A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

1, 1, 4, 11, 37, 101, 328, 909, 2801

Offset: 0

Author

Gus Wiseman, Sep 09 2022

Keywords

• nonn
• more

Comments

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

Examples

			The a(1) = 1 through a(3) = 11 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
		

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.

Gapless multisets are counted by A034296, ranked by A073491.

Other conditions: A035310, A063834, A330783, A356934, A356938, A356954.

Other types: A356233, A356941, A356942, A356944.

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

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Table[Length[Select[Join@@mps/@strnorm[n],And@@nogapQ/@#&]],{n,0,5}]