OFFSET
0,3
EXAMPLE
The a(1) = 1 through a(5) = 18 multiset partitions:
{{1}} {{2}} {{3}} {{4}} {{5}}
{{1,1}} {{1,2}} {{1,3}} {{1,4}}
{{1},{1}} {{1,1,1}} {{2,2}} {{2,3}}
{{1},{1,1}} {{1,1,2}} {{1,1,3}}
{{1},{1},{1}} {{2},{2}} {{1,2,2}}
{{1,1,1,1}} {{1,1,1,2}}
{{1},{1,2}} {{1},{1,3}}
{{1},{1,1,1}} {{2},{1,2}}
{{1,1},{1,1}} {{1,1,1,1,1}}
{{1},{1},{1,1}} {{1},{1,1,2}}
{{1},{1},{1},{1}} {{1,1},{1,2}}
{{1},{1,1,1,1}}
{{1,1},{1,1,1}}
{{1},{1},{1,2}}
{{1},{1},{1,1,1}}
{{1},{1,1},{1,1}}
{{1},{1},{1},{1,1}}
{{1},{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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n], Length[csm[#]]==1&]], {n, 8}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 11 2018
STATUS
approved