Constructive Mathematica program for A007297

nn=6;
croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{2}],croXQ[{#1,#2}]&],And[Union@@#==Range[n],Length[csm[#]]<=1]&]],{n,0,nn}] (* Gus Wiseman, Feb 19 2019. Returns a(0) = 1 and a(1) = 0 which is arguably correct. *)