OFFSET
0,1
COMMENTS
An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge-sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.
EXAMPLE
The a(0) = 2 through a(4) = 17 antichains:
{} {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{}} {{1},{2,3}} {{1},{2,3,4}}
{{2},{1,3}} {{2},{1,3,4}}
{{3},{1,2}} {{3},{1,2,4}}
{{4},{1,2,3}}
{{1,2},{1,3,4}}
{{1,2},{2,3,4}}
{{1,3},{1,2,4}}
{{1,3},{2,3,4}}
{{1,4},{1,2,3}}
{{1,4},{2,3,4}}
{{2,3},{1,2,4}}
{{2,3},{1,3,4}}
{{2,4},{1,2,3}}
{{2,4},{1,3,4}}
{{3,4},{1,2,3}}
{{3,4},{1,2,4}}
MATHEMATICA
stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]], SubsetQ[#1, #2]||Length[#1]==Length[#2]&], Union@@#==Range[n]&];
Table[Length[cleq[n]], {n, 0, 6}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 18 2019
EXTENSIONS
a(8) from Andrew Howroyd, Aug 13 2019
STATUS
approved