A356938 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers with weakly decreasing multiplicities.

1, 1, 3, 7, 18, 41, 101, 228, 538, 1209

Offset: 0

Author

Gus Wiseman, Sep 09 2022

Keywords

• nonn
• more

Comments

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

Examples

			The a(1) = 1 through a(4) = 18 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1,2}}    {{1},{1,2,3}}
         {{1},{2}}  {{1},{2,3}}    {{1,2},{1,2}}
                    {{3},{1,2}}    {{1},{2,3,4}}
                    {{1},{1},{1}}  {{1,2},{3,4}}
                    {{1},{1},{2}}  {{4},{1,2,3}}
                    {{1},{2},{3}}  {{1},{1},{1,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{1,2}}
                                   {{1},{4},{2,3}}
                                   {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{3},{4}}
		

Links

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

Crossrefs

A000041 counts integer partitions, strict A000009.

A000670 counts patterns, ranked by A333217, necklace A019536.

A011782 counts multisets covering an initial interval.

Intervals are counted by A000012, A001227, ranked by A073485.

Other conditions: A035310, A063834, A330783, A356934.

Other types: A107742, A356936, A356937, A356939, A356943, A356954.

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

Programs

Mathematica

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@strnorm[n],And@@chQ/@#&]],{n,0,5}]