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A368726
Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.
2
1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560
OFFSET
0,3
LINKS
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
For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Sequence in context: A293937 A291857 A300672 * A052407 A233174 A185350
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 06 2024
STATUS
approved