cp's OEIS Frontend

A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.

%I A382430 #5 Apr 01 2025 10:27:12
%S A382430 0,0,1,1,2,3,5,6,9,12,17,22,32
%N A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.
%C A382430 We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.
%e A382430 The a(2) = 1 through a(7) = 6 multisets:
%e A382430   {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}  {1,1,1,1,1,1,1}
%e A382430                   {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}  {1,1,1,1,1,1,2}
%e A382430                              {1,1,1,2,2}  {1,1,1,1,2,2}  {1,1,1,1,1,2,2}
%e A382430                                           {1,1,1,1,2,3}  {1,1,1,1,1,2,3}
%e A382430                                           {1,1,1,2,2,2}  {1,1,1,1,2,2,2}
%e A382430                                                          {1,1,1,1,2,2,3}
%t A382430 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A382430 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382430 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A382430 Table[Length[Select[strnorm[n],Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]],{n,0,5}]
%Y A382430 Twice-partitions of this type are counted by A279785, strict A358914.
%Y A382430 The strict version is A292444.
%Y A382430 Factorizations of this type are counted by A381633, strict A050326.
%Y A382430 Normal multiset partitions of this type are counted by A381718, strict A116539.
%Y A382430 For integer partitions we have A381990, ranks A381806, complement A381992, ranks A382075.
%Y A382430 The strict version for integer partitions is A382078, ranks A293243, complement A382077, ranks A382200.
%Y A382430 The normal version is A382202, complement A382216, strict A292432, complement A382214.
%Y A382430 The complement is counted by A382523, strict A381996.
%K A382430 nonn,more
%O A382430 0,5
%A A382430 _Gus Wiseman_, Apr 01 2025