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%I #7 Jun 30 2019 06:50:44
%S 1,0,1,0,1,5,22,119
%N Number of minimal topologically connected chord graphs covering {1..n}.
%C Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b. A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.
%e The a(4) = 1 through a(6) = 22 edge-sets:
%e {13,24} {13,14,25} {13,25,46}
%e {13,24,25} {14,25,36}
%e {13,24,35} {14,26,35}
%e {14,24,35} {15,24,36}
%e {14,25,35} {13,14,15,26}
%e {13,14,25,26}
%e {13,15,24,26}
%e {13,15,26,46}
%e {13,24,25,26}
%e {13,24,25,36}
%e {13,24,26,35}
%e {13,24,35,36}
%e {13,24,35,46}
%e {14,15,26,36}
%e {14,24,35,36}
%e {14,24,35,46}
%e {14,25,35,46}
%e {15,24,35,46}
%e {15,25,35,46}
%e {15,25,36,46}
%e {15,26,35,46}
%e {15,26,36,46}
%t croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t 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]]]]]]]]];
%t crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
%t Table[Length[fasmin[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[crosscmpts[#]]<=1]&]]],{n,0,5}]
%Y The non-minimal case is A324327.
%Y Minimal covers are A053530.
%Y Topologically connected graphs are A324327 (covering) or A324328 (all).
%Y Cf. A000108, A006125, A007297, A054726, A136653, A324169, A326210, A326293.
%K nonn,more
%O 0,6
%A _Gus Wiseman_, Jun 29 2019
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