cp's OEIS Frontend

A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

%I A381992 #14 Mar 29 2025 13:49:30
%S A381992 1,1,1,2,3,5,6,9,13,17,25,33,44,59,77,100,134,170,217,282,360,449,571,
%T A381992 719,899,1122,1391,1727,2136,2616,3209,3947,4800,5845,7094,8602,10408,
%U A381992 12533,15062,18107,21686,25956,30967,36936,43897,52132,61850,73157,86466,101992,120195
%N A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.
%C A381992 Also the number of integer partitions of n whose Heinz number belongs to A382075 (can be written as a product of squarefree numbers with distinct sums of prime indices).
%e A381992 There are 6 ways to partition (3,2,2,1) into sets:
%e A381992   {{2},{1,2,3}}
%e A381992   {{1,2},{2,3}}
%e A381992   {{1},{2},{2,3}}
%e A381992   {{2},{2},{1,3}}
%e A381992   {{2},{3},{1,2}}
%e A381992   {{1},{2},{2},{3}}
%e A381992 Of these, 3 have distinct block sums:
%e A381992   {{2},{1,2,3}}
%e A381992   {{1,2},{2,3}}
%e A381992   {{1},{2},{2,3}}
%e A381992 so (3,2,2,1) is counted under a(8).
%e A381992 The a(1) = 1 through a(8) = 13 partitions:
%e A381992   (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
%e A381992             (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
%e A381992                    (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
%e A381992                             (2,2,1)  (3,2,1)    (6,1)      (7,1)
%e A381992                             (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
%e A381992                                      (2,2,1,1)  (3,3,1)    (4,2,2)
%e A381992                                                 (4,2,1)    (4,3,1)
%e A381992                                                 (5,1,1)    (5,2,1)
%e A381992                                                 (3,2,1,1)  (6,1,1)
%e A381992                                                            (3,2,2,1)
%e A381992                                                            (3,3,1,1)
%e A381992                                                            (4,2,1,1)
%e A381992                                                            (3,2,1,1,1)
%t A381992 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381992 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A381992 Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,10}]
%Y A381992 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381992 Twice-partitions of this type are counted by A279785.
%Y A381992 Multiset partitions of this type are counted by A381633, zeros of A381634.
%Y A381992 For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
%Y A381992 Normal multiset partitions of this type are counted by A381718, see A116539.
%Y A381992 The complement is counted by A381990, ranked by A381806.
%Y A381992 These partitions are ranked by A382075.
%Y A381992 For distinct blocks instead of sums we have A382077, complement A382078.
%Y A381992 For a unique choice we have A382079.
%Y A381992 A000041 counts integer partitions, strict A000009.
%Y A381992 A050320 counts multiset partitions of prime indices into sets.
%Y A381992 A050326 counts multiset partitions of prime indices into distinct sets.
%Y A381992 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381992 A382201 lists MM-numbers of sets with distinct sums.
%Y A381992 Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
%Y A381992 Cf. A293243, A293511, A358914, A381078, A381441, A381454.
%K A381992 nonn
%O A381992 0,4
%A A381992 _Gus Wiseman_, Mar 16 2025
%E A381992 a(21)-a(50) from _Bert Dobbelaere_, Mar 29 2025