cp's OEIS Frontend

A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

%I A382458 #8 Mar 31 2025 21:55:36
%S A382458 1,1,0,2,1,3,0,7,3,11,18,9
%N A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.
%C A382458 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 A382458 The normal multiset {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
%e A382458   {{2},{1,2},{2,3},{2,3,4}}
%e A382458   {{2},{2,3},{2,4},{1,2,3}}
%e A382458   {{2},{3},{1,2},{2,3},{2,4}}
%e A382458 so is not counted under a(8).
%e A382458 The a(1) = 1 through a(7) = 7 normal multisets:
%e A382458   {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
%e A382458           {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
%e A382458                               {1,2,3,3,3}     {1,1,1,2,3,3,3}
%e A382458                                               {1,2,2,2,2,3,4}
%e A382458                                               {1,2,2,2,3,3,3}
%e A382458                                               {1,2,3,3,3,3,4}
%e A382458                                               {1,2,3,4,4,4,4}
%t A382458 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
%t A382458 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t A382458 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
%t A382458 Table[Length[Select[allnorm[n], Length[Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]
%Y A382458 For constant instead of strict blocks we have A000045.
%Y A382458 Factorizations of this type are counted by A050326, with distinct sums A381633.
%Y A382458 For the strong case see A292444, A382430, complement A381996, A382523.
%Y A382458 MM-numbers of sets of sets are A302494, see A302478, A382201.
%Y A382458 Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.
%Y A382458 For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).
%Y A382458 With distinct sums we have A382459.
%Y A382458 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.
%Y A382458 Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.
%Y A382458 Set systems: A050342, A296120, A318361.
%Y A382458 Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
%Y A382458 Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.
%K A382458 nonn,more
%O A382458 0,4
%A A382458 _Gus Wiseman_, Mar 30 2025