cp's OEIS Frontend

A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.

%I A382523 #6 Apr 01 2025 10:27:08
%S A382523 1,1,2,3,4,6,9,13,18,25,34,45
%N A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.
%C A382523 First differs from A381996 at a(12) = 45, A381996(12) = 47.
%C A382523 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 A382523 First differs from A381996 in not counting the following under a(12):
%e A382523   {1,1,1,1,1,1,2,2,3,3,4,5}
%e A382523   {1,1,1,1,2,2,2,2,3,3,3,3}
%e A382523 The a(1) = 1 through a(6) = 6 multisets:
%e A382523   {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  {1,1,1,2,2,3}
%e A382523               {1,2,3}  {1,1,2,3}  {1,1,2,2,3}  {1,1,1,2,3,4}
%e A382523                        {1,2,3,4}  {1,1,2,3,4}  {1,1,2,2,3,3}
%e A382523                                   {1,2,3,4,5}  {1,1,2,2,3,4}
%e A382523                                                {1,1,2,3,4,5}
%e A382523                                                {1,2,3,4,5,6}
%t A382523 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A382523 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382523 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A382523 Table[Length[Select[strnorm[n],Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]
%Y A382523 Twice-partitions of this type are counted by A279785, strict A358914.
%Y A382523 Factorizations of this type are counted by A381633, strict A050326.
%Y A382523 Normal multiset partitions of this type are counted by A381718, strict A116539.
%Y A382523 For integer partitions we have A381992, ranks A382075, complement A381990, ranks A381806.
%Y A382523 The strict version is A381996.
%Y A382523 The strict version for integer partitions is A382077, ranks A382200, complement A382078, ranks A293243.
%Y A382523 The labeled version is A382216, complement A382202, strict A382214, complement A292432.
%Y A382523 The complement is counted by A382430, strict A292444.
%K A382523 nonn,more
%O A382523 0,3
%A A382523 _Gus Wiseman_, Apr 01 2025