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

0,3

Links

Table of n, a(n) for n=0..9.

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

Example

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}}

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}]

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.

Sequence in context: A183038 A141023 A242172 * A126982 A256281 A034739

Adjacent sequences: A356951 A356952 A356953 * A356955 A356956 A356957

Keyword

nonn,more

Author

Gus Wiseman, Sep 09 2022

Status

approved