cp's OEIS Frontend

A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

%I A382459 #7 Apr 03 2025 20:34:46
%S A382459 1,1,0,2,1,3,2,7,4,10,19
%N A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.
%C A382459 We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.
%e A382459 The normal multiset {1,2,2,2,2,3,3,4} has only one multiset partition into a set of sets with distinct sums: {{2},{1,2},{2,3},{2,3,4}}, so is counted under a(8).
%e A382459 The a(1) = 1 through a(7) = 7 multisets:
%e A382459   {1}  .  {112}  {1122}  {11123}  {111233}  {1111234}
%e A382459           {122}          {12223}  {122233}  {1112223}
%e A382459                          {12333}            {1112333}
%e A382459                                             {1222234}
%e A382459                                             {1222333}
%e A382459                                             {1233334}
%e A382459                                             {1234444}
%t A382459 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A382459 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382459 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382459 Table[Length[Select[allnorm[n],Length[Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]]==1&]],{n,0,5}]
%Y A382459 Twice-partitions of this type are counted by A279785, A270995, A358914.
%Y A382459 Factorizations of this type are counted by A381633, A050320, A050326.
%Y A382459 Normal multiset partitions of this type are A381718, A116540, A116539.
%Y A382459 Multiset partitions of this type are ranked by A382201, A302478, A302494.
%Y A382459 For at least one choice: A382216 (strict A382214), complement A382202 (strict A292432).
%Y A382459 For the strong case see: A382430 (strict A292444), complement A382523 (strict A381996).
%Y A382459 Without distinct sums we have A382458.
%Y A382459 For integer partitions we have A382460, ranks A381870, strict A382079, ranks A293511.
%Y A382459 Set multipartitions: A089259, A296119, A318360.
%Y A382459 Normal multiset partitions: A034691, A035310, A255906.
%Y A382459 Set systems: A050342, A296120, A318361.
%Y A382459 Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
%Y A382459 Cf. A000110, A000670, A007716, A255903, A275780, A317532, A321469, A326519, A381078, A381441, A382428.
%K A382459 nonn,more
%O A382459 0,4
%A A382459 _Gus Wiseman_, Apr 01 2025