cp's OEIS Frontend

A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

%I A381994 #6 Mar 18 2025 22:33:26
%S A381994 0,0,0,0,1,3,3,9,12,17,27,43,46,82,103,133,181,258,295
%N A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.
%e A381994 For y = (3,3,1,1) we have {{1,3},{1,3}}, so y is not counted under a(8).
%e A381994 For y = (3,2,2,1), although we have {{1,3},{2,2}}, the block {2,2} is not a set, so y is counted under a(8).
%e A381994 The a(4) = 1 through a(8) = 12 partitions:
%e A381994   (2,1,1)  (2,2,1)    (4,1,1)      (3,2,2)        (3,3,2)
%e A381994            (3,1,1)    (3,1,1,1)    (3,3,1)        (4,2,2)
%e A381994            (2,1,1,1)  (2,1,1,1,1)  (5,1,1)        (6,1,1)
%e A381994                                    (2,2,2,1)      (3,2,2,1)
%e A381994                                    (3,2,1,1)      (4,2,1,1)
%e A381994                                    (4,1,1,1)      (5,1,1,1)
%e A381994                                    (2,2,1,1,1)    (2,2,2,1,1)
%e A381994                                    (3,1,1,1,1)    (3,2,1,1,1)
%e A381994                                    (2,1,1,1,1,1)  (4,1,1,1,1)
%e A381994                                                   (2,2,1,1,1,1)
%e A381994                                                   (3,1,1,1,1,1)
%e A381994                                                   (2,1,1,1,1,1,1)
%t A381994 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381994 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
%t A381994 Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#], And@@UnsameQ@@@#&&SameQ@@Total/@#&]]==0&]],{n,0,10}]
%Y A381994 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381994 Twice-partitions of this type are counted by A279788.
%Y A381994 Interchanging "constant" with "strict" gives A381717, see A381635, A381636, A381991.
%Y A381994 Normal multiset partitions of this type are counted by A381718, see A279785.
%Y A381994 These partitions are ranked by A381719, zeros of A382080.
%Y A381994 For distinct instead of equal block-sums we have A381990, ranked by A381806.
%Y A381994 For constant instead of strict blocks we have A381993.
%Y A381994 A000041 counts integer partitions, strict A000009.
%Y A381994 A050320 counts factorizations into squarefree numbers, see A381078, A381454.
%Y A381994 A050326 counts factorizations into distinct squarefree numbers.
%Y A381994 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381994 A381633 counts set systems with distinct sums, see A381634, A293243.
%Y A381994 Cf. A002846, A047966, A279786, A279789, A293511, A299202, A317142, A358914, A381992.
%K A381994 nonn
%O A381994 0,6
%A A381994 _Gus Wiseman_, Mar 17 2025