cp's OEIS Frontend

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

%I A382216 #10 Mar 31 2025 21:55:23
%S A382216 1,1,1,3,5,11,23,48,101,208,434
%N A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.
%C A382216 We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.
%e A382216 The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
%e A382216   {{2,3},{1,2,3}}
%e A382216   {{2},{3},{1,2,3}}
%e A382216   {{2},{1,3},{2,3}}
%e A382216   {{1},{2},{3},{2,3}}
%e A382216 so is counted under a(5).
%e A382216 The multisets counted by A382214 but not by A382216 are:
%e A382216   {1,1,1,1,2,2,3,3,3}
%e A382216   {1,1,2,2,2,2,3,3,3}
%e A382216 The a(1) = 1 through a(5) = 11 multisets:
%e A382216   {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
%e A382216               {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
%e A382216               {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
%e A382216                        {1,2,3,3}  {1,1,2,3,4}
%e A382216                        {1,2,3,4}  {1,2,2,2,3}
%e A382216                                   {1,2,2,3,3}
%e A382216                                   {1,2,2,3,4}
%e A382216                                   {1,2,3,3,3}
%e A382216                                   {1,2,3,3,4}
%e A382216                                   {1,2,3,4,4}
%e A382216                                   {1,2,3,4,5}
%t A382216 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A382216 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382216 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382216 Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]
%Y A382216 Twice-partitions of this type are counted by A279785, without distinct sums A358914.
%Y A382216 Factorizations of this type are counted by A381633, without distinct sums A050326.
%Y A382216 Normal multiset partitions of this type are counted by A381718, A116539.
%Y A382216 The complement is counted by A382202.
%Y A382216 Without distinct sums we have A382214, complement A292432.
%Y A382216 The case of a unique choice is counted by A382459, without distinct sums A382458.
%Y A382216 For Heinz numbers: A293243, A381806, A382075, A382200.
%Y A382216 For integer partitions: A381990, A381992, A382077, A382078.
%Y A382216 Strong version: A382523, A382430, A381996, A292444.
%Y A382216 Normal multiset partitions: A034691, A035310, A255906.
%Y A382216 Set systems: A050342, A296120, A318361.
%Y A382216 Set multipartitions: A089259, A270995, A296119, A318360.
%Y A382216 Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.
%K A382216 nonn,more
%O A382216 0,4
%A A382216 _Gus Wiseman_, Mar 29 2025