OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
FORMULA
Inverse Euler transform of A320663.
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
{{1}} {{1,1}} {{1},{1,1}} {{1,1},{1,1}} {{1},{1,1},{1,1}}
{{1,2}} {{2},{1,2}} {{1,2},{1,2}} {{1},{1,2},{2,2}}
{{1},{1}} {{1},{1},{1}} {{1,2},{2,2}} {{2},{1,2},{1,2}}
{{1,3},{2,3}} {{2},{1,2},{2,2}}
{{1},{1},{1,1}} {{2},{1,3},{2,3}}
{{1},{2},{1,2}} {{3},{1,3},{2,3}}
{{2},{2},{1,2}} {{1},{1},{1},{1,1}}
{{1},{1},{1},{1}} {{1},{2},{2},{1,2}}
{{2},{2},{2},{1,2}}
{{1},{1},{1},{1},{1}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s, Flatten[MapIndexed[Table[#2, {#1}]&, #]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]], {n, 0, 8}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 06 2024
STATUS
approved