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A324327 Number of topologically connected chord graphs covering {1,...,n}.

%I A324327 #6 Feb 22 2019 21:16:53
%S A324327 1,0,1,0,1,11,257
%N A324327 Number of topologically connected chord graphs covering {1,...,n}.
%C A324327 A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.
%C A324327 Covering means there are no isolated vertices.
%H A324327 Gus Wiseman, <a href="/A324327/a324327.png">The a(5) = 11 topologically connected chord graphs.</a>
%H A324327 Gus Wiseman, <a href="/A324327/a324327_1.png">The a(6) = 257 topologically connected chord graphs.</a>
%F A324327 Inverse binomial transform of A324328.
%e A324327 The a(0) = 1 through a(5) = 11 graphs:
%e A324327   {}  {{12}}  {{13}{24}}  {{13}{14}{25}}
%e A324327                           {{13}{24}{25}}
%e A324327                           {{13}{24}{35}}
%e A324327                           {{14}{24}{35}}
%e A324327                           {{14}{25}{35}}
%e A324327                           {{13}{14}{24}{25}}
%e A324327                           {{13}{14}{24}{35}}
%e A324327                           {{13}{14}{25}{35}}
%e A324327                           {{13}{24}{25}{35}}
%e A324327                           {{14}{24}{25}{35}}
%e A324327                           {{13}{14}{24}{25}{35}}
%t A324327 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324327 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 A324327 crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
%t A324327 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[crosscmpts[#]]<=1]&]],{n,0,5}]
%Y A324327 Cf. A000108, A000699 (the case with disjoint edges), A001764, A002061, A007297, A016098, A054726, A099947, A136653 (the case with set-theoretical connectedness also), A268814.
%Y A324327 Cf. A324167, A324169 (non-crossing covers), A324172, A324173, A324323, A324328 (non-covering case).
%K A324327 nonn,more
%O A324327 0,6
%A A324327 _Gus Wiseman_, Feb 22 2019