A356954 Number of multisets of multisets, each covering an initial interval, whose multiset union is of size n and has weakly decreasing multiplicities.

1, 1, 3, 6, 15, 30, 71, 145, 325, 680

Offset: 0

Author

Gus Wiseman, Sep 09 2022

Keywords

• nonn
• more

Examples

			The a(1) = 1 through a(4) = 15 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
         {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}
                    {{1},{1,1}}    {{1,1,2,3}}
                    {{1},{1,2}}    {{1,2,3,4}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1,1},{1,1}}
                                   {{1},{1,1,2}}
                                   {{1,1},{1,2}}
                                   {{1},{1,2,2}}
                                   {{1},{1,2,3}}
                                   {{1,2},{1,2}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{1,2}}
                                   {{1},{1},{1},{1}}
		

Links

• Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Crossrefs

For unrestricted multiplicities we have A034691.

A000041 counts integer partitions, strict A000009.

A000670 counts patterns, ranked by A333217, necklace A019536.

A011782 counts multisets covering an initial interval.

Other conditions: A035310, A063834, A330783, A356934, A356938, A356943.

Other types: A055932, A089259, A356945, A356955.

Cf. A055887, A072233, A270995, A304969, A349050, A349055, A356942.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],And@@normQ/@#&]],{n,0,5}]