OFFSET
0,3
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.
LINKS
EXAMPLE
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}}
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}]
CROSSREFS
A011782 counts multisets covering an initial interval.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 09 2022
STATUS
approved