OFFSET
1,2
EXAMPLE
Non-isomorphic representatives of the a(3) = 21 multiset partitions of multiset partitions:
{{{1,1,1}}}
{{{1,1,2}}}
{{{1,2,3}}}
{{{1},{1,1}}}
{{{1},{1,2}}}
{{{1},{2,3}}}
{{{2},{1,1}}}
{{{1},{1},{1}}}
{{{1},{1},{2}}}
{{{1},{2},{3}}}
{{{1}},{{1,1}}}
{{{1}},{{1,2}}}
{{{1}},{{2,3}}}
{{{2}},{{1,1}}}
{{{1}},{{1},{1}}}
{{{1}},{{1},{2}}}
{{{1}},{{2},{3}}}
{{{2}},{{1},{1}}}
{{{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];
dubnorm[m_]:=First[Union[Table[Map[Sort, m/.Rule@@@Table[{Union[Flatten[m]][[i]], Union[Flatten[m]][[perm[[i]]]]}, {i, Length[perm]}], {0, 2}], {perm, Permutations[Union[Flatten[m]]]}]]];
Table[Length[Union[dubnorm/@Join@@mps/@Join@@mps/@strnorm[n]]], {n, 5}]
PROG
(PARI) \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=sExp(symGroupSeries(n))); NumUnlabeledObjsSeq(sCartProd(A, sExp(A)-1))} \\ Andrew Howroyd, Dec 30 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 29 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020
STATUS
approved