OFFSET
0,1
COMMENTS
An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.
EXAMPLE
The a(0) = 2 through a(4) = 22 antichains:
{} {} {} {} {}
{{}} {{}} {{}} {{}} {{}}
{{1}} {{1}} {{1}} {{1}}
{{2}} {{2}} {{2}}
{{1,2}} {{3}} {{3}}
{{1,2}} {{4}}
{{1,3}} {{1,2}}
{{2,3}} {{1,3}}
{{1,2,3}} {{1,4}}
{{3},{1,2}} {{2,3}}
{{2,4}}
{{3,4}}
{{1,2,3}}
{{1,2,4}}
{{1,3,4}}
{{2,3,4}}
{{1,2,3,4}}
{{3},{1,2}}
{{4},{1,3}}
{{1,4},{2,3}}
{{2,4},{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]]]];
cleqset[set_]:=stableSets[Subsets[set], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]], {n, 0, 5}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 18 2019
EXTENSIONS
a(9) from Andrew Howroyd, Aug 13 2019
STATUS
approved