 Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,10}]
%Y A381990 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381990 Twice-partitions of this type are counted by A279785.
%Y A381990 For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
%Y A381990 Normal multiset partitions of this type are counted by A381718, see A116539.
%Y A381990 These partitions are ranked by A381806, zeros of A381634 and A381633.
%Y A381990 The complement is counted by A381992, ranked by A382075.
%Y A381990 For distinct blocks we have A382078, complement A382077, unique A382079.
%Y A381990 MM-numbers of these multiset partitions (strict blocks with distinct sum) are A382201.
%Y A381990 A000041 counts integer partitions, strict A000009.
%Y A381990 A050320 counts multiset partitions of prime indices into sets.
%Y A381990 A050326 counts multiset partitions of prime indices into distinct sets.
%Y A381990 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381990 Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
%Y A381990 Cf. A293243, A293511, A358914, A381078, A381441, A381454.
%K A381990 nonn
%O A381990 0,5
%A A381990 _Gus Wiseman_, Mar 15 2025
%E A381990 a(21)-a(50) from _Bert Dobbelaere_, Mar 29 2025