OFFSET
0,3
COMMENTS
These are covering loop-graphs with at most one cycle (unicyclic) in each connected component.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
FORMULA
First differences of A369145.
EXAMPLE
Representatives of the a(1) = 1 through a(4) = 18 loop-graphs (loops shown as singletons):
{{1}} {{1,2}} {{1},{2,3}} {{1,2},{3,4}}
{{1},{2}} {{1,2},{1,3}} {{1},{2},{3,4}}
{{1},{1,2}} {{1},{2},{3}} {{1},{1,2},{3,4}}
{{1},{2},{1,3}} {{1},{2,3},{2,4}}
{{1},{1,2},{1,3}} {{1},{2},{3},{4}}
{{1},{1,2},{2,3}} {{1,2},{1,3},{1,4}}
{{1,2},{1,3},{2,3}} {{1,2},{1,3},{2,4}}
{{1},{2},{3},{1,4}}
{{1},{2},{1,3},{1,4}}
{{1},{2},{1,3},{2,4}}
{{1},{2},{1,3},{3,4}}
{{1},{1,2},{1,3},{1,4}}
{{1},{1,2},{1,3},{2,4}}
{{1},{1,2},{2,3},{2,4}}
{{1},{1,2},{2,3},{3,4}}
{{1},{2,3},{2,4},{3,4}}
{{1,2},{1,3},{1,4},{2,3}}
{{1,2},{1,3},{2,4},{3,4}}
MATHEMATICA
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n], {1, 2}]], Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]], {n, 0, 4}]
CROSSREFS
This is the covering case of A369145.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 23 2024
EXTENSIONS
a(7) onwards from Andrew Howroyd, Feb 02 2024
STATUS
approved