OFFSET
0,3
COMMENTS
It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.
FORMULA
a(n) = A070166(n, n). - Andrew Howroyd, Jan 09 2024
EXAMPLE
Non-isomorphic representatives of the a(0) = 1 through a(4) = 17 set-systems:
{} {{1}} {{1},{2}} {{1},{2},{3}} {{1},{2},{3},{4}}
{{1},{1,2}} {{1},{2},{1,2}} {{1},{2},{3},{1,2}}
{{1},{2},{1,3}} {{1},{2},{3},{1,4}}
{{1},{1,2},{1,3}} {{1},{2},{1,2},{1,3}}
{{1},{1,2},{2,3}} {{1},{2},{1,2},{3,4}}
{{1,2},{1,3},{2,3}} {{1},{2},{1,3},{1,4}}
{{1},{2},{1,3},{2,3}}
{{1},{2},{1,3},{2,4}}
{{1},{3},{1,2},{2,4}}
{{1},{1,2},{1,3},{1,4}}
{{1},{1,2},{1,3},{2,3}}
{{1},{1,2},{1,3},{2,4}}
{{1},{1,2},{2,3},{3,4}}
{{2},{1,2},{1,3},{1,4}}
{{4},{1,2},{1,3},{2,3}}
{{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 /@ Subsets[Subsets[Range[n], {1, 2}], {n}]]], {n, 0, 5}]
PROG
(PARI) a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024
CROSSREFS
The labeled version is A014068.
The covering case is A368599.
Row sums of A368836.
A000085 counts set partitions into singletons or pairs.
A001515 counts length-n set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 05 2024
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Jan 09 2024
STATUS
approved